Software tools for quantum control: Improving quantum computer performancethrough noise and error suppression

Harrison Ball,1 Michael J. Biercuk,1, ∗ Andre Carvalho,1 Rajib Chakravorty,1 Jiayin Chen,1 Leonardo A. de

Castro,1 Steven Gore,1 David Hover,1 Michael Hush,1 Per J. Liebermann,1 Robert Love,1 Kevin Nguyen,1 Viktor S.

Perunicic,1 Harry J. Slatyer,1 Claire Edmunds,2 Virginia Frey,2 Cornelius Hempel,2 and Alistair Milne2

1Q-CTRL, Sydney, NSW Australia & Los Angeles, CA USA2ARC Centre for Engineered Quantum Systems, The University of Sydney, NSW Australia

(Dated: January 14, 2020)

Effectively manipulating quantum computing hardware in the presence of imperfect devices andcontrol systems is a central challenge in realizing useful quantum computers. Susceptibility tonoise in particular limits the performance and algorithmic capabilities experienced by end users.Fortunately, in both the NISQ era and beyond, quantum control enables the efficient execution ofquantum logic operations and quantum algorithms exhibiting robustness to errors, without the needfor complex logical encoding. In this manuscript we introduce the first commercial-grade softwaretools for the application and integration of quantum control in quantum computing research from Q-CTRL, serving the needs of hardware R&D teams, algorithm developers, and end users. We surveyquantum control and its role in combating noise and instability in near-term devices; our primaryfocus is on quantum firmware, the low-level software solutions designed to enhance the stabilityof quantum computational hardware at the physical layer. We explain the benefits of quantumfirmware not only in error suppression, but also in simplifying higher-level compilation protocolsand enhancing the efficiency of quantum error correction. Following this exposition, we providean overview of Q-CTRL’s classical software tools for creating and deploying optimized quantumcontrol solutions at various layers of the quantum computing software stack. We describe oursoftware architecture leveraging both high-performance distributed cloud computation and localcustom integration into hardware systems, and explain how key functionality is integrable withother software packages and quantum programming languages. Our presentation includes a detailedtechnical overview of central product features including a multidimensional control-optimizationengine, engineering-inspired filter functions for high-dimensional Hilbert spaces, and a new approachto noise characterization. Finally, we present a series of case studies demonstrating the utility ofquantum control solutions derived from these tools in improving the performance of trapped-ionand superconducting quantum computer hardware.

CONTENTS

I. Introduction 2

II. Quantum control for quantum computing 3A. Quantum firmware: embedding control at the

physical layer 4B. The benefits of quantum control across the

quantum computing stack 5

III. Q-CTRL product architecture and integrations 6A. Product overview 7B. Cloud-compute architecture 8C. Quantum programming language integration

via Python 9D. Quantum computer hardware integration 9

IV. Technical functionality overview 11A. General quantum-control setting 11

1. Optimal quantum control 112. Robust quantum control 12

∗ Also ARC Centre for Engineered Quantum Systems, The Uni-versity of Sydney, NSW Australia

B. Performance evaluation for arbitrarycontrols 121. Modelling noise and error in D-dimensional

systems 132. Multi-dimensional filter functions in the

frequency domain 13C. Optimization tools to maximize control

performance 141. Optimizer framework 152. Constrained Optimizations 163. Optimizer performance benchmarking 16

D. Time-domain simulation tools for realistichardware error processes 161. Technical details of simulation

functionality 182. Simulation example 20

E. Characterizing hardware noise 20

V. Quantum control case studies 22A. Open-loop control benefits demonstrated in

trapped-ion QCs 23B. Simultaneous leakage and noise-robust

controls for superconducting circuits 25C. Robust control for parametrically-driven

superconducting entangling gates 25

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D. Experimental noise characterization ofmultiqubit circuits on an IBM cloud QC 29

E. Crosstalk-resistant circuit compilation 31

VI. Conclusion and outlook 33

Acknowledgements 33

A. Appendix: Technical Definitions 341. Fourier transform 342. Frobenius inner product 343. Frobenius norm 344. Approximations 345. Power spectral density 35

B. Control Hamiltonian 351. Generalized formalism 362. Control solutions 363. Control segments 364. Control coordinates 37

C. Multidimenional filter function derivations 381. Magnus expansion 382. Spectral representation of Magnus term 383. Leading order robustness infidelity 394. Leading-order filter functions 39

D. Technical details on implementation of the SVDnoise reconstruction 401. Measurement uncertainties for SVD

reconstruction 402. Filter function normalization 403. Calculating weights due to sensitivity 404. Singular value truncation to prevent numeric

instabiity 41

E. Methods for experimental demonstrations ofquantum control benefits 411. Quasi-static error robustness 412. Suppression of time-varying noise 423. Error homogenization characterized via

10-qubit parallel randomized benchmarking 424. Molmer-Sorensen drift measurements 42

F. Product Features Summary 431. BLACK OPAL 432. BOULDER OPAL 433. OpenControls 444. Devkit 44

G. BLACK OPAL Visualizations of noise andcontrol in quantum circuits 44

H. Details on cloud architecture 451. Presentation tier 462. Logic tier 463. Data tier 484. Infrastructure 48

References 49

I. INTRODUCTION

The emergence of NISQ-era [1] quantum computinghardware at the scale of a few tens of qubits has ledto an explosion of interest from end-users and softwaredevelopers. Today’s boom in quantum software is pre-dominantly focused on application mapping and algo-rithmic development. A cultural shift in the communityaway from exclusively seeking algorithms which provideprovable asymptotic scaling advantages to identifying ap-plications providing any commercially relevant computa-tional advantage [2] has also served as an accelerant forthe nascent quantum software industry. But as in con-ventional software engineering there is much that lies be-neath the most visible layers of the software stack [3] inproviding the ultimate functionality and computationaladvantages the quantum information community is seek-ing (Fig. 1).

The key challenge we consider is the efficient manip-ulation of quantum computing hardware with the ob-jective of extracting optimal performance at the device,circuit, and algorithmic levels. The central impedimentis the influence of noise and error in quantum computinghardware; electromagnetic noise in its various forms is re-sponsible for both diminished coherent lifetimes throughthe process of decoherence and reduction in the fidelityof quantum logic operations when quantum devices aremanipulated by faulty classical hardware. Both, takentogether, ultimately limit the range of achievable com-putations on quantum coherent hardware, measured viametrics such as circuit depth or quantum volume [4].

Susceptibility to noise and error remains the Achillesheel of quantum computers, and stabilizing hard-ware against noise via e.g. quantum error correction(QEC) [5], has been a major driver of research in the field.The complexity of QEC, however, motivates considera-tion of alternative techniques in the near-term which donot rely on full fault-tolerant encoding. Even beyondthe current era, QEC, for its virtues in underpinningthe asymptotic scaling of quantum computers, appearsa resource intensive tool for error mitigation in isolation.Realizing practical, functional machines in the NISQ-eraand beyond thus requires continued development of tech-niques to augment the noise and error robustness of quan-tum computing hardware, outside of QEC.

In the classical domain, working to stabilize an unsta-ble system against disturbance from its environment isfrequently the domain of control engineering [6, 7]. Con-trol engineering, as a modern discipline, is fundamen-tal to nearly all of our technological industries. Con-trol allowed the Wright brothers to achieve manned,powered, controlled flight at a time when others couldnot [8]. The concepts they introduced to the crowdedaviation field—notably three-axis control and deformableairfoils—would be considered control engineering contri-

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butions today. Similar stories have played out in au-tonomous vehicles, walking robots, and even the opera-tion of computer memory [9]. Control has built industriesbefore, and in our view quantum control will be just asessential in the development of the emerging quantumcomputing industry.

Considerable effort has already been taken to adaptthe concepts of control engineering to the strictures ofquantum coherent devices in research and in practice [10–19]. Adoption of many of the physical concepts un-derlying quantum control is already ubiquitous acrossthe field; every team that employs a spin-echo to elim-inate unwanted couplings between qubits or to mitigateslowly fluctuating ambient magnetic fields is leveragingthe physics of coherent averaging [20]. Similarly, optimalcontrol has been begun to emerge as a powerful tech-nique to manipulate complex Hilbert spaces [21–23], oroptimize quantum hardware performance [24–26].

Increasing the uptake and utility of these techniquesas the quantum computing community grows and di-versifies requires access to effective, user-friendly, profes-sionally engineered support software. Users with limitedexpertise in the research discipline of quantum controlmust be able to use these tools to apply and integratequantum control techniques into quantum computing re-search. Moreover, much like critical security software forconventional cloud computing, it is essential that profes-sionally engineered solutions are available from specialistproviders; the functionality provided is essential for busi-ness continuity, risk mitigation, and maintainability.

Such tools, however, have been lacking in the commu-nity, with most academic teams writing their own special-ized code or hacking existing software packages derivedfrom other fields such as NMR. This approach is cost-inefficient, leads to inconsistent results between teams,fails to deliver on the most up-to-date knowledge fromresearch community, and has substantial negative conse-quences as students and staff inevitably move on fromcurrent roles. Professionally engineered quantum con-trol software will become as important to the emergingquantum computing industry as security software is tomodern cloud computing; current DIY approaches areunsustainable.

In this manuscript we introduce a set of professionallyengineered infrastructure software tools aimed at build-ing and integrating error-robust quantum control strate-gies in the quantum computing stack. One key objectiveis the deterministic suppression of the influence of noisein quantum logic operations and quantum algorithms,providing greater functionality from fixed computationalresources (measured in qubits, gates, and compute run-time). In our presentation we describe how quantumcontrol suppresses errors in hardware, but also how itcan enable simplification of compilation at higher levelsof the stack by mitigation of performance variations inspace and time, and builds compatibility with quantumerror correction in the long term.

In support of the integration of these capabilities in the

design and operation of quantum computers, we intro-duce a suite of classical software tools which allow usersto create and deploy quantum control solutions acrossthe software stack—from the physical layer through toalgorithmic design. These tools, engineered leveraginga modern cloud-compute architecture, ensure that com-putationally complex tasks such as the optimization ofmulti-dimensional unitary transformations are handledefficiently while ensuring compatibility with other partsof the computational stack and long-term maintainabil-ity. We describe how these tools may be integrated withconventional programming workflows in Python, cloud-based quantum computers, and custom quantum com-puting hardware. Our presentation includes a detailedtechnical discussion of new algorithmic approaches tocontrol design and optimization, performance evaluationand validation, and control hardware characterization.These functions are validated and demonstrated througha series of case studies tied to real problems in quantumcomputing hardware.

The remainder of this manuscript is organized as fol-lows. In Sec. II we introduce the fundamental conceptsand utility of quantum control as it pertains to the chal-lenges building useful quantum computers. This is fol-lowed by an overview of Q-CTRL’s classical softwareproducts for the development and deployment of quan-tum control in Sec. III. We then move on to present atechnical, mathematical treatment of novel functionalitywe have developed in Sec. IV before presenting a series oftechnical case studies deploying these tools and capabil-ities to solve real problems in Sec. V. We then concludewith a brief summary and future outlook of forthcomingproduct and feature development.

II. QUANTUM CONTROL FOR QUANTUMCOMPUTING

Control engineering provides powerful tools permittingthe abstraction of complex physical systems away frommicroscopic models, and permitting efficient and stablemanipulation of hardware. In the quantum domain manyof the familiar concepts at work for control of classi-cal technologies must be amended, and a simple one-to-one translation of textbook control engineering conceptsrarely succeeds. For instance, quantum systems of in-terest to quantum computing are in general nonlinear(control over qubits is formally bilinear), noise-processesof interest in the laboratory are generally colored, andmeasurement has strong back-action on the plant (wewill focus exclusively on the control of qubits rather thanlinear degrees of freedom such as harmonic oscillators).In this section we will describe how quantum control canbring benefits to quantum computing via integration atthe physical layer as well as higher layers in the QC stack.

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USER INTERFACE, QAAS,& OPERATING SYSTEM

QUANTUM ALGORITHMS & APPLICATIONS

LOGICAL-LEVEL COMPILATION& CIRCUIT OPTIMIZATION

QUANTUM ERROR CORRECTION

HARDWARE-AWARE QUANTUM COMPILER

PHYSICAL QUBIT HARDWARE

Simplified Quantum Computing Stack

QUANTUM FIRMWARE: HARDWARE ERROR MITIGATION

D e v k i t

B L A C K O P A L

B O U L D E R O P A L

Discover how quantum control reduces noise via intuitive visualizations

Student, researchers, newcomers

Python tools for quantum control R&D and hardware integration

R&D Professionals

Optimize algorithmic performance on today’s noisy quantum computers.

Algorithm developers and front-end users.

FIG. 1. The integration of Q-CTRL software products in the quantum computing stack. Left: A notional depiction of the keyelements of the software stack in a quantum computer or Quantum as a Service (QAAS) cloud offering. Most quantum softwaredevelopment today spans the top two layers of the stack, with an emphasis on applications, algorithms, and programminglanguages. Infrastructure software spans all aspects of the middle of the stack from circuit compilation through to quantumfirmware which exists at the classical-quantum interface. Right: Q-CTRL’s core products span the quantum compute stack.Most contribute capabilities at the quantum firmware layer, however, in addition to providing higher level benefits, productslike the Q-CTRL Devkit help deploy control-theoretic insights and capabilities in error suppression to users such as applicationdevelopers.

A. Quantum firmware: embedding control at thephysical layer

The stabilization of quantum devices or quantum logicoperations against external perturbations is primarily de-signed to suppress hardware faults which contribute tocomputational errors. Decoherence during idle opera-tions, for instance, results in loss or randomization ofinformation stored in a qubit. Similarly, external insta-bilities in e.g. radiofrequency fields used to mediate mul-tiqubit entanglement can result in reduced operational fi-delity. Appropriate deployment of various forms of quan-tum control at the physical layer can therefore mitigatethe deleterious effects of these perturbations and improvethe logical error rate for quantum hardware. Becausethese techniques generally involve direct actuation on thedevices used in performing computations, they interactwith the physical layer in a generic quantum computingsoftware stack [27], prior to any form of logical encodingused in fault-tolerant quantum error correction (Fig. 1).We refer to this class of protocols loosely as firmware as

they are usually software-defined but embedded at thephysical layer and invisible to higher layers of abstraction(Quantum Error Correction is treated in the next subsec-tion). Moreover, these techniques bear resemblance toother forms of firmware in computer engineering, such asDRAM refresh protocols [9] employed to stabilize classi-cal memory hardware against charge leakage.

When considering quantum control in the context ofhardware stabilization for qubits in quantum comput-ers, we are generally presented with three interrelatedcontrol-engineering-inspired approaches:

• Measurement-free open-loop control

• Destructive closed-loop feedback via projectivemeasurements

• Indirect closed-loop feedback which preserves quan-tum coherence via weak measurements or the useof ancillae.

Open-loop control refers to feedback-free actuation ona system, as in the case of a timed irrigation system which

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can maintain a lawn without information on soil moistureor rainfall. It provides the substantial benefit that it isresource efficient and in the quantum domain has provento be remarkably effective in stabilizing quantum devicesboth during free evolution and during nontrivial logicoperations. In this framework, quantum logic operationsare redefined to provide the same mathematical transfor-mation within the computational space, but in a way thatprovides robustness against various error-inducing noiseprocesses. The implementation of open-loop quantumcontrol involves temporal modulation of incident controlfields employed to enact manipulation of physical devices.Considerable literature exists in the NMR and EPR com-munity concerned with geometric interpretations of howcontrol design can be used to cancel errors or decouple un-wanted Hamiltonian terms through dynamic decouplingsequences or composite pulse techniques [20, 28]. Due toits simplicity, ubiquity, and laboratory-validated perfor-mance we will focus substantially on this set of techniquesin our technical discussion.

The use of feedback control, in which actuation is de-termined based on measurements of the system, is highlyconstrained [29] by the destructive nature of projectivemeasurement in quantum mechanics [30]. While novelcontrol strategies can be developed for predictive con-trol [18, 31] during unsupervised periods, direct destruc-tive measurements on data qubits are not commonly usedwithin quantum computing architectures. These chal-lenges may be mitigated through the use of indirect mea-surements on embedded sensor ancilla qubits [19, 32, 33].Such techniques use physically proximal qubits as trans-ducers to detect local noise in the device with actuationon unsupervised data qubits. Weak measurements alsoprovide a means to stabilize data with minimal pertur-bation to the encoded quantum information, and haveseen adoption in superconducting circuits [34] and NVdiamond [35].

B. The benefits of quantum control across thequantum computing stack

There has been much effort driven both from experi-mental hardware and software engineering teams to buildtools addressing the multitude of practical problems in-herent to running a functional quantum computer in theNISQ era and beyond. Nonetheless, the abstraction lay-ers that have emerged in the quantum computing stackhave remained reasonably isolated, each being developedlargely independent of its neighbors. This is relevantfor quantum-computer-science studies, but may presentmissed opportunities in the context of practical quantum-computer engineering. Ultimately, enabling practical,commercially relevant systems that provide computa-tional advantages even in the presence of imperfect hard-ware requires consideration of broader benefits of quan-tum control, and the interface between abstraction layersacross the QC software stack. Quantum control impacts

quantum computer system design and higher-level soft-ware abstractions through a variety of benefits (Fig. 2),and insights from this field have the potential to shapethe practical implementation of many higher-level ab-stractions such as compilation and QEC.

As we have seen above, the generally accepted targetfor the deployment of physical-layer quantum control isthe reduction of error rates in individual quantum logicoperations and devices. The error suppression affordedby quantum control is, however, complemented by er-ror homogenization in space and time. In publicly avail-able cloud devices, for instance, it is common to observemore than an order of magnitude deviation between best-case and worst-case qubit error rates across a device [36].These can arise from fabrication variances as well as vari-ations in coupling to the ambient electromagnetic envi-ronment. Likewise, cloud devices typically go out of cal-ibration over a timescale that is typically much less thansix hours, and device performance can vary substantiallyfrom day to day [36].

The use of appropriate robust-control solutions ensuresend-users attain stable and repeatable algorithmic per-formance with minimal overhead. First, it minimizeshardware performance variations across a device, driv-ing error rates towards the best-case performance, evenin cases when the best case performance in a device isnot improved (due e.g. to T1 limits). In addition, thesame control solutions ensure that performance remainsapproximately constant throughout an inter-calibrationwindow, rather than permitting performance to degradenear the end of the calibration cycle. Together, thesebenefits help to mitigate the need for complex error-aware compilation needed to route around poorly per-forming devices [36, 37]. See Sec. V A for a case-studydemonstrating these benefits in real quantum computinghardware.

In the long-term these concepts also have critical im-pacts on the functioning of logical encoding for quantumerror correction. To start, while not commonly expressedin this way, it becomes apparent that the entire conceptof Quantum Error Correction can also be expressed asa form of feedback stabilization using a specialized andhighly constrained sensor model. Thus importing in-sights from quantum control engineering can improve thepractical implementation of quantum error correction.

At the interface of QEC and quantum firmware, errorsuppression has the potential to improve the efficiency ofquantum error correction. The physics of both feedbackstabilization and open-loop control at the physical layerexploit quantum coherence and spatio-temporal correla-tions in noise in order to offer robustness to quantumlogic [38]. For instance, coherent averaging allows a netUnitary transformation to be performed with reducedsensitivity to error. Truly stochastic processes such asT1 energy relaxation are not typically correctable usingthese strategies. Immediately we see that physical-layerquantum control serves as a complement to higher-levellogical encoding, maximally depressing hardware error

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FIG. 2. The benefits of quantum control for quantum computing. Benefits ranging from 10×−100× improvement have beenexperimentally demonstrated. Technical details on these capabilities and associated experimental demonstrations are providedthroughout this manuscript with a special focus on Sec. V.

rates to stochastic limits below fault-tolerant thresholds.QEC can then be deployed to identify and correct resid-ual errors.

Next, the homogenization of error rates in space andtime closes the gap between best and worst-case deviceperformance, again ensuring fault-tolerant error rates aremaintained across a logical block. Finally, the actionof quantum control as a low-frequency noise filter re-duces spatio-temporal correlations between residual er-rors [38], improving the resource-efficiency and efficacyof quantum error correction. This holds for both open-loop control (see Sec. IV B), and also closed-loop sta-bilization. In the latter, the innovations property [6]states that an ideal feedback loop extracts maximum in-formation from the process being stabilized, leaving un-correlated measurement-residuals; this has been exper-imentally demonstrated using trapped-ion qubits and amachine-learning-derived feedback correction [31]. Ex-tracting all useful information from a correlated noiseprocess ideally conditions errors for QEC [39].

III. Q-CTRL PRODUCT ARCHITECTURE ANDINTEGRATIONS

The Q-CTRL product suite is the first commer-cially available software specifically designed to improve

hardware performance in the quantum computing stackthrough access to quantum control. Q-CTRL tools in-corporate the most advanced capabilities relevant to sta-bilizing quantum systems against hardware errors:

• Error-robust control selection, creation, and opti-mization for quantum logic, quantum circuits, andquantum algorithms

• Predictive error-budgeting and simulation of hard-ware and circuit performance in realistic laboratoryenvironments

• Hardware characterization and calibration at themicroscopic level to identify sources of noise andimperfection.

Q-CTRL delivers a comprehensive toolset via a mod-ern cloud-compute architecture; unlike traditional soft-ware that takes the form of locally installed and executedcode, Q-CTRLs cloud-based solutions enable customersto take advantage of extremely high performance withthe flexibility to draw computational resources as needed.All tools are architected around an application program-ming interface (API), coded in Python. In certain cir-cumstances we also employ special-purpose programmingframeworks such as TensorFlow and Cython as appropri-ate in order to provide performance enhancements in thecore codebase. Processing and memory resources used

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FIG. 3. Relationships between Q-CTRL products, connectivity via API to the core cloud-compute engine, and demonstrationof various means of user interaction with Q-CTRL software. For instance, BOULDER OPAL is commonly accessed through aPython interface, or in combination with the last-mile-integration package may be built directly into user quantum hardware.Meanwhile core functionality may be accessed by users building custom tools via the Q-CTRL API.

in the execution of a computation are scaled automati-cally by the software, and specialized accelerators such asGPUs are accessed on demand for e.g. optimizations ex-ecuted using our TensorFlow-based tools (see Sec. IV C).In addition users enjoy the benefits of access to new fea-tures as they become available—without the need to up-date software locally.

A. Product overview

Here we describe the central software products de-signed by Q-CTRL, specifically BOULDER OPAL,BLACK OPAL, OpenControls, and the Q-CTRL De-vkit, each of which delivers the capabilities and advan-tages of quantum control to users with different back-grounds, interests, and objectives. While our core focusis on the lowest level of the QC software stack, controlprovides benefits at higher levels as well (Fig. 1). Theseinfrastructure-software tools, their relationship to eachother, and the way users interact with them is repre-sented in Fig. 3. A more detailed summary of featuresfor individual products is provided in App. F.

BOULDER OPAL offers an advanced Python-basedprofessional-grade toolkit for R&D teams to develop anddeploy quantum control in their hardware or theoreticalresearch. All technical features and core capabilities of

Q-CTRL products described in Sec. IV are accessible viaBOULDER OPAL, making this the core toolkit in our of-fering. In order to facilitate integration into conventionalprogramming environments, BOULDER OPAL includesa light Python package wrapper that is downloaded lo-cally and orchestrates calls to the web API. All compu-tationally intensive tasks remain the responsibility of thecore computational engine in the cloud.

BLACK OPAL helps students, new users, and re-searchers to learn about quantum control by taking ad-vantage of a graphical interface with interactive visual-izations. It is designed to assist in building intuition forcomplex concepts in quantum control, such as the mean-ing of entanglement in quantum circuits (Fig. 4). Theproduct is delivered as a web-based API service provid-ing users with a graphical front-end interface completewith guided tours, configuration wizards, and integratedhelp; an example of an interactive visualization is shownin Fig. 4. The front-end prepopulates common systemconfigurations for superconducting and trapped-ion pro-cessors, or custom configurations may be input, and con-tains predefined libraries of known control solutions. Thefunctions selected for inclusion in the front-end interfaceare designed to assist in the process of learning about therole and capabilities of control, and to determine relevantcontrol solutions for a given hardware problem.

OpenControls is an open-source Python package that

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FIG. 4. Screenshot of BLACK OPAL interface demonstrating visualization of a circuit performed on two qubits. The trajectoriesof individual qubit observables depicting separable states are represented on an interactive 3D Bloch sphere. The bloch vectorgoes to zero as qubits become entangled through the action of the CNOT gate. Three interactive entanglement tetrahedratrack correlations between the nine pairs of observables enabling visual representation of complete two-qubit state evolution.Details of the visualization packages within BLACK OPAL are presented in App. G.

includes established error-robust quantum control proto-cols from the open literature. The aim of the packageis to be a comprehensive library of published and testedquantum control techniques developed by the commu-nity, with easy-to-use export functions allowing users todeploy these controls on custom quantum hardware, pub-licly available cloud quantum computers, or other partsof the Q-CTRL product suite.

The Q-CTRL Devkit allows quantum algorithm de-signers and end-users of quantum computers to deriveenhanced performance for their applications. The De-vkit enables developers to both analyze algorithmic per-formance in the presence of realistic time-varying noise,and to build in robustness through the structure of thecompiled circuit. This toolkit is compatible with othercompilers and can provide deterministic error robust-ness without the need for additional overhead such asrepetition when adding engineered error in zero-noise-

extrapolation schemes. Due to its cross-compatibility itcan augment circuit performance when used in combina-tion with higher-level circuit compilation or Pauli-framerandomization schemes. Because error robustness comesnot only from circuit construction but also the specifictiming of operations within a circuit, Devkit protocolsmust be used as the last stage of circuit compilation, fol-lowing translation to hardware architectural constraintsor circuit compression. See Sec. V E for an example casestudy.

B. Cloud-compute architecture

These products and the back-end infrastructure sup-porting them encapsulate a fully-engineered, tested, andsecured cloud-based platform for efficiently implementingthe tasks set out in Sec. IV. Once the client has entered

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relevant inputs, this information is sent to the back-endand processed through the API. The client’s data is takenthrough to the Python module, which performs the rel-evant computations and outputs objects based on thesystem inputs. The interface with the API varies basedon the product in use as described below (Fig. 3), andeven allows for custom application development by theuser.

Our back-end software architecture employs estab-lished and lightweight web interfaces (OpenAPI specifica-tions, REST APIs and JSON), as well as performant andscalable architectural designs such as a three-tiered ap-plication with dedicated worker pools and work queues.Q-CTRLs use of open standards enables our entire ap-plication stack to be deployed on any cloud—public, pri-vate, hybrid or on-premises—ensuring users are able tochoose the right balance between price, performance andprivacy. For instance, a remotely managed on-premises-cloud instance can be provided to users seeking to main-tain full control over all sensitive data, while still ensuringthe advantages of a cloud-compute architecture.

Our choice of Python for the API incorporates sim-plicity, speed of development, and support for collabo-ration, e.g. with external developers. Both quantumscientists and programmers are typically familiar withPython, having used its libraries for tasks ranging frominstrument control and advanced computation to webdesign. Building in Python also leverages compatibilitywith huge global resources of open-source code, and web-based frameworks like Django, bringing embedded secu-rity features to safeguard against attacks such as SQLinjection, request forgery or cross-site scripting, cruciallyimportant for companies entrusting third parties withcritical hardware tasks. Further technical details on thecloud-compute architecture are available in App. H.

C. Quantum programming language integrationvia Python

A core mode of accessing Q-CTRL products comesfrom a lightweight Python wrapper or SDK which en-ables access to the API from within a standard Pythoninterface. This approach brings the added advantage ofcompatibility with a wide variety of programming lan-guages commonly employed in the quantum computingresearch community. As a complement, Q-CTRL pro-vides Python adaptors for all open-source Python-basedquantum computing languages, allowing integration ofadvanced control solutions into conventional program-ming workflows and execution on cloud hardware plat-forms.

The qctrl-qiskit package provides export functions ofQ-CTRL-derived control solutions or protocols to qiskit.Q-CTRL dynamical decoupling sequences (used for im-plementing the identity operator in preservation of quan-tum memory) can be converted into qiskit quantum cir-cuits using these methods, accounting for approximations

made in qiskit circuit compilation and ensuring circuitsare not compactified in such a way that undermines per-formance. Furthemore, Q-CTRL pulses can be exportedfrom both BLACK OPAL and BOULDER OPAL in theOpenPulse format [40] which provides analog-level pro-gramming of microwave operations performed on hard-ware. As a complement, Q-CTRL has also developedcalibration routines for IBM hardware to account for thetruly open nature of the OpenPulse API. Such appro-priately formatted controls and calibration routines canthen be immediately run on IBM’s quantum computinghardware with an IBM Q account.

The qctrl-pyquil package provides export functions toRigetti’s pyQuil, and qctrl-cirq allows export to GoogleCirq. At present, the absence of analog-level control ac-cess limits export functionality to timed sequences ofstandard control operations handled natively in theseplatforms. In both cases the translation layer ensuresthat the integrity of the sequence structure and timingis preserved within approximations made in sequencingin these two languages. pyQuil integration currently per-mits the execution of Q-CTRL sequences using Rigetti’sQuantum Cloud Service on real quantum hardware.

D. Quantum computer hardware integration

Q-CTRL products have been designed to integratewith a wide variety of hardware systems—from lo-cal laboratory-based quantum hardware to quantum-compute cloud engines. This requires an architecturewhich is flexible and platform agnostic, and also man-dates compatibility and integrability with various hard-ware and software packages.

The Last-Mile Integration (LMI) toolset integrates thefull suite of Q-CTRL cloud-based products with a user’sexisting experimental hardware control system and soft-ware stack. The LMI is delivered via a customized, local-instance Python package that runs in parallel with theuser’s experimental control stack to automate and sched-ule key tasks in control definition, calibration, and op-timization. A schematic overview of the relationshipbetween BOULDER OPAL, the Last-Mile IntegrationPackage, and the experimental hardware/software stackis presented in Fig. 5.

Hardware integration is most tightly coupled to the de-termination of appropriate outputs from hardware signalgenerators such as arbitrary waveform generators, directdigital synthesizers, and vector signal generators. In theLMI package, all controls are written in a format tailoredto hardware constraints such as sample rates, amplituderesolution, and data formats. Custom control pulses areexported into a format (e.g. CSV or JSON [40]) easilyread by the experimental control stack. Q-CTRL hasalready partnered with a number of original equipmentmanufacturers serving the quantum-computing market,providing pre-built formatting scripts to translate con-trol output into machine-compatible formats.

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Closed-Loop Control OptimizationProduct: Custom Integration

Hardware Feedback StabilizationProduct: Custom Integration

Real-Time Tracking & PredictionProduct: Custom Integration

Automated Mesoscale TuneupProduct: Custom Integration

Hardware CalibrationProduct: BOULDER OPAL Last-Mile Integration

Control Formatting for StackProduct: BOULDER OPAL Last-Mile Integration

Control OptimizationProduct: BOULDER OPAL

Q-CTRL Protocol LibraryProduct: BOULDER OPAL

Abst

ract

ion

Host PC/Instrument Control

Scheduling &Synchronization

FPGA Data Processing

AWGs/Synths

Quantum Hardware

Control Evaluation & Error BudgetingProduct: BOULDER OPAL

Hardware Noise CharacterizationProduct: BOULDER OPAL

Scripting: Automation & SchedulingProduct: BOULDER OPAL Last-Mile Integration

Scripting: Control Tuneup/ValidationProduct: BOULDER OPAL Last-Mile Integration

FIG. 5. Schematic overview of quantum control functions, accessibility via Q-CTRL products, and integration into a user’scustom classical experimental control hardware. As tasks are abstracted further away from the details of experimental hardware,the point of integration similarly rises in the experimental control stack.

As an example, Q-CTRL has partnered with Quan-tum Machines, providing a direct interface betweenQ-CTRL protocols and QM’s Quantum OrchestrationPlatform stack. The Quantum Orchestration Platformis a software-hardware solution whose software inter-face is Quantum Machines programming language called“QUA”, and an advanced hardware system allowing or-chestration of QUA programs in real-time (e.g waveformgeneration, waveform acquisition, classical data process-ing and real-time control-flow). This integration natu-rally permits control formatting matched to QUA, butalso exploits parametric encoding to enable rapid hard-ware tuneup and calibration. For instance, by parame-terizing the duration of an experiment or a DDS phasevalue it becomes trivial to calibrate control amplitudesand phases. Similarly, scripting in QUA allows exploita-tion of low-latency FPGA-based computation for the exe-cution of real-time machine-learning routines or Bayesianupdates via tools offered jointly by Q-CTRL and QM.Using this framework we have recently demonstrateddephasing-robust single-qubit operations in a tuneabletransmon device, indicating that the combination of Q-CTRL software with the Quantum Machines quantumorchestration platform stack permits faithful output ofcomplex modulated control protocols. Beyond officialQ-CTRL partners, custom scripts cover commonly en-countered hardware solutions, leveraging the flexibilityof Python programming.

An essential aspect of this integration is efficient hard-ware calibration permitting determination of the analogvoltages or digital commands required to achieve out-put signals with appropriate phase and amplitude val-ues. This helps account not only for residual amplitudemodulation in the presence of other forms of modulation(e.g. PM), but also cross-coupling and signal distortions

encountered due to room-temperature hardware such asmixers. The LMI allows efficient tuneup of control hard-ware and produces a local lookup table allowing rapidtranslation of target phase and amplitude values for anoptimized control to control-hardware instructions.

Hardware calibration can also leverage quantum con-trol in order to gain access to information about signaldistortions and transmission-line nonlinearities withinexperimental systems that are not easily characterizedthrough conventional means. This approach is widelyemployed in most laboratories through basic proto-cols for qubit frequency determination such as Ramseyspectroscopy, drive-amplitude calibration through Rabimeasurements, and more advanced protocols to esti-mate microwave phases [41] or identify quadrature cross-couplings [42, 43]. Further, we offer time-domain impulseresponse characterization [44] in order to determine thenonlinear response of transmission lines and passive ele-ments in the signal propagation pathway.

In addition to characterization of classical hardware,the LMI package maximizes automation via a suite ofautomated Python scripts for the efficient tuneup ofqubit hardware. These scripts permit extraction of stan-

dard parameters (T1, T1ρ, T2, T(echo)2 , Ω(Rabi)) as well

as calibration of key parameters relevant to high-fidelitycontrol implementation. The package permits sched-uled noise sensing and reconstruction in order to detectchanges in dominant noise power spectra (e.g. controlnoise or dephasing). Again, all computationally inten-sive calculations are handled by the distributed cloud-compute engine with only scripting and experimental-control-software integration handled locally. Experimen-tal calibration and noise characterization results are alsologged on the cloud server and accessible to users.

Moving further down the experimental control hard-

11

ware stack it becomes possible to implement a variety oflow-latency real-time processing tasks via custom hard-ware integration. Capabilities are based on core routinesand techniques developed by Q-CTRL and customizedfor a user’s or vendor’s hardware system (e.g. QUAdevelopment with Quantum Machines). An example isclosed-loop optimization of control solutions in order toreoptimize error-robust controls as system noise sourcesdrift. Numerically optimized controls may be used asa seed for optimization based on experimental measure-ments based on user-defined cost functions such as ran-domized benchmarking survival probabilities.

IV. TECHNICAL FUNCTIONALITYOVERVIEW

In this section we provide an overview of key techni-cal capabilities afforded by quantum control as accessedthrough the products introduced in Sec. III. Our pre-sentation focuses on tasks relevant to building error ro-bustness into quantum computer hardware, though manyother applications exist in quantum sensing, data fusion,and advanced medical imaging. In Sec. V we providedetailed case studies to illustrate how these features areemployed in real research settings.

A. General quantum-control setting

Here we establish the general control-theoretic settingfor both the optimal and robust optimization of con-trol solutions in multi-dimensional quantum systems. Wewrite the total Hamiltonian as the sum of dynamical con-tributions from both control and noise interactions

Htot(t) = Hctrl(t) +Herr(t). (1)

We assume that Hctrl(t) is a deterministic componentof Htot(t) containing both an intrinsic drift term (e.g.the frequencies of the qubits) and controllable partsof the system (e.g. microwave drives or clock shifts).See App. B for generalized definitions of these terms.The error Hamiltonian Herr(t) captures the influence ofnoise, is assumed to be small, and may also be a func-tion of control Hamiltonian Hctrl(t). The typical controlproblem may be split into two distinct tasks:

1. Design a control solution for a D-dimensionalHilbert space such that Hctrl(t) implements a tar-get unitary Utarget at time τ .

2. Design a control solution such that Hctrl(t) is ro-bust against noise interactionsHerr(t) over duration[0, τ ].

The first task is typically referred to as an optimal con-trol problem while the second is called robust control;Q-CTRL provides tools for both.

1. Optimal quantum control

The optimal control setting reduces the following prob-lem. Given the Schrodinger equation

iUctrl(t) = Hctrl(t)Uctrl(t) (2)

the aim is to find Hctrl(t) such that

Utarget = Uctrl(τ) (3)

where Uctrl(τ) is the evolved unitary at time τ and Utarget

is the target operation. This control problem, gener-ally, can be considered a bilinear control problem as thecontrollable element of the equation of motion (Hctrl(t))linearly multiplies the state. This is in contrast to themuch more common linear control problems from clas-sical control where the controllable element is linearlyadded to the state. Bilinear control problems typicallydo not have analytically tractable solutions, and insteadmust be solved numerically.

We define a measure of optimal control using a Fro-nenius inner product Eq. (A3) to evaluate the operator-distance between Uctrl(τ) and Utarget. Specifically

Foptimal(τ) =

∣∣∣∣1

D

⟨Utarget, Uctrl(τ)

⟩F

∣∣∣∣2

. (4)

This measure is bounded between [0, 1], with perfectimplementation of the target Eq. (3) corresponding toFoptimal(τ) = 1. We define a corresponding infidelitymeasure as

Ioptimal(τ) = 1−Foptimal(τ). (5)

A simple modification of this fidelity measure enablesthe optimal condition to be evaluated on a subspace ofinterest. Specifically

FPoptimal(τ) =

∣∣∣∣1

Tr (P )

⟨PUtarget, Uctrl(τ)

⟩F

∣∣∣∣2

(6)

where P defines a projection matrix, enabling optimalcontrol to be evaluated on a target subspace. Similarly,achieving high-fidelity state-transfer |ψinitial〉 → |ψfinal〉is equivalent to maximizing the state fidelity defined as

Fψoptimal(τ) = |〈ψinitial|Uctrl(τ)|ψfinal〉|. (7)

As discussed in detail in Sec. IV C, crafting numeric so-lutions for these control problems may be convenientlycast as cost-minimization; we provide specific numerictools addressing this challenge. These measures strictlydescribe whether the ideal Hamiltonian Hctrl implementsthe target evolution. They do not incorporate errors aris-ing from stochastic noise processes. This is the subjectof robust control, described below.

12

2. Robust quantum control

The robust control setting presents the multi-objectiveproblem of achieving both tasks 1 and 2. In this case weconsider the stochastic evolution of the total HamiltonianHtot(t)

iUtot(t) = Htot(t)Utot(t) (8)

(9)

Noisy dynamics contributed by Herr(t), therefore distortthe final operation Utot(τ) away from the ideal Uctrl(τ).This effect may be isolated by expressing the total prop-agator as

Uerr(τ) = Utot(τ)Uctrl(τ)†. (10)

The residual operator Uerr(τ) defined in Eq. (10) is re-ferred to as the error action operator. This unitary sat-isfies the Schrodinger equation in an interaction pictureco-rotating with the control, which we call the controlframe. Specifically,

Uerr(τ) = T exp

[−i∫ τ

0

Herr(t)dt

](11)

where T is the time-ordering operator, and

Herr(t) ≡ Uctrl(t)†Herr(t)Uctrl(t) (12)

defines the control-frame Hamiltonian. An analogousconcept originally appeared in average Hamiltonian the-ory developed for NMR [45], but was called a togglingframe in that context because it considered only instan-taneous operations.

Using the definition of Uerr(τ) in Eq. (10), the robustcontrol problem may be formalized in terms of the dualconditions

Utarget = Uctrl (13)

Uerr = I (14)

where I is the identity operation on the control system.The robust control problem therefore consists of aug-menting the optimal control problem with the additionalcondition Eq. (14), describing how susceptible the sys-tem is to noise interactions under a given control Hamil-tonian. We define the corresponding measure

Frobust(τ) =

⟨∣∣∣∣1

D

⟨Uerr(τ), I

⟩F

∣∣∣∣2⟩, (15)

where the angle brackets 〈·〉 denote an ensemble averageover realizations of the noise processes, and 〈·, ·〉F de-notes a Frobenius inner product Eq. (A3). Robustnessis therefore evaluated as the noise-averaged operator dis-tance between the error action operator Eq. (10) and theidentity. This measure is bounded between [0, 1], withthe robustness condition Eq. (14) perfect implemented

when Frobust(τ) = 1. We define a corresponding infi-delity measure as

Irobust(τ) = 1−Frobust(τ) (16)

As above, a simple modification of this measure enablesthe robustness condition to be evaluated on a subspaceof interest. Specifically

FProbust(τ) =

⟨∣∣∣∣1

Tr (P )

⟨PUerr(τ), I

⟩F

∣∣∣∣2⟩

(17)

where P defines a projection matrix, enabling robust con-trol to be evaluated on a target subspace. And again,finding a robust control for a state transfer problem maybe expressed

Fψrobust(τ) = 〈|〈ψinitial|Uerr|ψinitial〉|〉. (18)

Note Eq. (15) does not include any information aboutwhether Uctrl(τ) implements a particular target gateUtarget. In this sense, the robustness criterion is target-independent. Once again, solving these conditions re-quires a numerical approach subject to a multiobjectiveoptimization routine. In practice, with sufficient control,it is always possible to satisfy both of these conditionsand find a robust control that achieves the desired targetoperation with high fidelity. We tackle this problem us-ing a number of novel approaches developed in Sec. V B,and demonstrated in real case studies in Sec. IV C.

B. Performance evaluation for arbitrary controls

The evaluation of any measure for the fidelity of arobust-control operation as in Eq. (15) requires the com-putation of Eq. (11), which is generally challenging ascontrol and noise Hamiltonians need not commute at dif-ferent times. Characterizing control robustness and per-formance in realistic laboratory settings—especially foroperations performed within large interacting systems—therefore requires simple, easily computed heuristics thataid a user in gaining intuition into control performance.

Here we introduce generalized multi-dimensional filterfunctions which serve as an engineering-inspired heuristicto determine noise susceptibility for arbitrary unitarieswithin high-dimensional Hilbert spaces. These objectsexpress the noise-admittance of a control as a functionof noise frequency, and reduce control selection to theexamination of an easily visualized object similar to theBode plot in classical engineering. Noise may be consid-ered over a wide range of parameter regimes, from qua-sistatic (noise slow compared to Hctrl(t)) to the limit inwhich the noise fluctuates on timescales comparable to orfaster than Hctrl(t). We build on past single-qubit stud-ies [16, 17, 46–48], assuming quasi-classical noise chan-nels, to produce an explicit basis independent compu-tational form incorporating all leading-order filter func-tions and extensible to higher-dimensional quantum sys-tems [49] with enhanced computational efficiency.

13

1. Modelling noise and error in D-dimensional systems

The error action operator Uerr(τ) for non-dissipativesystem-bath dynamics is treated as the unitary generatedby an effective Hamiltonian H(eff) = Φ(τ)/τ , such that

Uerr(τ) ≡ e−iΦ(τ), Φ(τ) ≡∞∑

α=1

Φα(τ). (19)

We obtain an arbitrarily accurate approximation forthe unitary evolution using a Magnus series expan-sion [50, 51]. where the αth Magnus term, Φα(τ), iscomputed as the sum of time-ordered integrals over per-mutations of αth-order nested commutators of Herr(tj),for j ∈ 1, ..., α (see App. C 1).

Consider an arbitrary D-dimensional quantum systemdefined on the Hilbert space H, and let the total controlHamiltonian in Eq. (1) have control and error terms ofthe form

Hctrl(t) =

n∑

j=1

αj(t)Cj , (20)

Herr(t) =

p∑

k=1

βk(t)Nk(t). (21)

The control Hamiltonian, Hctrl(t), captures a target evo-lution generated by n participating control operators,Cj ∈ H. The noise Hamiltonian, Herr(t), captures inter-actions with p independent noise channels. Distortionsin the target evolution are generated by the noise op-erators Nk(t) ∈ H, formally time-dependent such thatNk(t) = 0 for t /∈ [0, τ ]. The noise fields βk(t) are as-sumed to be a classical zero-mean wide-sense stationaryprocesses with associated noise power spectral densitiesSk(ω). The toggling-frame [16, 49] Hamiltonian takes theform

Herr(t) =

p∑

k=1

βk(t)Nk(t) (22)

where

Nk(t) ≡ Uctrl†(t)Nk(t)Uctrl(t) (23)

defines the noise operators in this toggling-frame. Usinga Magnus expansion as in Eq. (19), the noise actionoperator may then be approximated to the desired order.

For the purpose of calculating the error action oper-ator Eq. (11) we are free to choose any gauge trans-

formation of the form H ′err = Herr + gI which, up toa global phase, leaves the the dynamical evolution un-changed. With this freedom it is convenient to de-fine the transformed Hamiltonian with the property thatTr(PH ′err) = 0, namely tracelessness on the subspaceassociated with the projection matrix P , by choosingg = −Tr(PHerr)/Tr(P ). From Eq. (22), using the linear-ity of the trace and observing the noise variables βk(t)

are scalar-valued for classical noise, we obtain

H ′err(t) =

p∑

k=1

βk(t)N ′k(t) (24)

where we define the traceless noise operators in the tog-gling frame as

N ′k(t) ≡ Nk(t)−Tr(PNk(t)

)

Tr (P )I. (25)

Assuming the noise fields βk(t) are sufficiently weak,we truncate the Magnus expansion Eq. (19) at leadingorder and approximate the error action operator as

Uerr(τ) ≈ exp[−iΦ1(τ)]. (26)

Substituting into Eq. (17) the leading-order infidelitymeasure for robust control is approximated as

Irobust(τ) ≈ 1

Tr (P )Tr(P⟨

Φ1(τ)Φ†1(τ)⟩)

. (27)

To obtain this expression we perform a Taylor expansionon Eq. (26), retain terms consistent with the leading-order approximation, and use the inherited propertyfrom Eq. (25) that Tr(PΦ1) = 0 (see App. C 3).

To compute the first order Magnus term we substi-tute Eq. (24) into Eq. (C1), yielding

Φ1(τ) =

p∑

k=1

∫ ∞

−∞dtβk(t)N ′k(t) (28)

where we formally extend the limits of integration to±∞,noting that N ′k(t) = 0 for t /∈ [0, τ ].

2. Multi-dimensional filter functions in the frequencydomain

In order to efficiently compute Magnus contributionsto the infidelity we move to the Fourier domain, and re-express contributions to error in a D-dimensional system

Φ1(τ) =

p∑

k=1

∫ ∞

−∞

dω

2πGk(ω)βk(ω) (29)

where the Fourier-domain functions

Gk(ω) ≡ F N ′k(t) (−ω) (30)

βk(ω) ≡ F βk(t) (ω) (31)

are defined according to the conventions set outin App. A 1.

Substituting Eq. (29) into Eq. (27) then yields a com-pact expression for the leading-order robustness infidelityin the frequency-domain

Irobust(τ) ≈ 1

2π

p∑

k=1

∫ ∞

−∞dωFk(ω)Sk(ω) (32)

14

Here, each noise channel k contributes a term computedas an overlap integral between the noise power spec-trum Sk(ω) and a corresponding filter function, Fk(ω).An approximation to the inclusion of higher-order Mag-nus terms for the infidelity may be obtained by expo-nentiating this expression, due to the similarity of thepower-series expansion for an exponential function andthe structure of the Magnus series [17].

The critical element for capturing the action of the con-trol is the filter function, Fk(ω), relative to the kth noisechannel. The explicit form of the filter function with re-spect to the projection matrix P is defined (see App. C 4for details) as

Fk(ω) =1

Tr (P )Tr(PGk(ω)G†k(ω)

). (33)

This expression may be simply recast in a form that iseasily computed numerically, an essential task in softwareimplementations. Let pl be the lth diagonal element of

Q-CTRL solutions

Frequency

Resp

onse

Low

Susc

eptib

le

Standard control

Colored noise Whitened noise

Inse

nsiti

ve

High

Typical noise

Control applies

FIG. 6. Overview of the action of control as a noise filterat the operator level. Upper: An example of how colorednoise enters expressions for the infidelity of a control opera-tion as the overlap integral of the noise power spectrum andfilter function for the control. A colored spectrum is thuswhitened by the control through the physics of coherent av-eraging. Lower: Example filter function for an appropriatelyconstructed noise-suppressing/filtering control. Such filtersare low-frequency-noise suppressing; by reducing the filterfunction magnitude in a spectral range where the noise powerspectrum is large, the fidelity of the operation is improved.

P , then the filter function may be expressed

Fk(ω) =1

Tr (P )

D∑

l=1

pl

D∑

q=1

∣∣∣∣Gk(ω)

lq

∣∣∣∣2

. (34)

That is, take the Fourier transform of each matrix el-ement of the time-dependent operator N ′k(t), sum thecomplex modulus square of every element, weighted bythe diagonal elements pl, and divide through by Tr(P ),the dimension of the quantum system subspace. Withthis computationally simple definition, we enable the ef-ficient calculation of filter functions for single and multi-qubit gates, arbitrary high-dimensional systems, andcomplete circuits composed of multiple qubits and manyoperations. Thus we have a new computational deviceallowing the calculation of noise susceptibility for a widerange of elements relevant to quantum computation.

Given a noise power spectral density which representsrealistic time-varying noise for a target noise operator(e.g. dephasing ∝ σz), one may use the filter functionto simply estimate operational fidelity; the net fidelity isgiven by the overlap integral of these two quantities as afunction of frequency (Fig. 6). A high-fidelity control willminimize the filter function’s spectral weight in frequencyranges where the noise power spectral density for a par-ticular error channel is large. The predictive capabili-ties of this technique to control performance evaluationare experimentally validated for both single-qubit oper-ations [17] and higher-dimensional systems (e.g. MolmerSorensen gates [52]). An example of the predictive powerof the filter function is presented in Fig. 10d-e for a vari-ety of single-qubit controls.

C. Optimization tools to maximize controlperformance

Precise manipulation of quantum systems via shapedcontrol pulses has emerged as a key area of develop-ment for quantum physics and chemistry. In near-termquantum computers, for example, substitution of primi-tive operations with optimized noise-robust controls canyield improved fidelities for single- and multi-qubit gateswithout hardware modification, and offers a range of ad-ditional benefits, generally also providing reduced over-head in operation time. As discussed in Sec. IV A, forall but the simplest systems identifying such controls isanalytically intractable, and adapting these methods tohigher-dimensional systems relies on powerful numericaloptimization techniques.

Q-CTRL has developed a versatile optimization en-gine based on a GPU-compatible graph architecture,and leveraging where appropriate the efficient compu-tational heuristics introduced in Sec. IV B. This engineis purpose-built for rapidly creating high-fidelity con-trols spanning both optimal and robust control in high-dimensional Hilbert spaces. Creation of an optimizedcontrol solution may be undertaken for individual gates,

15

small interacting subcircuits, or complete algorithms. Allsuch circumstances are efficiently incorporated using thesystem definition introduced above without the need fora change in the underlying toolkit. Following system andtarget unitary definition, control optimization effectivelyrequires only a single command.

A technical description of the essential framework ispresented below, covering the core optimization engine,parameterization of control variables, definition of costfunctions, and incorporation of constraints. In Sec. V wedemonstrate these capabilities using higher-dimensionalsuperconducting systems as important case studies.

1. Optimizer framework

Mathematically we define the optimization problemas follows. Let C(v) denote the cost function for op-timization, where v = (v1, v2, ...) denotes an array ofgeneralized control variables parameterizing the con-trol Hamiltonian Hctrl(v). The evolved unitary aftertime τ is therefore parameterized as Uctrl(v, τ), and thecost function obeys the functional dependency C(v) =C (Uctrl(v, τ)). To benefit from gradient ascent methodsit is necessary to calculate all partial derivatives of thegradient function

~∇vC =(∂C∂v1

, ∂C∂v2

, . . .). (35)

In general this is a complex computation requiring manyapplications of the chain rule, and depends on the specificform of the cost function. These challenges are, how-ever, naturally overcome using TensorFlow as the op-timizer framework. This benefits from an in-built gra-dient calculator based on the underlying tensor map,and machine-learning algorithms for minimizing the cost-function. Moreover, TensorFlow permits the calculationof nonlinear gradients, which is particularly relevant incircumstances such as the parametrically driven gate forinteracting transmons where there the modulation ap-plied to a signal generator in the lab frame producesan effective operator experienced by the quantum sys-tem transformed by a Bessel function. We offer vari-ous methods for performing the optimization in orderto meet system constraints, including optimization overthe piecewise-constant control segments defined below,Runge-Kutta optimization, and CRAB optimization [53](see Sec. IV C 2 for further information).

In order to perform the optimization, the specific con-trol parameterization and cost functions must be defined.As described in App. B 3 the composition of the controlHamiltonian is formalized in terms of drive and shift con-trol pulses. For efficient optimizations, control pulses aretypically parameterized as piecewise-constant functionsdivided into m segments of uniform duration

τi = τ/m, i ∈ 1, ..,m. (36)

In this case the full control operation is determined byspecifying only the segment magnitudes on each of the m

Opt.Type

Cµ(v): functionalform

Description

Optimal IoptimalNoise-free target unitary

over D-dimensions

Robust 12πFk(0)

Quasi-static noise/constant-offset

Robust 12πFk(ω)

Fixed frequency noisesuppression at ω

Robust∫ ω2

ω1Sk(ω)Fk(ω) dω

2π

Broadband noisesuppression over [ω1, ω2]

TABLE I. Component cost functions to be included as de-sired in Eq. (45) for an optimization task. Here Fk(ω) andSk(ω) are the filter function and noise power spectral densityrespectively, associated with the kth noise channel.

segments. For a system accommodating d drive pulses,and s shift pulses, we define vector arrays

Ωj =[

Ω1,j , Ω2,j , . . . Ωm,j], j ∈ 1, ..., d (37)

φj =[φ1,j , φ2,j , . . . φm,j

], j ∈ 1, ..., d (38)

αl =[α1,l, α2,l, . . . αm,l

], l ∈ 1, ..., s (39)

where Ωi,j and φi,j are the modulus and phase of thejth drive pulse, and αi,l is the amplitude of the lth shiftpulse, on the ith segment. For efficient optimization itis convenient to work with normalized, non-dimensionalvariables. We therefore define

Ω′j ≡ Ωj/Ωmax,j j ∈ 1, ..., d (40)

φ′j ≡ (φj/π + 1)/2 j ∈ 1, ..., d (41)

α′l ≡ (αl/αmax,l + 1) /2 l ∈ 1, ..., s (42)

where Ωmax,j and αmax,l are the upper bounds permit-ted for the jth drive modulus and lth shift amplituderespectively. These transformed variables satisfy

v ∈ [0, 1], ∀v ∈ Ω′i,j , φ′i,j , α′i,l, (43)

j ∈ 1, ..., d, l ∈ 1, ..., s, i ∈ 1, ...,m.

With these definitions, we compactly express the full ar-ray of variables in the (2d+ s)m-length vector of gener-alized control variables

v =[

Ω′1, . . . Ω′d, φ′1, . . . φ

′d, α

′1, . . . α

′s

]. (44)

For a given total duration τ , the control Hamiltonianis therefore functionally expressed as Hctrl(v), and op-timization tasks are implemented on the search spacespanned by v.

16

2. Constrained Optimizations

In general an unbounded optimization can lead to ei-ther unphysical or impractical control solutions. Limit-ing the search space via implementation of appropriatelyconstructed constraints can assist in ensuring that con-trols meet hardware limitations, and also dramaticallyimprove the general efficiency of the optimization prob-lem. To generate optimized controls subject to multiplecompeting constraints, we generalize the cost function asa linear combination

C(v) =∑

µ

wµCµ(v) (45)

where each component Cµ(v) measures a distinct aspectsof the target performance as a function of the general-ized control variables v defined in Eq. (44), and the con-stants wµ weight the relative importance of these contri-butions in the optimized result. The optimizer iterativelysearches over suitably represented control parameters vin order to minimize the cost function defined above. Thefunctional forms of some important cost function compo-nents are outlined in Table I.

This framework allows physically motivated con-straints to be incorporated into the optimization prob-lem which dramatically expands the range of problemswhich can be addressed. In particular we have focusedon meeting the demands imposed by physical hardwarelimitations, as detailed in Table II.

One of the most important constraints to be consideredis smoothing of pulse waveforms to accommodate band-width limits and finite response times from hardware. Ingeneral smoothed solutions can be achieved by increasingthe segment number in an optimization, constraining theeffective time-derivative (bounding the magnitude of thecontrol’s changes between segments), or ensuring that alloptimized waveforms include linear-time-invariant filterssuch as RC filters or filters defined by a tanh function.Such filtered optimization ensures that a distorted wave-form passed through e.g. a transmission line, will stillexhibit near-optimal performance at the qubit. An exam-ple RC-filtered pulse waveform is highlighted in Fig. 13d.Additionally, the Q-CTRL optimization package sup-ports optimization of evenly-sampled smooth functions.In this mode, optimized pulses are represented as evenly-spaced samples from a smooth function and propagatedusing a Runge-Kutta integrator, which leads to a modelof system dynamics with greater accuracy in the presenceof physical hardware limitations.

Finally, we also offer a CRAB-type [53, 54] optimiza-tion in which a waveform is selected from a superposi-tion in the Fourier basis and discretized in time. Thistruncates the effective search space by limiting it to theassociated Fourier coefficients, and is therefore indepen-dent of the granularity of the piecewise-constant dis-cretization. The Q-CTRL optimization package containsa flexible CRAB implementation that allows a variety of

0100

101

102

103

100 200 300 400 500

Number of Segments

NumPy QuTiP Q-CTRL

Tim

e: 2

0 O

ptim

izat

ions

(s)

FIG. 7. Head-to-head local-instance performance benchmark-ing of Q-CTRL optimization tools vs optimization complex-ity (measured by number of segments). Time-to-solution ispresented on a logarithmic scale, demonstrating an approxi-mately order-of-magnitude enhancement in performance overQuTiP, and ∼ 100× improvement over local-instance NumPy.Additional performance benefits are achieved via parallel pro-cessing in a cloud architecture that are not captured here.

CRAB techniques (e.g. bases with randomized frequen-cies, fixed frequencies or optimizable frequencies) to beincorporated alongside other constraint options.

3. Optimizer performance benchmarking

The Q-CTRL optimization engine provides perfor-mance which outstrips competitive packages in both flex-ibility and time to solution. In head-to-head performancebenchmarking in a local-instance implementation we findgreater than order-of-magnitude performance improve-ments over NumPy optimization tools and approximately5× advantage relative to optimization tools in QuTiP.We note, however, that the general flexibility to performconstrained optimization is often not possible at all withother tools. Moreover, the cloud-architecture of Q-CTRLproducts provides an additional parallelization advantagewhich translates to approximately 9× improvement intime-to-solution over a local-instance implementation ona modern PC. This ultimately translates to nearly ∼ 50×speedup in optimizer performance over the broadly avail-able competitive packages.

D. Time-domain simulation tools for realistichardware error processes

An independent approach for analysing the dynamicsof an algorithm or gate in the presence of noise is viatime-domain simulation. Provided the noise-free evolu-tion of the system is well-understood, simulation may be

17

Optimizer Constraint Description Technical Details

Smooth Controls:The temporal variation in a control

waveform is bounded to ensure smoothtransitions and limit discontinuous

transitions requiring high bandwidths.

A “smooth” waveform may be obtained via one of multiplemethods:

(i) Limiting the effective time-derivative for any drives orshifts, e.g. |Ω(τi)− Ω(τi−1)| < (δΩ)Max.

(ii) Composing candidate control waveforms as superpositionsin the Fourier basis before discretized sampling in time.

(iii) Employing a novel smooth-function sampling routinebased on Runge-Kutta integration.

These approaches ensure that for any targeted channel the rateof change of controls between segments is bounded, and are

equivalent, by the properties of the Fourier transform, toimposing a maximum frequency cutoff on the control.

Filtered Controls:Control pulses are transformed by a

linear-time-invariant filter

To account for hardware bandwidth limitations, ideal pulsesegments may be transformed into filtered waveforms usingarbitrary linear-time-invariant filters such as RC filters with

specified high-frequency cutoff, ωmax, tanh, or equivalent filterscapturing transmision-line dynamics.

Bounded-Strength Controls:For any control pulse, the magnitude ofthe pulse waveform may be bounded.

This ensures the optimization does notexceed physically motivated bounds on

control parameters.

For any drive pulse j ∈ 1, .., d, or shift pulse l ∈ 1, .., s,upper bounds on pulse waveforms may be enforced, such that

Ωj(t) ≤ Ωmax,j

|αl(t)| ≤ αmax,l

where Ωmax,j and αmax,j define the maximum (positive)permissible value of drive modulus or shift amplitude

waveforms respectively.

Fixed-Control Waveforms:For any individual control, the pulse

waveform may be held fixed andeffectively frozen out of the variational

search.

For any drive pulse j ∈ 1, .., d, or shift pulse l ∈ 1, .., s, set

Ωj(t) = fj(t)φj(t) = gj(t)αl(t) = hl(t)

where fj(t), gj(t), hl(t) are specific functions of time. For thesegmentation described in App. B, this corresponds to setting

Ωj , φj , αl → constant vector

Concurrent vs Interleaved Controls:Control pulses on different drives and

shifts are executed sequentially orsimultaneously.

In certain physical systems it is not possible to implement allcontrols simultaneously. In this case the control Hamiltonianmust be constructed from appropriately sequenced drive/shiftoperations. This is achieved by transforming the optimizationvariables in Eq. (44) as v → v · b, where b is a binary mask

enforcing the required structure of interleaved operations. Forb set to unity, controls may be applied concurrently.

TABLE II. Constraint options for Q-CTRL’s numerical optimization toolkit.

used to investigate system dynamics in the presence ofdifferent noise sources.

Simulation packages based on Schrodinger integrationand matrix multiplication are commonly found in manysoftware packages, including free repositories such as

Qiskit [55] and GST [56]. However, the integration ofnoise to simulate realistic environments is typically re-stricted to a small set of quasi-static offsets or the incor-poration of fully stochastic depolarizing noise.

Q-CTRL provides several unique tools for simulating

18

quantum systems exhibiting correlated and colored noisechannels as typically encountered in realistic laboratoryenvironments. These capabilities significantly expand auser’s ability to understand and predict the impact ofnoise and hardware imperfection on algorithmic perfor-mance. These tools are designed to be efficient and togive access to information which is not typically avail-able through conventional simulation packages.

The core Q-CTRL simulation module accepts a controlHamiltonian (expressed as drives, shifts and drifts, as de-scribed in App. B), together with any number of arbitrarypiecewise-constant time-domain noise processes. Thesenoise processes can multiplicatively perturb the moduliof the drive or shift controls, or contribute additively tothe system Hamiltonian. From this information, the sim-ulation module produces an overall piecewise-constantsystem Hamiltonian, and solves the Schrodinger equa-tion via matrix exponentials to compute the unitary timeevolution operator for the system at arbitrary times.

This package provides several key functions that enableefficient and useful simulation in noisy environments:

• Creation of a time-domain noise process from aninput user-defined noise power spectral density

• Incorporation of a user-defined time-series into asimulation, including data-series interpolation

• Automated homogenization of time-segmentationof all input and software-defined time series in orderto permit Schrodinger integration from data setsexpressing different temporal discretization

• Forward propagation of an initial input state sub-ject to calculated time-evolution operators includ-ing noise

• Calculation of ensemble-averaged density matricesover independent but statistically identical noise re-alizations.

In the remainder of this subsection we provide a tech-nical exposition on each of these capabilities.

1. Technical details of simulation functionality

The first objective in this simulation framework is toconvert a noise process which is conveniently expressedas a power spectral density (e.g. oscillator phase noiseor magnetic field noise) to a time-domain representa-tion. Consider a (one-sided) user-defined noise powerspectral density S(ω), sampled at discrete frequenciesSk = S(k∆ω) (for 0 ≤ k ≤ N − 1 and spacing ∆ω).

Samples from the corresponding two-sided spectrumare defined by symmetrizing and rescaling the one-sidedspectrum:

S(2)k :=

S0 k = 0

0.5Sk 1 ≤ k ≤ N − 1

0.5S2N−1−k N ≤ k ≤ N − 2.

(46)

Next, a discrete amplitude spectral density, xk, is gen-erated from the two-sided power spectral density using

S(2)k = |xk|2, meaning the phases of the xk may be chosen

arbitrarily. Accordingly, the amplitude spectral densityis defined as

xk := eiφk√S

(2)k , (47)

where φk is chosen uniformly at random for 1 ≤ k ≤N − 1, and we fix φ0 := 0 and φk := φ2N−1−k forN ≤ k ≤ 2N − 2 (which ensures Hermitian symmetryof xk). Finally, the real time series corresponding tothe generated amplitude spectral density is produced viaa suitably-normalized discrete inverse Fourier transform:

xk =√

∆ω

N−1∑

l=0

xle2πi klN . (48)

This process yields a single random realization of a realtime-domain signal with a power spectrum matching theinput spectrum S(ω).

Given this form of a discrete, real, time series gen-erated from a noise power spectral density (created asabove or provided directly by the user), it may be desir-able to perform simulation using a higher sampling ratethan that native to the data (for example if only low-frequency noise is specified). The Q-CTRL simulationpackage enables this upsampling via Whittaker-Shannoninterpolation. This produces a continuous-time functionxk that interpolates the initial time series, with a bandlimit set by the Nyquist frequency of the series

x(t) =

∞∑

k=−∞xksinc

(t− k∆t

∆t

), (49)

where ∆t is the time spacing between samples. To ap-proximate the infinite sum, the simulation package auto-matically performs periodic extension of the input seriesand truncation of the sum to accuracy within the domainof the original time series.

With discretized time-series data in hand it becomespossible to simulate the time evolution of a system via in-tegration of the Schrodinger equation. However, in manycases the natural temporal discretizations will vary be-tween different fields within the system. For example,rapidly-fluctuating noise sources may be defined on sig-nificantly shorter time scales than control fields, whilequasi-static noise processes could be defined on longertime scales.

To enable simulation in such cases, all discretizationsare automatically resampled on a shared grid prior tointegration. This enables a user to simply input data se-ries as-is and the package will handle all homogenizationissues. For example, if a drive control pulse is definedon two segments of duration τ/2 by [Ω1,Ω2], but a noiseprocess is defined on three segments of duration τ/3 by[β1, β2, β3], the shared discretization has six segments of

19

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)(r

ad)

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z)

0 15 30

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z)

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z)

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)(r

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z)

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z)

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abili

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100

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abili

ty

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100

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ulat

ion

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ronm

ent

Add

itive

deph

asin

g

Frequency (Hz) Time (ns) Time (ns)

Detu

ning

nois

ePh

ase

nois

e

Frequency domain Time domain Noisy system

Ideal Noisy

a

b

c

d

FIG. 8. Demonstration of time-series simulation for a Gaussian DRAG pulse implementing an X gate on a qutrit subject toleakage. The control corresponds to a Gaussian DRAG pulse including a Gaussian envelope on the X quadrature, its timederivative on Y, and a Gaussian-squared pulse on Z. (a-c) Left column: frequency-domain noise power spectra input intosimulation tool. Middle column: interpolated single-instance time-series derived from noise channels and associated controlchannels (where appropriate). Right column: Noisy control operators combining time-domain noise process and controls. Threedifferent noise processes are included relevant to implementation of DRAG: (a) phase noise on Y quadrature of microwave con-trol, (b) microwave detuning, (c) ambient dephasing. (d) time-domain evolution under ideal and noisy evolutions, representingpopulations as a function of time for states |0〉, |1〉, and |2〉. This simulation assumes perfect control on the X-quadrature withnoise as indicated above.

duration τ/6 with drive pulse [Ω1,Ω1,Ω1,Ω2,Ω2,Ω2] andnoise [β1, β1, β2, β2, β3, β3].

Following resampling, and letting the endpoints of theN piecewise-constant time segments be τk for 1 ≤ k ≤ N ,

the Hamiltonian can be written

H(t) =

H1 0 ≤ t ≤ τ1. . .

HN τN−1 < t ≤ τN .(50)

20

With this representation in place, calculation of the uni-tary time evolution operator U(t) is achieved by integrat-ing the Schrodinger equation via the matrix exponential:

U(t) = e−iHk(t−τk−1)e−iHk−1(τk−1−τk−2) . . . e−iH1τ1 ,

where k is chosen such that τk−1 < t ≤ τk.

The Q-CTRL simulation package provides a functionto propagate an initial state using the unitary time evo-lution operator (calculated via the integration processdescribed above). Specifically, for samples of the unitarytime evolution operator Ul = U(tl) (evaluated at arbi-trary times tl), propagation of an initial state |ψ0〉 tothe state |ψl〉 at time tl is given by

|ψl〉 = Ul|ψ0〉.

In general, however, calculating a single instance of thetemporal evolution of the state is insufficient to under-stand the target dynamics, and an ensemble average overdifferent noise realizations is required. The Q-CTRL sim-ulation package provides a function to compute the meandensity matrix associated with an ensemble of propa-gated state vectors. Given a set of state vectors |ψm〉(for 1 ≤ m ≤ M) produced from an ensemble of simu-lations corresponding to different noise realizations, themean density matrix ρ is given by

ρ =1

M

M∑

m=1

|ψm〉〈ψm|.

2. Simulation example

In Fig. 8 we demonstrate this capability in order tosimulate the impact of noise on a Gaussian DRAG pulseused to perform a leakage-suppressed single-qubit oper-ation in a superconducting circuit [57]. This particularexample focuses on the execution of a single quantumlogic operation subject to high-frequency noise; howeverwith this package it is easy to extend this simulation tocomplex multi-operation circuits experiencing noise on avariety of timescales.

This simulation includes time-varying Y -quadraturephase noise on a microwave drive, a time-varying mi-crowave detuning and an additive ambient dephasingfield applied to the qubit. In each case an input powerspectral density (left column) is converted to a time-seriesand combined with the relevant control channel (middlecolumn), resulting in a noisy system representation (rightcolumn). The simulated performance in the ideal case isshown in Fig. 8d, indicating high fidelity state transferfrom |0〉 → |1〉 with negligible population of the leakagelevel |2〉. However, in the presence of noise the fidelity ofstate transfer is reduced by approximately three ordersof magnitude and a leakage error emerges.

E. Characterizing hardware noise

Identifying opportunities for hardware improvementand deploying high-performance robust quantum con-trols benefits from access to microscopic informationabout the noise in user hardware, beyond that capturedin more generic error characterization routines [58]. Inthe filter function framework this is captured throughthe noise power spectral density, a statistical frequency-domain representation of noise processes such as ambientdephasing or control noise. Having access to this infor-mation is a key requirement for both the evaluation ofcontrol performance and control optimization, both de-scribed above, and serves as a complement to Hamilto-nian learning procedures [59, 60].

In general detailed information about noise powerspectral density is not easily determined when sourcesmay be local to a device and inaccessible to traditionalmeasurement protocols. This limit may be overcome byusing a qubit as a measurement device to directly probethe noisy environment and pursuing non-parameterizedspectral estimation. Consider a noise Hamiltonian Herr

as in Eq. (21), comprised of multiple independent noisesources, each described by a corresponding PSD. Ap-plication of appropriate time-domain modulation to thequbit can change the noise sensitivity (by tuning the filterfunction for the control), as well as the noise operator towhich the qubit is most sensitive. The problem treatedhere is how to reconstruct these PSDs from measurablequantities.

The process of noise characterization follows a simpleworkflow, highlighted schematically in Fig. 9:

• Create controls based on the target noise operatorand hardware limits such as maximum experimen-tal duration and timing resolution

• Output controls and perform measurements usingthe prescribed protocols

• Perform data fusion on measurement results to re-construct a noise power spectrum.

There is a tight integration between the type of noise tobe characterized, the available control protocols, and thedata fusion routine employed for spectrum reconstruc-tion. A range of routines is available for the character-ization of both multiplicative and additive Hamiltoniannoise channels, corresponding, e.g. to control-amplitudenoise and ambient dephasing, respectively. Users may ul-timately trade experimental simplicity against the com-plexity and (in)stability of numerical data-fusion rou-tines.

In cases where experimental simplicity is prioritized,dephasing-noise information can be obtained using timedsequences of simple driven rotations, often referred to aspulsed dynamical decoupling sequences. Here quantumbit flips are sequenced in order to produce a filter func-tion with a dominant peak at the frequency defined by

21

Noise source ReconstructionControls probe noise

Frequency

Power

Low

Stro

ngW

eak

High

Line frequency/ mains power

Microwaveinstability

Electronicnoise

Filter functionof “probe” pulse

FIG. 9. Overview of the noise characterization process usingmulti-dimensional filter functions and SVD spectral inversiontechnique. Upper: A noise source is probed by a sequenceof control solutions each providing sensitivity to a differentspectral range, as determined by the multi-dimensional filterfunction. Measurement results are then used to produce a re-construction of the actual spectrum experienced by the qubit,with degradation in fidelity determined by the available con-trols and numeric routine. Lower: Concept demonstratinghow an appropriately constructed filter function can serve asa narrowband probe of underlying noise processes, giving ac-cess to different technical components of the noise spectrum.Selecting an alterate control can shift the peak in the filterfunction in order to allow broadband sampling of the noisespectrum.

the inverse interpulse delay. These controls unfortunatelyrequire a numerically intensive data fusion proceduresbased on a large matrix inversion in order to appropri-ately account for the many harmonics appearing in thefilter function. Moreover, performing inversion mandatesvery strict requirements on sequence structure which canultimately induce limits on achievable frequency resolu-tion.

Shaped controls based on so-called Slepian waveformsare highly effective for the characterization of both con-trol noise and dephasing. These controls are provably op-timal in terms of spectral concentration, i.e. how muchspectral weight resides within a target band. Accord-ingly they mitigate issues of spectral leakage which causeunwanted out-of-band signals to contribute to the mea-surement as a form of interference. They can be thoughtof as mathematically optimal window functions applieddirectly to the qubit itself, restricting the qubit’s sensi-

tivity to noise. These approaches are powerful in theirability to suppress out of band interference and requirevery simple post-measurement processing, but require ar-bitrary pulse shaping.

We have focused on producing a novel spectrum re-construction protocol which allows a user to employ wellconditioned control sequences while relaxing both com-mon performance limits and the computational inten-sity of spectrum reconstruction. Any relevant set ofmeasurements may be employed to characterize a tar-get noise channel, with an overall frequency-dependentsensitivity function. Data fusion is conducted usinga machine-learning technique based on an efficient im-plementation of pseudo-inverse by singular value de-composition (SVD). This enables parameter-free estima-tion needed to perform reconstructions without a prioriknowledge of the underlying structure of the noise. Thisapproach is easily generalized beyond single qubits basedon the multi-dimensional filter-function formalism sum-marized in Sec. IV B, and as demonstrated experimen-tally in Sec. V D.

Consider a quantum system consisting of p indepen-dent noise sources

Herr(t) =

p∑

k=1

βk(t)Nk(t) (51)

where the βk(t) are stochastic scalar-valued noise fieldswith corresponding PSDs, Sk(ω). The structure ofHerr(t) may be probed by defining a set of c distinctcontrol protocols

Hctrl,j(t), j ∈ 1, ..., c (52)

and measuring the corresponding infidelities.From Sec. IV B, and assuming the noise is suffi-ciently weak, the infidelities may be approximatedas

p∑

k=1

∫ ∞

−∞

dω

2πF jk (ω)Sk(ω) ≈ I(j), j ∈ 1, ..., c (53)

where F jk (ω) is the leading-order filter function associ-ated with the jth control protocol and kth noise source.The filter functions may be computed using Eq. (34)given knowledge of the control Hamiltonians and dy-namical noise generators, while the infidelities may beobtained from experiment.

Let [ωmin,k, ωmax,k] denote the frequency domainspanned by the low- and high-frequency cutoffs, assum-ing they exist, for each PSDs in Eq. (53). We may thendefine the sample frequencies

ωk,` = ωmin,k + (`− 1)∆ωk, (54)

for samples ` ∈ 1, ..., s(k), incremented by frequencysteps

∆ωk =ωmax,k − ωmin,k

s(k)− 1(55)

22

on each domain k ∈ 1, ..., p. Assuming sufficientlylarge sample numbers, s(k), Eq. (53) may be recast asa discrete sum, with the integrals approximated usingthe trapezoidal rule. Specifically

Ij =∆ωk2π

F jk,`Sk,`

(1− δ`,1

2

)(1− δ`,s(k)

2

)(56)

where δij is the Kronecker delta[61], and the sum runsimplicitly over repeated tensor indices. Here we haveintroduced the following tensor notation for the varioussampled quantities

F jk (ωk,`) ≈ F jk,` ±∆F jk,` (57)

Sk(ωk,`) ≈ Sk,` ±∆Sk,` (58)

I(j) ≈ Ij ±∆Ij (59)

where at Q denotes the estimated value for the quantityQ, and ∆Q denotes its uncertainty. See App. D 1 for adiscussion of estimates and uncertainties for the respec-tive measured and calculated quantities.

The challenge, then, is to obtain estimates, Sk,`±∆Sk,`

for the power spectral densities by inverting the relation-ship defined by Eq. (56), given knowledge of the mea-

sured quantites Ij ± ∆Ij and computed values F jk,` ±∆F jk,`. To facilitate efficient numerical solutions we moveto a matrix representation. Namely,

F S = I, (60)

where F =[F 1 F 2 · · · F p

]is a horizontal concate-

nation of matrices of the form

F k =∆ωk2π

12 F

1k,1 F 1

k,2 . . . F 1k,s(k)−1

12 F

1k,s(k)

12 F

2k,1 F 2

k,2 . . . F 2k,s(k)−1

12 F

2k,s(k)

......

. . ....

...12 F

ck,1 F ck,2 . . . F ck,s(k)−1

12 F

ck,s(k)

,

(61)

the estimated PSDs are concatenated vertically as

S =

S1

S2

...

Sp

, Sk =

Sk,1

Sk,2

...

Sk,s(k)

, (62)

and the estimated infidelities are arranged as

I =

I1

I2

...

Ic

. (63)

The matrix dimensions therefore satisfy

dim(F)

= c× n (64)

dim(S)

= n× 1 (65)

dim(I)

= c× 1 (66)

where n =∑pk=1 s(k). From Eq. (60), performing noise

reconstruction thus reduces to solving the matrix inverse

problem S = F−1I, solutions to which are discussed in

the next section.Depending on the particular set of controls and noise

sources, and on the dimensions c, p, and n, the exact

matrix inverse F−1

may not exist. Generally, the systemmay be underdetermined or overdetermined, and the ma-trix F may be singular. An approximate solution may,however, be obtained via singular-value decomposition(SVD). This may be used to obtain a pseudo-inverse ifthe problem is undetermined, to perform regression if itis over-determined, or to calculate the exact inverse if itis in fact determined. Usefully, in all cases, the singularvalue decomposition of F takes the same general form:

F = UDV †, (67)

Here, U is a (c × c) unitary matrix, V † is a (n × n)unitary matrix, and D is a (c× n) rectangular diagonalmatrix with non-negative real numbers on the diagonal.The columns of U and V are the left- and right-singularvectors of F , and the diagonal elements of D, denotedsi, are known as the singular values. The final represen-tation is then given by

S = V D+U †I, (68)

where D+ is a diagonal matrix with entries 1/si for allnon-zero singular values, and zero otherwise. Furtherdetails on the SVD method, techniques to include mea-surement uncertainties, and our approaches to truncatingsingular values are presented in App. D.

V. QUANTUM CONTROL CASE STUDIES

In the sections above we described the role of quan-tum control in combating hardware error, and intro-duced new technical capabilities for the characterizationand optimization of quantum hardware performance. Inthis section we provide case studies to demonstrate theapplication of these capabilities in contemporary quan-tum computing hardware. First, we provide experimen-tal demonstrations of performance of open-loop controlsolutions in trapped-ion hardware, demonstrating error-robustness as well as error-rate homogenization in spaceand time. Second, we apply the numerical optimizationpackage described in Sec. IV C to generate single andmulti-qutrit gates optimized for robustness against leak-age and dephasing errors in a coupled-transmon system.Third, we apply the SVD spectral reconstruction soft-ware outlined in Sec. IV E to the IBM Q cloud-basedquantum processor to characterize noise affecting two-qubit cross-resonance gates. Finally, we present an ex-ample of optimizing the structure of a quantum circuit,producing a logically-equivalent compiled circuit exhibit-ing suppression of cross-talk errors arising from a con-stant ZZ interaction. We emphasize that these examples

23

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2000101 10

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1

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54321

pR

B(

10

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108642Qubit Number

Control Errors

Primitive Dephasing Suppressing Dual Control Suppressing

Ideal Gate 1st Order

1 10...

-0.1 0.0 0.1

1.0

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Rob

ustn

ess

Dephasing Errors

Infi

de

lity

( p|1

i)

2000

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2 qubit MS

Fractional Error Magnitude

ab

h

f

ed

c

g i

Primitive Control Suppressing

2P1/2

2S1/2

Experimental Setting: Trapped 171

Yb+ Noise & Error Suppression Error Homogenization in Space & Time

FIG. 10. Experimental validation of key capabilities in quantum firmware. (a) Schematic of experimental platform in use, basedon trapped Ytterbium ions as described in detail in [38, 52]. (b-c) Experimental demonstration of robustness to quasistatic errorsusing known open-loop dynamically corrected gates. The control suppressing gate is BB1, dephasing suppressing is CORPSE,and dual suppressing gate is CinBB. BB1 and CORPSE are single-axis error suppressing DCGs, designed to suppress over-rotation control errors and off-resonance dephasing errors respectively, and reduced CinBB is designed to suppress errors inboth quadratures. The control-noise-suppressing gate appears most robust to control errors, as indicated by a low measuredinfidelity as a function of applied overrotation error. Similarly, the dephasing suppressing gate shows similar performance inthe presence of dephasing errors, and the dual suppressing gate shows comparable performance in the presence of both formsof error. (d-e) Demonstration of robustness to time-varying noise for the same DCGs, matched to filter-function predictions.Again, low values of measured infidelity indicate robustness; all gates show an onset of error susceptibility at high frequenciesnear the inverse gate time, while error-suppressing gates show robustness at low noise frequencies (left of the graph). (f)Demonstration of error homogenization in space across a ten-qubit device using open-loop control, measured by randomizedbenchmarking. In this system, ions are simultaneously addressed by a global microwave control field to drive global single-qubitrotations. However, due to a gradient in the strength of the microwave field due to interference patterns, the qubits rotate witha spatially varying Rabi rate. Shading indicates the range of experimental outcomes while the mean error across the device isindicated by lines. Using DCGs the mean error is reduced ∼ 5× while the range (measured either by the standard deviationor the difference between minimum and maximum values) is reduced ∼ 10×. (g-h) Demonstration of improved stability of atwo-qubit Molmer-Sorensen gate using control solutions. As the order of error-suppression is increased by control construction,the range of measured values of ion populations is narrowed while the system operates continuously over approximately 3.5hours. Here black is P0, the measured probability of zero ions measured in |1〉, blue is P1, the probability of one ion in |1〉, andred is P2, the probability of two ions in |1〉. The ideal gate configuration is shown in panel (i) with P0 = P2 = 0.5 and P1 = 0.Colored shading represents the range of the associated data set over the measurement window. Further experimental detailsare available in Appendix E.

are not exhaustive representations of product capability,and that additional demonstrations for e.g. optimizingparallel Molmer-Sorensen gate implementation, or char-acterizing simultaneous noise processes in spin qubits willbe presented in separate manuscripts.

A. Open-loop control benefits demonstrated intrapped-ion QCs

Trapped-ion quantum computers already exist atmedium scales and provide an ideal platform for studiesof quantum control efficiency due to long intrinsic life-times, high-fidelity operations, and access to multiqubitdevices. We have used a trapped-ion quantum computercomposed of individual 171Yb+ ions in order to explorethe efficacy of quantum control and quantum control op-timization in real hardware.

We begin with demonstrations of error-robustness us-ing open-loop control solutions available in the OpenCon-trols package of driven single-qubit operations. In Fig. 10we probe the robustness of various composite control op-erations which induce a net bit flip X (equivalently a πpulse) to systematic quasistatic errors in the pulse areaand dephasing due to a detuning from the qubit fre-quency (Fig. 10b-c). Protocols designed to provide ro-bustness to the associated error channel reveal little de-viation from the baseline error rate achieved in the centerof the graph (zero induced error) while the measured infi-delity (I) increases rapidly in the presence of systematicerrors for non-robust controls. This difference is a keysignature of error-robust control solutions.

Similarly, using a so-called ‘system-identification’ tech-nique to probe control robustness to a time-varying per-turbation [17] we demonstrate that appropriately craftedcontrols suppress noise at frequencies slow compared with

24

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1.0

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0.5

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0.5

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Optimized

Primitive

Mea

sure

d In

fidel

ityPr

imiti

ve

0-

0

6

0

85Time (µs)

(rad

)(k

Hz)

Opt

imiz

ed

0-

0

6

0

425Time (µs)

(rad

)(k

Hz)

a b

c d

e f

FIG. 11. Demonstration of optimized controls in a trapped-ion quantum computer. (a) and (c) Bloch sphere representa-tion of primitive and numerically optimized robust controls,respectively. Robust control was constrained to ensure a fixedpulse amplitude and optimized to provide dual suppression ofboth detuning and amplitude errors. (b) and (d) Polar coor-dinate representation of waveforms used to implement a netX rotation. The optimized control implemented in (d) is con-strained to have a fixed amplitude and only undergoes phasemodulation. (e) and (f) demonstration of robustness againstquasistatic overrotation and detuning errors; low values cor-respond to small measured errors. The x-axis is expressedfractionally relative to the Rabi rate or qubit frequency re-spectively. Shading represents the net improvement in errorrobustness afforded by the optimized solution. Further detailsof the implementation are provided in App. E.

the control rate (Fig. 10d-e). Experimental measure-ments compare well with calculation of the control fil-ter function (solid lines), which is available through bothBLACK OPAL and BOULDER OPAL. In particular,these experiments demonstrate that it is possible to con-struct single-qubit logic operations robust to noise inboth the control amplitude and qubit-frequency detun-ing.

Moving beyond physical-layer benefits we can alsoprobe the manner in which these control operations inter-sect with higher levels of the quantum computing stack asdescribed in Sec. II B. For instance, in Fig. 10 f we demon-

strate homogenization of Parallel Randomized Bench-marking (RB) error rates across a 10-qubit quantum com-puter using error-suppressing open-loop gate construc-tions validated in Fig. 10 b-e. Here the dominant errorsource is a spatial gradient in the coupling of the qubitdrive field to the individual ions, due to reflections andinterference of the 12.6 GHz microwaves inside the ion-trap vacuum enclosure. We therefore select an control-noise suppressing solution and replace all gates in therandomized benchmarking procedure with their logicallyequivalent error-robust constructions [38].

In this experiment the best-performing qubit does notexhibit a net improvement in the measured RB error rate,pRB—a proxy measure for gate error—beyond measure-ment uncertainty due to other limiting error sources suchas laser-light leakage. However, all other qubits exhibitRB error rates that now approximately match the bestreported values and show a narrowing of the spread ofoutcomes across the device. Quantitatively, the standarddeviation of RB error rates across the 10-qubit array isreduced 10.2× using the appropriate open-loop controlsolution.

Moving beyond the application of control solutions forsingle-qubit gates, we examine the stability of two-qubitgates realized via the Mølmer-Sørensen interaction on apair of trapped ions as the system experiences drifts intime. In this experiment we are targeting the creation ofa Bell state (|00〉 − i |11〉) /

√2; ideally in this experiment

there is an equal probability of measuring two ions in|0〉 or |1〉, and one should never observe any experimentswith one ion each in these two states. Therefore ourkey proxy measure for gate performance is the measuredpopulation of zero, one, and two ions in state |1〉. Theexpected performance of these metrics is shown in Fig. 10i.

We compare two different gate constructions, one be-ing relatively susceptible to drifts and the other designedto reduce sensitivity via a modulation protocol availablein both BLACK OPAL and BOULDER OPAL, and ex-perimentally demonstrated first in [52, 62]. Repeatedlyperforming the same gate shows variations in the mea-sured ion-state populations, corresponding to reductionsin gate fidelity. However, by using the error-suppressinggate construction we observe a ∼ 3−4× reduced suscep-tibility to system drifts, indicated by arrows in Fig. 10 h,showing the reduced range of outcomes.

Finally, we demonstrate the efficacy of numerically op-timized single-qubit gates against various noise processesin trapped-ion qubits in Fig. 11. Specifically, we havefocused on the use of the optimization toolkit describedin Sec. IV C to produce gates that are simultaneouslyrobust against control noise and dephasing. Typicallythis requires a concatenated analytic construction whichdramatically extends the control duration by up to 24×relative to the primitive gate. For an X gate the opti-mizer returns solutions showing simultaneous robustnessto error with a gate duration reduced ∼ 5× relative tothis analytic approach. In the presence of quasistatic

25

errors the numerically optimized solutions provide showrobustness to error in the presence of up to 10% miscali-brations in qubit frequency and Rabi rate, similar to theresults of Fig. 10b-c. Similar results (not shown) havebeen obtained using superconducting circuits on IBMQ. The particular solution employed for the supercon-ducting circuit was implemented using IBM’s OpenPulseformat [40] through Qiskit, and was slew-rate-limited inorder to comply with band-limits in transmission lines(such constraints are generally not germane in trapped-ion systems due to the relatively long pulse durationsand use of microwave antennae rather than transmissionlines). Further details on the execution of these experi-ments is included in App. E.

B. Simultaneous leakage and noise-robust controlsfor superconducting circuits

For superconducting qubits, a two-level system is typ-ically singled out from the many levels of an anharmonicoscillator. When driven by naive single-qubit controls,the system is subject to off-resonant coupling to leakagelevels outside the qubit manifold, resulting in substantialleakage errors. In addition, these qubits face the com-mon challenges of decoherence from ambient dephasing,control-phase and control-amplitude noise.

Suppression of leakage errors has been the focus ofconsiderable research in the superconducting communityand has been demonstrated to improve gate performance.The standard approach at present employs an analyticoptimal control technique to implement target quantumoperations via so-called DRAG pulses [57], or variantsthereof. For example, half-DRAG is designed to suppressleakage out of the qubit subspace via dual-quadraturecontrol, typically involving a Gaussian pulse on σx, andits time derivative simultaneously applied on σy.

Unfortunately this technique does not combine suc-cessfully with other open-loop error-suppression strate-gies needed to combat decoherence from additional noisechannels, e.g. NMR-inspired composite pulses [28]. Forexample, concatenation of pulse segments defined byDRAG into an overall CORPSE structure, known to sup-press detuning noise (see Sec. V A), fails as the dual-axis DRAG control breaks the geometric constructionrequired to provide noise robustness.

Advancing on previous work, we present optimal androbust controls that simultaneously reduce sensitivity toboth leakage and dephasing errors by orders of magni-tude, using BOULDER OPAL optimization tools. Ourstarting point is the Hamiltonian for an anharmonicthree-level qutrit subject to dephasing noise and leakageto the lowest-lying excited state in the system (Fig. 12a):

H(t) =χ

2(a†)2(a2) + γ(t)a+ γ∗(t)a†

+∆(t)

2a†a+ βz(t)σz (69)

where a = |0〉 〈1|+√

2 |1〉 〈2|. We encode this anharmonicoscillator using a drift control with operator χ

2 (a†)2a2; amicrowave drive control with operator a and complexpulse envelope γ(t) = Ω(t)e−iφ(t); a clock shift controlwith operator a†a/2 with a real pulse envelope ∆(t); andan additive noise operator with Pauli operator σz (seeApp. B for further details on this representation).

We perform two robust-control optimizations subjectto different constraints (Table II). First, we implementa concurrent optimization allowing dual-quadrature con-trols similar in required controls to half DRAG (e.g. IQmodulation). Next we perform a fixed-waveform opti-mization that holds the amplitude of the control pulseassociated with the microwave drive fixed while allowingits phase to vary, as in phase-modulation. The resultingwaveforms are displayed in Fig. 12.

We compare performance in three distinct ways whichillustrate the simultaneous robustness to both leakageand dephasing errors in a pulse whose duration is thesame as the half-drag solution. First, we represent thedephasing-noise operator associated filter function. Wesee that the filter functions for the two optimized controlsshow a low-frequency-noise suppressing character similarto that illustrated in Fig. 6, while all other controls indi-cate broadband noise admittance. Next, we use the nu-merical simulation tools described in Sec. IV D to deter-mine control robustness to quasistatic dephasing errors.In this circumstance the two optimized solutions demon-strate a broad plateau of fixed detunings over which theinfidelity remains low, again following the experimentalresults of Fig. 10b-c. Finally, we simulate the full evolu-tion of the three states of the qutrit under application ofthe net X rotation and applied noise. Here we see thatdespite the complex dynamics at times less than the gatetime, at the conclusion of the gate the optimized solu-tions show the appropriate net state transfer. Other op-timization approaches such as RC-filtered and slew-rate-bounded controls have been used to demonstrate similarperformance. Overall these solutions represent totallynew controls that – for the first time – allow simultane-ous leakage-error and dephasing-noise suppression.

C. Robust control for parametrically-drivensuperconducting entangling gates

In this subsection we present novel parametrically-driven iSWAP gates optimized for error robustness.Parametric activation of entangling gates presents an al-ternative paradigm enabling tunable, high-fidelity two-qubit gates [63, 64]. This overcomes the scaling penaltyimposed by frequency crowding in conventionally coupled

26

FIG. 12. Comparison of standard and Q-CTRL optimized control solutions for simultaneous dephasing and leakage error sup-pression. (a) Level and control diagram for the transmon circuit under consideration. The driving field is applied concurrentlywith a time-varying noise process β(t). This time-varying field causes a shift in the qubit level spacing which results in aneffective detuning between the drive field and the qubit levels, resulting in dephasing. (b-d) Control solutions on the Blochsphere, in polar coordinates, and demonstrations of simulated probabilities of state occupancy during the gate application.An ideal gate results in perfect state transfer from |0〉 → |1〉, meaning the probabilities state populations should take valuesP0 → 0, P1 → 1, and P2 → 0 (leakage channel). In this presence of noise the Q-CTRL solutions suppress P0 and P2 at the endof the gate by orders of magnitude relative to Half DRAG. (e) Filter function representation of different controls for dephasingfield β(t) represented in the Fourier domain as Sz(ω). The Q-CTRL Filter functions, which are small at low frequencies,suppress noise and provide robustness. (f) Quasistatic error robustness of controls as a function of a fixed detuning betweenthe qubit frequency and microwave drive. Robustness is demonstrated by controls which stay near zero away from the centerof the graph, indicating improved performance of both optimized Q-CTRL solutions. Even if β(t)→ 0, the flat plateau for theQ-CTRL solutions shows superior robustness relative to Half DRAG.

transmons [65], though it suffers from decoherence chan-nels arising from control noise in the parametric drive.We perform first-principles analyses of dominant errorchannels and introduce novel gate structures incorporat-ing numeric optimization via BOULDER OPAL in order

to suppress the influence of these control errors.

Two-qubit parametrically-driven gates may be imple-mented between one fixed- and one tunable-frequencytransmon. A control flux drive Φ(t), with frequency ωpand phase offset θp, is applied to the tunable-frequency

27

control solution Coptimal Crobust Itot

Primitive 2.2× 10−1 2.6× 10−2 2.2× 10−1

Half-DRAG 9.1× 10−3 4.1× 10−2 1.1× 10−2

Q-CTRL: AM+ΦM 9.1× 10−9 1.6× 10−8 2.5× 10−7

Q-CTRL: fixed amp. 2.7× 10−9 4.9× 10−9 2.6× 10−7

TABLE III. Performance metrics for control solutions pre-sented in Fig. 12(d-g). Here Coptimal and Crobust are definedas in Table I. The total infidelity Itot is computed as thesum of two contributions: Ioptimal, and Irobust, with the in-tegral computed with respect to the dephasing PSD plottedFig. 12(b).

transmon resulting in a modulated transition frequencyof the form

ωT (t) = ωT + ωT cos (2ωpt+ 2θp) (70)

where ωT is the average shift in qubit frequency and ωT isthe amplitude of the modulation caused by the appliedflux drive. The Hamiltonian for the system under thismodulation, transforming to an interaction picture, takesthe form

Hint(t) = g(t)

∞∑

n=−∞Jn

(ωT2ωp

)e+i(2ωpt+2θp)n

×e−it∆ |10〉〈01|

+√

2e−i(∆+|ηF |)t |20〉〈11|+√

2e−i(∆−|ηT |)t |11〉〈02|+ 2e−i(∆+|ηF |−|ηT |)t |21〉〈12|+ H.C.

.

(71)

Here g(t) describes the capacitive coupling between thetransmon qubits; ηT (ηF ) are the positively-defined an-harmonicities for the tunable-frequency (fixed-frequency)transmons; ∆ = ωT −ωF is the detuning between the av-erage transition frequency of the tunable-frequency qubitand the fixed transition frequency of the fixed-frequencyqubit; and Jn(x) is the nth-order Bessel function of thefirst kind. A detailed description of the underlying phys-ical system and the derivation of the associated Hamil-tonians can be found in [63, 64, 66, 67].

Target entangling gates are activated by matching themodulation frequency ωp to the detuning between rele-vant energy levels of the capacitively-coupled transmons.For typical experimental parameters, the time-dependentphase factors on the Hamiltonian operators above leadto rapidly-oscillating terms in the system evolution thateffectively suppress the coupling rate to the associatedtransitions. Activation of a target transition is achievedby resonantly tuning the drive frequency to cancel theassociated phase factor. In particular:

iSWAP: |10〉 ↔ |01〉 2nωp = ∆ (72)

CZ20: |11〉 ↔ |20〉 2nωp = ∆ + ηF (73)

CZ02: |11〉 ↔ |02〉 2nωp = ∆− ηT (74)

The use of a parametric drive with a user-defined am-plitude, phase, and frequency introduces a new control-induced channel for errors in the gate. To account forthese control errors we assume the flux drive, and con-sequently the parametric drive in Eq. (70), experiencethree distinct error processes

modulation offset error: ωT → ωT + εT (75)

modulation amplitude error: ωT → ωT + εT (76)

modulation frequency error: ωp → ωp + εp (77)

where the ε are assumed to be small errors. These gen-erate additional Hamiltonian terms which, performing aTaylor expansion in the small offset parameters and mov-ing to the interaction picture, result in the noise Hamil-tonian Herr(t) = β(t)N , where β(t) captures the effectivenoise strength. We introduce a noise-operator of the form

N = IF ⊗ (Π1 + 2Π2) . (78)

Here Πi = |i〉〈i| defines the projection operator onto theith eigenstate of the tunable-frequency transmon, and IFis the identity on the fixed-frequency transmon.

The iSWAP interaction is activated by resonantly driv-ing the |10〉〈01| term, for example by setting the 1st-order(n = 1) resonance condition ωp = ∆/2. Assuming thisconfiguration we may therefore restrict attention to therelevant (4× 4) iSWAP subspace, spanned by the eigen-states

|00〉 , |10〉 , |01〉 , |11〉 . (79)

In this case the remaining rapidly-oscillating terms maybe ignored, and Eq. (71) reduces to

HiSWAP(t) ≈ 1

2Λ(t)e+iξ(t) |10〉〈01|+ H.C. (80)

where the parametric coupling rate Λ = 2g(t)J1

(ωT2ωp

)

and the parametric drive phase ξ = 2θp.On the iSWAP subspace the noise operator defined

in Eq. (78) reduces to

N =1

2IF ⊗ σz =

− 12 0 0 0

0 12 0 0

0 0 − 12 0

0 0 0 12

(81)

resembling dephasing on the subspace of the tunable-frequency qubit. This new analysis thus shows that thethree different channels for the introduction of noise viathe parametric drive are manifested at the Hamiltonianlevel as an effective dephasing process.

Our objective is now to craft control solutions whichare able to suppress this effective dephasing channel us-ing the control available to us. Incorporating the fixed-frequency transmon term

Hqubit-F(t) =

(1

2Ω(t)e+iφ(t) |0〉〈1|+ H.C.

)⊗ IT (82)

28

f

PrimitiveaΛ

/2π

Ω/2π

Λ/2

π

Analytic Designb

Q-CTRL: Fixed amp.c

π

π

π

π

φΩ

/2π

ξΛ

/2π

Sz Fzω ω

Primitive Analytic Fixed amp. Bandlimited

Sz(f)

e

Q-CTRL: Bandlimitedd

QP/2

πQ

T/2

πI T

/2π

I P/2

πFIG. 13. Optimized error-robust control solutions for two-qubit parametrically driven entangling gates in superconductingcircuits. (a) Primitive coupling gate (b) analytically designed Walsh-modulated gate, (c) Numerically optimized interleavedgate construction with fixed-amplitude constraints (phase modulation only) (d) Numerically optimized concurrent gate withband-limited 5 MHz RC filter integrated into optimization. (e) Filter functions for control solutions overlaid with an examplenoise power spectrum for ambient dephasing, showing noise-suppressing capabilities of Q-CTRL solutions. (f) quasi-static errorsusceptibility as a function of relative detuning error (measured relative to Λ.

the full control Hamiltonian is then written

Hctrl(t) = HiSWAP(t) +Hqubit-F(t). (83)

This Hamiltonian is then parameterized according to theprescription in App. B, Hctrl(t) = γ(t)C+ H.C. in termsof the drive pulses and operators

γ(t) =[Λ(t)e+iξ(t), Ω(t)e+iφ(t)

],

C =

[12 |10〉〈01|

12 |0〉〈1| ⊗ I

].

We introduce three novel solutions providing robust-ness against control errors in the parametrically activatedgate. All combine an iSWAP coupling drive Λ(t) withsingle-qubit rotations which yields full control over therelevant subspace. This may be examined by observingthat sandwiching an iSWAP operation between a pair ofsingle-qubit X gates permits the realization of the (Her-mitian) operator

Aeff ∝

0 0 0 10 0 0 00 0 0 01 0 0 0

, (84)

similar in structure to a general NOT gate. Solutionsneed not employ this particular gate, but leverage thefull controllability afforded by the combination of single-qubit and iSWAP control modulation.

The first solution we present is fully defined analyt-ically (Fig. 13b), recognizing that dephasing noise canbe mitigated by the action of the spin echo and Walsh-modulation on a driven operation. We combine single-qubit X operations with an iSWAP control envelope de-fined using a superposition of Walsh functions. Thisapproach has previously been used to craft dephasing-robust single-qubit driven operations [17, 68] and pro-vide a simple means to realize error robustness. In thiscase the gate construction strongly suppresses both qua-sistatic and time-varying control errors as expressed us-

29

ing our multi-dimensional filter functions and numericsimulation (Fig. 13e-f) .

In Fig. 13c-d we also present two representative numer-ically optimized solutions subject to constraints outlinedin Table II. First, we produce a fixed-amplitude solu-tion which combines a phase-modulated coupling drivewith an interleaved single-qubit control, compatible withsituations in which both controls cannot be applied si-multaneously. The single-qubit operations are enactedwith fixed amplitude, but variable durations (hence vari-able rotation angles) and variable phase. Similarly wepresent a band-limited, concurrent solution incorporat-ing these two drives. Here we have enforced, as an ex-ample, an RC-filter on the controls to match potentialband limits as may be experienced in a system with finitetransmission-line bandwidth. Again in these cases we ob-serve enhanced robustness to quasistatic errors, validatedby time-domain simulation, as well as time-varying noise,as captured through filter functions (Fig. 13e-f).

Overall these examples demonstrate how high-performance solutions may be achieved under a widerange of constraints for entangling gates in superconduct-ing circuits. We have achieved similar results for the CZgate and non-parametric cross-resonance gates.

control solution Coptimal Crobust Itot

Primitive 0.0 1.73 2.6× 10−4

Analytic 0.0 0.0 1.5× 10−6

Q-CTRL: fixed amp. 1.6× 10−8 2.0× 10−12 2.0× 10−6

Q-CTRL: bandlimited 2.0× 10−9 7.3× 10−9 4.9× 10−6

TABLE IV. Performance metrics for control solutions pre-sented in Fig. 13(i-l). Here Coptimal and Crobust are definedas in Table I. The total infidelity Itot is computed as the sumof two contributions: Ioptimal, and Irobust, with the integralcomputed with respect to the noise PSD plotted Fig. 13(g).

D. Experimental noise characterization ofmultiqubit circuits on an IBM cloud QC

Microscopic noise characterization is widely employedin the development of optimized control solutions for arange of devices including superconducting qubits [69].However, existing protocols have focused on the charac-terization of global fields measured at the single-qubitlevel [70–72]. Q-CTRL’s multi-dimensional filter func-tions (Sec. IV B) and flexible noise reconstruction algo-rithms (Sec. IV E) permit new insights to be gleaned fromreal experimental quantum computing hardware.

We have focused on the characterization of previ-ously unidentified microscopic noise sources present inentangling gates executed on cloud-based superconduct-ing quantum computers [55]. At present access to suchsystems is highly restricted, making the arbitrary ap-plication of complex modulated controls on subspaces

within the machines impossible. Accordingly, we havedeveloped and deployed a simplified probing sequenceconsisting of sequences of entangling gates for two-qubitnoise characterization. This probing sequence is readilyimplementable on the current IBM quantum computingplatform.

Our probing sequence is designed to characterize two-qubit dephasing noise defined by the noise operator N =12 (Za − Zb), where Za and Zb are the Pauli Z operatoron qubits a and b, respectively. This probing sequenceconsists of a fixed number M of single-qubit and two-qubit quantum gates, in which each quantum gate hasa fixed gate duration of Tg. Fixing M - and in partic-ular the number of two-qubit gates used - ensures thatsignatures arising from imperfect execution of the entan-gling gates do not vary between sequences and swampthe noise signals to be measured. In general, M can bechosen suitably based on the hardware specification andTg; here, we choose M = 66 to ensure the total dura-tion of the experiment is within the coherence time ofthe IBM NISQ computers and Tg = 110ns.

We first prepare the Bell state (|01〉+ |10〉)/√

2 by ap-plying a sequence of 3 quantum gates defined by

UE = HaCNOTabXb, (85)

where CNOTab is the controlled-not gate on control qubita and target qubit b, Ha is the Hadamard operator onqubit a and Xb is the Pauli X operator on qubit b. Theni identity gates are applied to both qubits followed by aswap operation defined as

SW =CNOTabHabCNOTabHab

CNOTabHabCNOTab, (86)

where Hab = Ha⊗Hb. A second SW operation is appliedafter an additional j identity gates. Then M−16−i−j =50−i−j identity gates are employed before reverting the

quantum state via the U†E operation. The schematic ofthe probing sequence is shown in Fig. 14b. This probingsequence effectively implements a series of π pulses forthe two-qubit system, mimicking the behaviour of CPMGsequences executed on a single-qubit system.

A set of generalized filter functions and the compos-ite spectral sensitivity function for the set of control se-quences can be constructed for each sequence as i, j vary(Fig. 14d). Subsequently, when the two-qubit system issubject to noise, a corresponding set of infidelity mea-surements can be obtained and used to infer the noisepower spectral density via the SVD noise reconstructionmethod described in Sec. IV E. We note that the fact thatthe sequences are highly constrained in structure (basedon the limits of the IBM hardware) makes standard ma-trix inversion approaches to spectrum reconstruction im-possible.

We implement the new probing sequence on the IBM Q5 Tenerife (ibmqx4 ) quantum computer [73]. Fig. 14adepicts the device chip layout and qubit couplings; here,we choose the qubit pair Q0 and Q2 for the experimental

30

Q0

Q2

Time

66 gate spaces

5 5 33

5

38

1318232833384348

9 13 17 21 25 29 33 37 41 45 49 53

0.3

0.4

0.5

Infid

elity

, position of the 2nd SW gate

, pos

ition

of t

he 1

st S

W g

ate

00.0

0.5

1.0

1.5

2.01e–11

50 100 150 200 2500

1

2

3

4

5

Frequency (kHz)

Filte

r Fun

ctio

ns (H

z–1)

Sens

itivi

ty (H

z–1)

1e–10

00

150

300

450

600

50 100 150 200 250

Frequency (kHz)Po

wer

Den

sity

(kW

•Hz–1

)

a

de

b c

FIG. 14. Noise characterization in the IBM Q 5 Tenerife (ibmqqx4) device. (a) Chip layout is shown identifying the twoqubits for which data are presented. (b) A Schematic of the probing sequence structure with time-varying dephasing noise;the convolution of these two processes in the time domain gives a product in the Fourier domain as captured via the multi-dimensional filter function. (c) Measured infidelity from all probing sequences specified by the index pair i, j denoting thelocation of the two swap gates employed in the sequence. (d) multi-dimensional filter functions for the target noise channel areshown for all sequences tested, as well as the net sensitivity function. (e) Reconstructed dephasing noise power spectral usingthe SVD reconstruction method with a 10−13 threshold for singular value truncation.

results displayed, but all pairs have been characterizedin detail yielding qualitatively similar results. For thisdevice, CNOT20 can be natively implemented betweenthese two qubits.

The values of i and j for the probing sequences arevaried to construct a set of filter functions with high sen-sitivity for a broad range of frequencies of interest, up to250kHz. For each chosen values of i, j, the correspondingquantum circuit was executed over 8192 shots, yieldingthe average infidelity depicted in Fig. 14c as a colorscaleheatmap. For a fixed i, as j increases until the secondSW operation is placed roughly half-way between the

first SW and the U†E operations, the average infidelitybecomes small, resembling an echo-effect similar to thatof CPMG sequences. However there is additional struc-ture present which breaks the symmetry of these graphs,indicating other noise contributions away from DC.

Utilizing the SVD method (Sec. IV E), a reconstructednoise power spectral density based on these results isshown in Fig. 14e. Our result shows that with highconfidence, the two-qubit dephasing noise exhibits alow-frequency noise component and a repeatable higher-frequency noise contribution in the range 150 − 200kHz. The breadth of spectral features observed is due

to the Fourier limits of the individual measurements em-ployed in the reconstruction routine, as confirmed by nu-merical simulations. We have observed similar perfor-mance across multiple qubit pairs, with variations in thestrength of the quasi-dc component. Numerical simula-tions have been used to demonstrate that similar resultsto those shown in Fig. 14e arise for a simple spectrumcomposed of a quasi-dc noise component and a singlefixed-frequency spur at higher frequencies. Similarly wehave confirmed by engineering numerically synthesizeddata used in the reconstruction that the presence of thefeature in the range 150 − 200 kHz does not appear tobe an artefact of either the measurement routine or thereconstruction method.

Our experimental results represent an early demon-stration of microscopic noise characterization within two-qubit gates. The identification of a high-frequency noisecomponent in a range commonly associated with elec-tronic noise provides guidance on system improvementand noise suppression strategies for these machines.

31

E. Crosstalk-resistant circuit compilation

In this subsection we demonstrate the use of optimalcontrol for algorithmic design and compilation. We con-sider a complex circuit composed of multiple interactingtransmons and subject to unwanted cross-coupling. Weuse Q-CTRL optimization tools in order to implement atarget circuit, subject to constraints on control and cir-cuit time, and optimized to combat always-on cross-talkerrors through the structure of the circuit itself (no gate-level optimization).

Due to the low anharmonicity of transmons, quantumcomputations can be designed to exploit the three lowest-energy levels. The relative detunings between the variousenergy levels in an ensemble of transmons gives rise to analways-on effective ZZ-type coupling Hamiltonian thatcan be exploited for generating entangling operations.However, this coupling also leads to residual cross-talkerrors that degrade algorithmic performance. Below wedescribe an example physical system and create an opti-mized circuit compilation which suppresses these residualcouplings.

We consider a linear arrangement of 5 qutrits, labelledq ∈ 1, ...5. For a given qutrit pair p = (q, q + 1), char-acteristic detunings between respective energy levels gen-erate relative phases on states |11〉, |12〉, |21〉, and |22〉.In this case the total coupling Hamiltonian is written

Hzz = H(1,2)zz ⊗ I⊗ I⊗ I

+ I⊗H(2,3)zz ⊗ I⊗ I

+ I⊗ I⊗H(3,4)zz ⊗ I

+ I⊗ I⊗ I⊗H(4,5)zz

(87)

where nearest-neighbour interactions between pair p aredescribed by

Hpzz = αp11|11〉〈11|+ αp12|12〉〈12|

+ αp21|21〉〈21|+ αp22|22〉〈22| (88)

and the αpij are effective coupling strengths, tabulatedin Table V for all pairs. Here I is the identity on a3-dimensional single-qutrit Hilbert space, Hp

zz operateson a 32-dimensional Hilbert space associated with thepth qutrit pair, and Hzz operates on the 35-dimensionalHilbert space associated total 5-qutrit system.

qutrit pairsp = (q, q + 1)

αp11(2π ·MHz)

αp12(2π ·MHz)

αp21(2π ·MHz)

αp22(2π ·MHz)

(1, 2) −0.27935 0.1599 −0.52793 −0.74297

(2, 3) −0.1382 0.15827 −0.33507 −0.3418

(3, 4) −0.276 −0.6313 0.24327 −0.74777

(4, 5) −0.26175 −0.49503 0.14497 −0.70843

TABLE V. Example ZZ-type coupling strengths betweennearest-neighbor qutrit pairs within a circuit.

As an example algorithm, we consider a circuit on this5-qutrit system which seeks to simultaneously executecontrolled-sum (CSUM) gates on qutrit pairs (1, 2) and(3, 4), while leaving the 5th qutrit unaffected. This maybe expressed formally as

Utarget = UCφ ⊗ UCφ ⊗ I (89)

where UCφ is a 2-qutrit phase gate, locally equivalent toa CSUM, defined as

UCφ =

11

11ω∗

ω1ωω∗

(90)

and ω ≡ e2πi/3.We consider a physically motivated restricted control

basis spanning only single-qutrit operations coupling thelevels |0〉 ↔ |1〉 and |1〉 ↔ |2〉. We assume that these maybe implemented instantaneously and in parallel across allqutrits within the circuit.

The total control Hamiltonian may be written

Hctrl(Ω,φ) =

5∑

q=1

∑

ν∈01,12Hqν (Ωqν , φ

qν) (91)

where

Ω =(Ω1

01, . . . Ω501,Ω

112, . . . Ω5

12

), (92)

φ =(φ1

01, . . . φ501, φ

112, . . . φ

512

). (93)

The assumption of instantaneous single-qutrit operationsallows us to absorb the duration ∆t over which the cor-responding unitary is implemented, yielding a total evo-lution operator

Uctrl(θ,φ) = exp [−iL(θ,φ)] , θ = ∆tΩ (94)

where

L(θ,φ) =

5∑

q=1

∑

ν∈01,12θqν exp [+iφqν ]Cqν + H.C.. (95)

Here we refer to driven operations on the qth qutrit andνth transition, with Rabi rate Ωqν and phase φqν .

Within this formulation we have defined single-qutritdrive operators

C10 =1

2|1〉〈0| = 1

2

0 0 01 0 00 0 0

(96)

C21 =1

2|2〉〈1| = 1

2

0 0 00 0 00 1 0

(97)

32

Q1

1.51-0.68

2.271.5

2.65-0.8

2.631.82

4.41-0.09

3.581.14

2.41-0.57

1.172.05

0.222.96

0.462.33

1.090.76

1.11-1.01

3.5-2.4

5.60.69

0.871.27

1.31-1.51

5.59-3.03

2.84-2.15

0.92-0.7

0.850.92

5.34-0.59

0.00.0

1.091.29

2.611.74

0.7-1.32

1.36-2.33

2.850.41

1.07-1.32

5.631.17

4.221.71

5.620.45

2.132.71

1.74-1.29

5.981.98

2.09-1.72

5.83-1.98

5.361.97

3.171.42

1.72-1.03

6.083.0

3.37-2.18

3.741.7

3.22-2.54

1.38-1.65

6.22-0.29

4.26-2.68

4.13-1.68

1.241.0

1.030.1

0.670.33

Q2

3.132.58

0.00.0

4.442.59

4.461.77

5.21.78

2.83-3.06

1.63-0.92

1.99-1.28

4.21.77

2.310.91

1.411.37

1.96-2.03

0.00.0

1.24-0.58

5.062.45

1.02-2.1

6.280.44

0.00.0

4.482.34

1.862.54

2.81-0.02

5.911.85

5.84-2.23

4.16-0.77

0.91-2.78

2.43-1.27

1.09-3.13

2.310.81

6.232.67

4.45-0.75

3.171.22

3.32.17

3.831.74

1.3-2.35

1.631.01

2.81-1.35

4.332.39

0.00.0

5.991.28

6.250.01

2.751.75

0.24-0.62

3.791.52

0.31-0.56

0.00.0

4.81-0.0

0.562.97

4.52-1.64

0.00.0

1.642.97

Q3

0.00.0

0.00.0

3.723.11

5.090.33

2.35-1.78

6.282.45

3.872.26

3.682.06

4.55-0.55

0.433.02

3.21-2.39

0.47-0.71

4.41.9

1.28-1.6

6.12-0.07

2.29-3.11

4.19-2.8

4.682.45

1.832.6

0.17-1.27

1.59-0.86

5.86-1.12

4.951.03

4.32-2.81

5.66-0.18

0.00.0

4.19-0.89

4.12.08

4.081.87

0.00.0

5.14-1.91

1.592.58

2.583.01

2.31.59

1.66-0.33

0.00.0

0.570.29

4.11-1.18

3.89-0.93

6.05-0.81

2.54-0.26

4.72.48

5.52.08

3.680.43

0.240.39

4.191.62

2.251.5

2.06-2.16

0.53.07

4.512.91

Q4

0.00.0

0.00.0

0.00.0

3.142.14

0.75-0.32

5.79-1.36

0.561.47

5.41-2.1

3.69-1.19

1.38-2.24

3.940.55

3.86-2.84

2.06-0.39

3.48-3.02

4.530.35

2.03-0.29

0.00.0

3.14-2.3

1.36-1.34

6.230.14

2.130.79

4.81-0.71

3.531.79

5.61.61

0.00.0

5.082.75

3.52-0.43

4.812.71

5.51-1.01

5.512.71

0.880.73

0.26-2.64

6.253.11

1.61-0.34

5.690.49

2.052.3

0.00.0

0.00.0

0.590.26

3.01-0.58

4.692.67

2.38-0.18

1.5-1.33

2.22-2.19

1.392.77

3.420.68

2.94-1.34

5.491.63

4.571.83

6.03-1.82

Q5

5.31-0.49

2.11-1.32

2.88-1.89

2.651.04

2.24-1.49

0.00.0

0.51-0.46

5.751.12

4.52-1.73

3.821.34

0.79-0.17

1.45-2.88

2.371.16

0.710.98

6.26-1.85

2.080.35

3.63-1.78

1.260.78

5.26-1.05

2.24-1.74

3.59-1.05

0.541.24

5.61-2.25

3.441.61

4.36-2.3

1.070.01

6.11-1.23

1.370.18

5.062.89

4.63.11

1.251.29

2.532.0

0.54-1.9

3.862.64

5.69-2.69

3.47-1.7

3.92-0.08

5.482.62

5.360.56

5.43-0.81

5.511.11

3.31-2.37

2.011.64

4.72.3

0.00.0

1.77-3.05

6.22.66

0.00.0

1.03-1.96

2.54-0.61

1 2 3 4 5 6

0.07 µs 0.12 µs 0.20 µs 0.33 µs 0.37 µs 0.18 µs

inst. inst. inst. inst.

Q5

Qk

0101

1212

= exp

"i

Pk=01,12

keik

Ck + H.C.

#

18

FIG. 15. Optimized 5-qutrit circuit. Each qutrit is controlled via |0〉 ↔ |1〉 and |1〉 ↔ |2〉 transitions. Control on each transition iscaptured via the phasor γν = θνeiφν , comprising rotation, θν , and phase angles, φν , where ν ∈ 01, 12. As shown in the inset, eachsingle-qutrit operation is indicated via the block of angles [θ01, φ01|θ12, φ12], with corresponding colours. Consecutive sequences of blockarrays indicate product operations of the form Pj defined in Eq. (99). Blocks of unitary operations are separated by durations τj . Allselected phasor components on the control operations and values of τj are returned via the circuit optimization procedure. The optimalcost for this circuit compilation is Ioptimal = 5.678e−3.

which are generalized for the qth qutrit within the multi-qutrit system

Cqν = I⊗(q−1) ⊗ Cν ⊗ I⊗(n−q−1), ν ∈ 01, 12. (98)

Our approach to circuit-level optimization for residual-cross-talk suppression is through the integration of de-terministic dynamic decoupling. We partition the circuitimplementing Eq. (89) into m periods of free evolutionunder the always-on coupling Hamiltonian in Eq. (87),segregated by products of k distinct control unitaries ofthe form defined by Eq. (94)

Pj =

k∏

`=1

Uctrl(θj,`,φj,`). (99)

starts and Here, the jth period, of duration τj , startsand ends with instantaneous unitaries Pj−1 and Pj re-spectively. This structure corresponds to a generalizeddynamic decoupling sequence and augments the control-lability of the 5-qutrit system due to the non-commutingterms in the operator products.

This generalized dynamic decoupling sequence struc-ture must now be optimized in order to return the targetcircuit functionality expressed in Eq. (89), which nec-essarily entails cancellation of the unwanted ZZ cross-coupling terms. Optimization of the control structure

is performed on the search space spanned by the tim-ing variables τ = (τ1, ..., τm), rotation variables θ`,j ,and phase variables φ`,j for ` ∈ 1, ..., k and j ∈1, ...,m+ 1.

Following the procedure described in Sec. IV C 1, thesethen form the basis for defining the array of generalizedcontrol variables v, appropriately normalized for efficientTensorFlow optimization. We also introduce an experi-mentally motivated constraint in that the circuit mustnot exceed the qutrit coherence time, set by T1. Wetherefore compose the cost function in Eq. (45) as

C(v) = Ioptimal(v) + Cduration(τ ) (100)

where Ioptimal(v) is defined in Sec. IV A and Cduration(τ )imposes a penalty for exceeding the upper limit chosenfor the circuit duration. We set a threshold of ∼ 1.5 µs,chosen assuming a qutrit coherence time of ∼ 20 µs.

The optimizer returns variations on the circuit struc-ture composed of compound rotations on different qutritlevels with variable timing between these operations. Thecombination of driven rotations and their timing in thesequence is essential in performing the target net unitarywith low infidelity; compactifying the circuit structurein order to reduce nominal dead time changes the cross-talk-suppressing nature of the circuit. In the exampleoptimization realized in Fig. 15 we are able to improve

33

the cross-talk limited fidelity (1− Ioptimal) in the targetunitary from ∼ 2.2% (under the simplest compilation,without any form of dynamic cross-talk suppression) to99.4%.

This demonstration validates the underlying premise ofcircuit-level optimization in order to realize deterministicerror robustness. Treating circuit compilation as a chal-lenge in optimal control - for either deterministic errorsuppression or decomposition of a complex circuit into aconstrained set of physical-layer controls - is a feature setincorporated in the Q-CTRL Devkit.

VI. CONCLUSION AND OUTLOOK

In this manuscript we have provided an overview of therole of quantum control in advancing the field of quantumcomputing and identified quantum firmware, integratingconcepts from this field, as a natural part of the quan-tum computing stack. We have provided an overviewof various theoretical underpinnings of quantum control,with a focus on open-loop error suppression, and haveoutlined key tasks in quantum control which bring per-formance benefits to the functioning of quantum com-puters. This has included experimental evidence derivedfrom real quantum computing hardware demonstratingthe key benefits of hardware noise and error suppression,error homogenization in space, hardware stabilization intime, and modification of error correlations for integra-tion with higher-levels of abstraction such as QEC.

In addition to this technical background we introducedQ-CTRL’s suite of classical software designed for the de-velopment and deployment of quantum control in hard-ware R&D and quantum algorithm development. Q-CTRL tools span users of various levels of expertise andrange from intuitive web interfaces with interactive vi-sualizations through to advanced python toolkits for in-tegration into professional programming tasks. We con-cluded our presentation with a set of case studies demon-strating the utility of these software capabilities in solv-ing challenging problems in quantum control, and reveal-ing new information about the performance of real quan-tum computing hardware.

By combining technical rigor with professional softwareengineering and a focus on intuitive product interfaces,we believe this toolkit will prove a valuable resource fora wide range of users, including

• Hardware research and development teams

• Quantum algorithm developers & software engi-neers

• Students of quantum computing

• Conventional developers entering QC

• Quantum consultancies & businesses seeking to bequantum ready

Moreover, our approach to producing a specialized,highly maintainable, professionally engineered solutionfor quantum control will provide major benefits to theresearch and business communities, much like the intro-duction of specialist cloud security software has acceler-ated many aspects of cloud-service businesses.

Q-CTRL is continually rolling out new features in ourproduct suite, responding to customer and partner re-quests for new capabilities tailored to their needs. Amongforthcoming features will be a new framework for opti-mizer functionality in which a user assembles a set ofpredefined “building blocks” to set up their optimizationproblem (in code or using a visual interface); for instancethis may include the incorporation of band-limits, specificconstraints on control solutions, or nonlinearities.

Similarly, we are soon releasing a number of time-seriesanalysis packages which enable the identification and ex-traction of system dynamics from discretized measure-ment records. It’s common practice in quantum comput-ing experiments to simply average together large datasets in order to obtain probabilistic information aboute.g. quantum-state populations in algorithms. This pro-cedure, while common, is confounded by the presenceof large-scale temporal drifts in hardware that can intro-duce dynamics in measurements that are not captured bysimple averaging procedures. The tools we are buildinginclude features for Gaussian Process Regression and Au-toregressive Kalman Filtering [18], targeting both datafitting for the removal of background dynamics and alsopredictive estimation for feedforward control stabiliza-tion of qubits and clocks [31]. These time-domain an-alytic frameworks are also useful for data fusion incor-porating multiple measurement streams from sensors ormeasured qubits.

The most substantial development activities over thecoming year will leverage the power of control engineer-ing to produce automated and adaptive strategies for thetuneup, calibration, and optimization of mesoscale sys-tems, moving beyond the brute-force strategy of indepen-dent calibration of all devices. In this space, advancedmachine learning and robotic control concepts providenew opportunities to facilitate rapid, autonomous bringup of devices in a way that will grow in importance as sys-tem sizes increase. We already have considerable effortin this area, taking inspiration from autonomous roboticcontrol to facilitate adaptive measurement and data in-ference on large qubit arrays [19, 33], and will be invest-ing heavily in this area in the future.

ACKNOWLEDGEMENTS

Q-CTRL efforts supported by Data Collective, Hori-zons Ventures, Main Sequence Ventures, Sequoia Cap-ital (China), Sierra Ventures, and SquarePeg Capital.Development of multi-dimensional filter functions andSVD spectrum inversion technique by Q-CTRL sup-ported by the US Army Research Office under Contract

34

W911NF-12-R-0012. Q-CTRL is grateful to I. Siddiqiand D. Santiago for provision of device data which in-spired circuit optimization results. Experimental workusing trapped ions at USYD partially supported by theARC Centre of Excellence for Engineered Quantum Sys-tems CE170100009,the Intelligence Advanced ResearchProjects Activity (IARPA) through the US Army Re-search Office Grant No. W911NF-16-1-0070, and a pri-vate grant from H. & A. Harley.

Appendix A: Appendix: Technical Definitions

1. Fourier transform

In this paper we exclusively use the non-unitary angular-frequency convention for Fourier transform pairs, defin-ing

Q(ω) ≡∫ ∞

−∞dte−iωtQ(t) (A1)

Q(t) ≡ 1

2π

∫ ∞

−∞dωeiωtQ(ω) (A2)

where Q(t) denotes any scalar-, matrix- or operator-valued function of time, and Q(ω) is its Fourier trans-form, implemented element-wise for matrices. For ease ofnotation we reuse the same symbol and simply change theargument to distinguish time- or frequency-domain trans-forms. To avoid confusion we also write F Q(t) (ω) ≡Q(ω) and F−1 Q(ω) (t) ≡ Q(t) .

2. Frobenius inner product

For matrices A,B ∈ Cm×n, the Frobenius inner productis defined as

〈A,B〉F =∑

i,j

A∗ijBij = Tr(A†B

)(A3)

3. Frobenius norm

The inner product in Eq. (A3) induces a matrix norm.For a matrix A ∈ Cm×n, the Frobenius norm is definedby

‖A‖F =√〈A,A〉F =

√∑

i,j

|Aij |2 =√

Tr (A†A) (A4)

4. Approximations

The measure for robustness may be approximated us-ing filter function framework. Specifically

Irobust ≈p∑

k=1

Irobust,k (A5)

where

Irobust,k ≡1

2π

∫ ∞

−∞Sk(ω)Fk(ω)dω (A6)

is the contribution from the kth noise channel, expressedas an overlap integral between the noise power spectraldensity, Sk(ω), and the associated filter function Fk(ω).

35

5. Power spectral density

Here we develop the relationship between noise pro-cesses in the time-domain and their frequency-domainrepresentations. Let βk(t) for k ∈ 1, ..., n denote setof scalar-valued noise fields. Using the definition forthe Fourier transform set out in App. A 1, we establishthe following relationships between time- and frequency-domain variables

βk(t) =1

2π

∫ ∞

−∞dωeiωtβk(ω), (A7)

βk(ω) =

∫ ∞

−∞dte−iωtβk(t). (A8)

We assume the noise fields are independent[74], zero-mean random variables. The cross-correlation functionsconsequently vanish, namely

〈βj(t1)β∗k(t2)〉 = 0, j 6= k ∈ 1, .., n (A9)

where the angle brackets denote an ensemble average overthe stochastic variables. The frequency-domain variables

inherit the equivalent property, namely

〈βj(ω1)β∗k(ω2)〉 = 0, j 6= k ∈ 1, .., n, (A10)

which may be shown by substituting in Eq. (A8), andinvoking Eq. (A9). We further assume the noise processesare wide sense stationary, implying the autocorrelationfunctions, defined as

Ck(t2 − t1) ≡ 〈βk(t1)β∗k(t2)〉, i ∈ 1, .., n, (A11)

depend only on the time difference τ = t2 − t1. Un-der these conditions the autocorrelation function for eachnoise field may be related to its power spectral densitySi(ω) using the Wiener-Khinchin Theorem [75]. Specifi-cally,

Ck(t2 − t1) =1

2π

∫ ∞

−∞Sk(ω)eiω(t2−t1)dω, (A12)

which is consistent with defining the power spectral den-sity as

Sk(ω) ≡ 1

2π

⟨|βk(ω)|2

⟩. (A13)

To show this observe

〈βk(ω1)β∗k(ω2)〉 =

⟨(∫ ∞

−∞dt1e

−iω1t1βk(t1)

)(∫ ∞

−∞dt2e

−iω2t2βk(t2)

)∗⟩(A14)

=

∫ ∞

−∞dt1

∫ ∞

−∞dt2 〈βk(t1)β∗k(t2)〉 eiω2t2e−iω1t1 (A15)

=

∫ ∞

−∞dt1

∫ ∞

−∞dt2

(1

2π

∫ ∞

−∞Sk(ω)eiω(t2−t1)dω

)eiω2t2e−iω1t1 (A16)

=1

2π

∫ ∞

−∞dωSk(ω)

∫ ∞

−∞dt1e

−i(ω+ω1)t1

∫ ∞

−∞dt2e

i(ω+ω2)t2 (A17)

=1

2π

∫ ∞

−∞dωSk(ω)

(2π · δ(ω + ω1)

)(2π · δ(−ω + ω2)

)(A18)

=

0 ω1 6= ω2

2πSk(ω1) ω1 = ω2(A19)

Appendix B: Control Hamiltonian

As outlined in the main text, the central objectives of quantum control is to enhance the performance of a quantumsystem by leveraging the available controls against the influence of relevant noise sources. Delivering this for arbitraryquantum systems (qubits, qutrits, multi-qubit ensembles, etc.) requires a generalized formalism for describing thecontrol Hamiltonian. In this appendix we introduce this formalism.

36

1. Generalized formalism

Let H be a d-dimensional Hilbert space for the controlled quantum system. The control Hamiltonian is written

Hc(t) =

d∑

j=1

γj(t)Cj + H.C.

+

s∑

l=1

αl(t)Al + D (B1)

in terms of the control operators Al, Cj , D ∈ H and control pulses (waveforms) αl(t) ∈ R and γj(t) ∈ C. To unpackthis notation we introduce the nomenclature of drives, shifts and drifts, useful for mapping generalized quantumcontrol concepts to common physical control variables. These are detailed in Table VI. In this framework, systemevolution under Hc may be viewed as a combination of generalized rotations, driven by control pulses (real or complex)about effective control axes, defined by the associated operators.

control term operator pulse

drive Cj non-Hermitian γj(t) C: complex

shift Al Hermitian αl(t) R: real

drift D Hermitian - -

symbol type symbol type

TABLE VI. Decomposition of control Hamiltonian into generalized drive, shift and drift terms. Drive terms are defined bynon-Hermitian operators, Cj 6= C†j , and complex-valued control pulses γj(t). Shift terms are defined by Hermitian operators,

Al = A†l , and real-valued control pulses αl(t). The operator D is a time-independent Hermitian operator, which we refer to asthe drift Hamiltonian.

2. Control solutions

Assuming the operator basis defined above, the control Hamiltonian may be expressed more compactly as

Hc(t) =(γ(t)C + H.C.

)+α(t)A+D (B2)

in terms of the vectorized objects defined by

drive terms: γ(t) =[γ1(t), γ2(t), . . . γd(t)

], C =

C1

C2

...Cd

, t ∈ [0, τ ] (B3)

shift terms: α(t) =[α1(t), α2(t), . . . αs(t)

], A =

A1

A2

...As

, t ∈ [0, τ ] (B4)

with drive and shift pulses listed as complex and real row vectors respectively, and corresponding drive and shiftoperators listed as column vectors. Given the operator-basis defined by A and C, the most general description ofcontrol is therefore specified by the set of functions α(t) and γ(t), defined on the time interval t ∈ [0, τ ], defining theduration over which the control is applied. We refer to this structure as a control solution.

3. Control segments

It is often more useful to specify the form of the control Hamiltonian, or functional form of the control pulses γ(t)and α(t), locally in time. In this case the time-domain t ∈ [0, τ ] is partitioned into a series of intervals, or segments.The functional form of the control pulses are then defined on each segment. This is illustrated below for a shift pulseαj(t). The time domain t ∈ [0, τ ] has been formally partitioned into m subintervals

[ti−1, ti], i ∈ 1, ...,m, t0 ≡ 0, tm ≡ τ (B5)

37

where ti−1 and ti are respectively the start and end times of the ith segment, and

τi = ti − ti−1 (B6)

is its duration. The shift pulse αj(t) is piecewise-constant, defined to take the constant value αi,j on the ith segment,t ∈ [ti−1, ti].

// t

t0 tm

θ1,j

α1,j

θ2,j

α2,jθ3,j

α3,j

θm−1,j

αm−1,j θm,j

αm,j

0 t1 t2 t3 tm−2 tm−1 τ

τ1 τ2 τ3 τm−1 τm

Aj :

FIG. 16. Segmentation of control amplitude αj(t) for control axis Aj into m segments. The area under the ith segment is givenby the variable θi,j ≡ αi,jτj .

4. Control coordinates

Here we introduce notation conventions followed by Q-CTRL to define drive pulses, their decomposition, and theirrelationship to Hermitian and non-Hermitian control operators. Since the following structure applies to every pulse-operator pair (γj(t), Cj), we drop the subscript j for simplicity. The complex-valued pulse γ(t) ∈ C may be writtenin polar or Cartesian form. Namely,

Polar form: γ(t) = Ω(t)e+iφ(t) (B7)

Cartesian form: γ(t) = I(t) + iQ(t) (B8)

allowing us establish the following control forms

modulus: Ω(t) = |γ(t)| (B9)

phase: φ(t) = Arg (γ(t)) (B10)

in-phase: I(t) = Re (γ(t)) = Ω(t) cos(φ(t)) (B11)

in-quadrature: Q(t) = Im (γ(t)) = Ω(t) sin(φ(t)) (B12)

where the drive phase φ(t) = +Arg(γ(t)) is defined as the positive argument. The drive term in the control Hamiltonianis therefore expressed

γ(t)C + H.C. = γ(t)C + γ∗(t)C† (B13)

=(I(t) + iQ(t)

)C +

(I(t)− iQ(t)

)C† (B14)

= I(t)(C + C†

)+Q(t)

(iC − iC†

)(B15)

= I(t)AI +Q(t)AQ (B16)

where we have defined the Hermitian operators

AI = C + C†, AQ = i(C − C†). (B17)

Each drive term therefore decomposes into a pair of shift-terms (I(t), AI) and (Q(t), AQ) in the familiar form ofquadrature controls. These are related to the non-Hermitian operator as

C =1

2(AI − iAQ) (B18)

38

The modulus Ω(t) sets the rotation rate, while the phase φ(t) sets the direction of rotation, oriented between thecontrol axes defined by (AI , AQ).

Appendix C: Multidimenional filter function derivations

1. Magnus expansion

The first few Magnus terms are computed as [50, 51]

Φ1(τ) =

∫ τ

0

dtHerr(t) (C1)

Φ2(τ) = − i2

∫ τ

0

dt1

∫ t1

0

dt2[Herr(t1), Herr(t2)

](C2)

Φ3(τ) =1

6

∫ τ

0

dt1

∫ t1

0

dt2

∫ t2

0

dt3[Herr(t1),

[Herr(t2), Herr(t3)

]]+[Herr(t3),

[Herr(t2), Herr(t1)

]](C3)

...

The Magnus series establishes a framework to define error cancellation order. Implementing a control with fidelitydefined in Eq. (17) up to order α means choosing controls such that Φk(τ) ≈ 0 for all k ≤ α. Below we describe auseful framework for computing the Φα(τ) in the Fourier domain using filter functions.

2. Spectral representation of Magnus term

Here we derive the frequency-domain form for the leading-order Magnus term in Eq. (29). Using the definition forthe Fourier transform set out in App. A 1, we have

βk(t) =1

2π

∫ ∞

−∞dωeiωtβk(ω), (C4)

N ′k(t) =1

2π

∫ ∞

−∞dωeiωtN ′k(ω). (C5)

Substituting into Eq. (28) we therefore obtain

Φ1(τ) =

p∑

k=1

∫ ∞

−∞dt

(1

2π

∫ ∞

−∞dω1e

iω1tN ′k(ω1)

)(1

2π

∫ ∞

−∞dω2e

iω2tβk(ω2)

)(C6)

=

(1

2π

)2 p∑

k=1

∫ ∞

−∞dω1

∫ ∞

−∞dω2N

′k(ω1)βk(ω2)

∫ ∞

−∞dteiω1teiω2t (C7)

=

(1

2π

)2 p∑

k=1

∫ ∞

−∞dω1

∫ ∞

−∞dω2N

′k(ω1)βk(ω2) · 2πδ(−ω2 − ω1) (C8)

where δ(x) is the Dirac delta function. Consequently

Φ1(τ) =1

2π

p∑

k=1

∫ ∞

−∞dω2N

′k(−ω2)βk(ω2) (C9)

=1

2π

p∑

k=1

∫ ∞

−∞dωGk(ω)βk(ω) (C10)

39

where we have defined

Gk(ω) ≡ N ′k(−ω) ≡∫ ∞

−∞dteiωtN ′k(t). (C11)

3. Leading order robustness infidelity

The leading-order error action operator Eq. (26) may be Taylor expanded as

Uerr(τ) = I− iΦ1 −1

2Φ2

1 + ... (C12)

= I− iΦ1 −1

2Φ1Φ†1 + ... (C13)

where we have used the property that the Magnus terms are Hermitian. Substituting into Eq. (17) the leading ordercontribution to the robustness fidelity takes the form

Frobust(τ) =

⟨∣∣∣∣1

Tr (P )Tr

(P

I− iΦ1 −

1

2Φ1Φ†1 + ...

)∣∣∣∣2⟩

(C14)

≈⟨∣∣∣∣

1

Tr (P )

Tr (P )− iTr (PΦ1)− Tr

(1

2PΦ1Φ†1

)∣∣∣∣2⟩. (C15)

Due to our choice of gauge transformation in Eq. (25), we additionally use the property that Tr (PΦ1) = 0, yielding

Frobust(τ) =

⟨∣∣∣∣1−1

Tr (P )Tr

(1

2PΦ1Φ†1

)∣∣∣∣2⟩

(C16)

=

⟨[1− 1

Tr (P )Tr

(1

2PΦ1Φ†1

)]∗ [1− 1

Tr (P )Tr

(1

2PΦ1Φ†1

)]⟩(C17)

=

⟨1− 2

Tr (P )Tr

(1

2PΦ1Φ†1

)+O

(|Φ1|4

)⟩(C18)

where the last line uses the result that Tr(PΦ1Φ†1

)is real-valued, following from the Hermiticity of Φ1. Ignoring

terms beyond O(|Φ1|2

), and observing the ensemble average over noise-realizations only affects terms dependent on

Φ1, we therefore obtain

Irobust(τ) = 1−Frobust(τ) (C19)

≈ 2

2Tr (P )

⟨Tr(PΦ1Φ†1

)⟩(C20)

=1

Tr (P )Tr(P⟨

Φ1Φ†1

⟩). (C21)

4. Leading-order filter functions

Substituting Eq. (29) into Eq. (27) we obtain

⟨Φ1(τ)Φ†1(τ)

⟩=

p∑

i=1

p∑

j=1

(1

2π

)2 ∫ ∞

−∞dω1

∫ ∞

−∞dω2Gi(ω1)G†j(ω2)

⟨βi(ω1)β∗j (ω2)

⟩(C22)

=

p∑

k=1

(1

2π

)2 ∫ ∞

−∞dω1

∫ ∞

−∞dω2Gk(ω1)G†k(ω2) 〈βk(ω1)β∗k(ω2)〉 (C23)

=

p∑

k=1

1

2π

∫ ∞

−∞dωGk(ω)G†k(ω)Sk(ω) (C24)

40

where in the second line we invoke the independence property of the frequency-domain variables βi,j(ω) definedby Eq. (A10), and in the third line we use Eq. (A19). Substituting into Eq. (27) we therefore obtain

Irobust(τ) ≈p∑

k=1

∫ ∞

−∞

dω

2π

1

Tr (P )Tr(PGk(ω)G†k(ω)

)Sk(ω) (C25)

Appendix D: Technical details on implementation ofthe SVD noise reconstruction

In this appendix we provide additional technical de-tails pertaining to our implementation of the SVD noisereconstruction algorithm.

1. Measurement uncertainties for SVDreconstruction

All quantities captured in Eq. (56) must be estimatedin order to perform a computationally valid spectral esti-mate. Formally, I(j) is the expectation value over a prob-ability density function dependent on all p noise PSDs,and the jth control protocol. An estimate may be ob-tained from experiment by averaging over a set of mea-sured infidelities obtained from repeated application ofthe jth control protocol. We denote these estimates asIj ± ∆Ij , where ∆Ij denotes the experimental uncer-tainty, e.g. estimated via the standard error.

In principle, the uncertainties ∆Sk,` will have contri-butions from the uncertainties both in F jk,` and in I(j).

Since F jk,` are calculated directly from the knowledge ofthe set of controls and noise operators, their uncertain-ties stem from numerical errors. In what follows, we ne-glect these contributions and consider only the dominantstatistical error coming from the measured infidelities.Under this assumption, the solution for the uncertainties∆Sk,` can obtained straightforwardly from the knowl-edge of the covariance matrix for I, namely

ΣI = diag(σ2I,1, σ

2I,2, · · · , σ2

I,c), (D1)

where σI,j is the standard deviation for the measuredinfidelity subjected to the control sequence j. Us-ing Eq. (68), we can write the covariance matrix for Sas

ΣS = V D+U †ΣIUD+V †, (D2)

and obtain the individual standard deviations explicitlyas the square root of its diagonal elements

σS,k =

√diag(ΣS)k. (D3)

2. Filter function normalization

For a system of multiple noise sources, the SVD of theraw filter functions, F , may present two issues. First, all

filter functions in the matrix F must have the same units,otherwise the SVD transformation would be invalid. Thesecond issue is that different filter functions in the ma-trix F (corresponding to, in principle, controls of widelyvarying lengths or amplitudes), can differ in scale by or-ders of magnitude. In this case, the singular values willalso present large variations, affecting the stability of theinversion process.

To account for these scenarios, we normalize the filterfunctions prior to performing the SVD. We define a c× cdiagonal scaling matrix

Gi,j = δi,j

√∑

k,l

|F jk,l|2, (D4)

which quantifies the strength of the filter functions forthe j-th control, as it sums up the contributions for allnoise channels and frequencies, represented by indices kand l, respectively. We then use this scaling factor todefine a dimensionless filter-function matrix F :

F = G−1F . (D5)

Given the SVD of this matrix:

F = UDV †, (D6)

we can write the original filter function matrix as

F = GUDV †. (D7)

The reconstruction solution given by Eq. (68) then be-comes

S = V D+U †G+I, (D8)

where, as before, a matrix A+ corresponds to a diagonalmatrix with entries 1/Ai for all non-zero Ai, and zerootherwise.

3. Calculating weights due to sensitivity

One of the important issues with the reconstructionprocedure is to ensure that the frequency range wherethe PSD to be reconstructed is appreciable is also wellcovered by the filter functions used in the inversion pro-cedure. This means that the predictions for the noisePSD will diminish in relevance when the filter functionsensitivity in a given frequency region is close to zero. In

41

order to capture this, we introduce an additional vari-able called a weight associated with each element in thenoise predictor. We employ these weights in order toidentify spectral regions where the validity of the SVDbreaks down and uncertainties in the returned estimatesbecome large.

The weights are defined as the sum of all c filter func-tions for a given noise k, normalized by their maximumvalue:

wk =

∑j,lF

jk,l

maxk(∑

j,lF jk,l)

. (D9)

4. Singular value truncation to prevent numericinstabiity

When the system of linear equations is over-determined, depending on the condition of the matrixF , the SVD reconstruction method can become numer-ically unstable. Intuitively, when some singular valuesof F are small, they introduce instability in the matrixpseudo-inverse as their inverses diverge.

To mitigate this, we truncate the singular values witha preset threshold χ. The appropriate value of this pre-set threshold can be found by the following heuristic: Foreach experiment, run the circuits multiple times and re-construct the noise spectrum using the measured results.The threshold of the singular values is set to be the valuesuch that the reconstructed noise spectra from differentconverge within uncertainties, prior to the onset of po-tential numeric instability.

Suppose that F is a n×m matrix, with n < m. Sup-pose the singular values in D are ordered in the descend-ing order such that the i+ 1-th entry of D is less than χ.Let D′ denotes the n×m diagonal matrix obtained fromD with the i + 1, . . . n diagonal elements set to be zero.Then the SVD of the truncated filter function matrix isF′

= UD′V ∗. We then use the truncated F′

to performthe pseudo-inverse.

The last essential issue in the numerical reconstructionis the distribution of singular values in the decompositionof F . Significant numerical instability can arise when thesmallest singular value is much smaller than the largestone. This is due to the inversion in the reconstruction:

S = V D+U †G+I. (D10)

If the ratio Dmax/Dmin between the maximum and min-imum singular values, known as the condition number, isvery large, then any noise in I will be strongly amplified,leading to a numerically unstable inversion process. Toavoid this issue, we implement a truncation by consider-ing only the first TD values in D. Using the truncatedmatrix DT , the estimation of the PSD is given by:

S = V D+TU†G+I (D11)

ΣS = G+V D+TU†ΣIUD

+TV

†G+. (D12)

We have implemented three different methods to selectthe truncation number TD that are available when callingthe reconstructor:

• Ratio-based truncation: The user can specify atruncation ratio rT corresponding to the ratio be-tween the maximum and minimum (truncated) sin-gular values. TD is the number of singular valuesthat satisfy the specified ratio, that is, D1/DTD <rT .

• Fixed truncation: In this method, the user canspecify the exact number TD of singular values tobe used for estimating noise PSD.

• Entropy truncation: In this method we estimatethe truncation number TD based on the entropy ofthe singular values defined as

HD = −∑

i

Di log2(Di), (D13)

with Di = Di∑j Dj

. Note that the entropy assumes

that singular values that are zero have already beenignored, consistently with the use of D+ in the so-lution of the inversion problem. The number oforthogonal linearly independent vectors can thenbe estimated using the entropy. This is done in thestandard way in which the information is estimatedfrom the entropy, that is, by defining

ND = 2HD . (D14)

This provides a real number that quantifies thenumber of linearly independent vectors that thetransformation captures. In order to get an inte-ger truncation number, we define it as being thefloor of ND, i.e., TD = bNDc.

.

Appendix E: Methods for experimentaldemonstrations of quantum control benefits

1. Quasi-static error robustness

In Fig. 10b,c we demonstrate the robustness of vari-ous dynamically corrected gate protocols implementinga net πX rotation to quasistatic errors in both the rota-tion angle and qubit frequency. For this demonstration, asingle trapped ion qubit is prepared in |0〉, 10 microwaveπ-rotations are applied, and the difference between thefinal and target qubit states is measured. The measuredquantity p|1〉 is the probability of finding the qubit in |1〉;it can be considered as the sequence infidelity, I, return-ing zero for error-free rotations. Four different pulse con-structions are utilized: primitive gates (red), BB1 (pur-ple), CORPSE (cyan), and reduced CinBB (blue).

42

In Fig. 10b, an over-rotation control error is engineeredby scanning the pulse length around the appropriatelytuned value. A sequence of 10 π-rotations about the +Xaxis is applied for each error strength; this sequence am-plifies the effect of over-rotation errors. In Fig. 10c, anengineered off-resonance error is created using an offsetbetween the frequency of the qubit and that of the mi-crowave drive. The absolute frequency detuning is nor-malized by the Rabi rate to quantify a relative error thatis varied between ±10%. To amplify the effect of theoff-resonance error, the 10 π-rotations are alternated be-tween the +X and −X axes.

2. Suppression of time-varying noise

The experimental filter function reconstructions shownin Fig.Fig. 10d,e in the main text were performed throughthe application of a single frequency disturbance at ωsid

added to either the control or the dephasing quadrature.We denote these time-dependent noise fields with βk(t)with k ∈ Ω, z and our corresponding Hamiltonian reads

H(t) = HC(t)(1 + βΩ(t)) + βz(t)σz, (E1)

where HC(t) is the control Hamiltonian that representsthe driven evolution through the microwave field, βΩ(t)is the amplitude noise and βz(t) is the dephasing noise.Typically, we write HC(t) in a rotating frame with re-spect to the qubit splitting such that it takes the form ofa time-dependent x- or y-rotation with

HC(t) =Ω(t)

2(cosφ(t)σx + sinφ(t)σy), (E2)

where Ω(t) is the time-dependent Rabi rate and φ(t) isthe control phase (see [72] for further details on the ex-perimental system and the control synthesis). The noisefields take the explicit form of

βk(t) = αk cos(ωsidt+ ϕ), for k ∈ Ω, z, (E3)

where αk is a constant factor to set the modulation depth.Through averaging over phase parameter ϕ ∈ 0, 2π,this form of modulation produces a δ-function like noisespectrum Sk(ω) ≈ δ(ω − ωsid) which, using the rela-tionship χ ∝

∫dωS(ω)F (ω) (cf. XXX), allows us to

directly extract the value of the filter function at the fre-quency ωsid [17, 72]. For the experiments here, we usedαΩ = 0.25 and αz = 0.7 and the points were averagedover 5 values of ϕ spaced linearly between 0 and 2π.

Experimentally, this is achieved via amplitude and fre-quency modulation of the microwave field that drives thequbit transition. The amplitude noise is added digitallyto the I/Q control waveforms before upload to the mi-crowave signal generator (Keysight E8267D), such thatwe obtain a noisy drive waveform with Ω(t) → Ω(t)(1 +

βΩ(t)). The dephasing noise is engineered through anadditive term in the phase of the control Hamiltonian(Equation E2), such the total phase component gets splitinto a control and noise term: φ(t) → φC(t) + φN (t). Aframe transformation with respect to this dephasing noise

term via eiφN (t)σz/2HC(t)e−iφN (t)σz/2 − φN (t)σz2 yields

H(t) =− Ω(t)(1 + βΩ(t))

2(cosφ(t)σx + sinφ(t)σy)

− φN (t)σz2

. (E4)

Using our notation from Equation E1, we can identifythe dephasing noise term βz(t) = φN (t)/2. The corre-sponding dephasing noise waveforms are generated us-ing an external arbitrary waveform generator (Keysight33600A), whose output is fed into the analog FM portof the microwave generator, which produces the desireddephasing noise term. For more details, see chapter 2 in[72].

3. Error homogenization characterized via 10-qubitparallel randomized benchmarking

In Fig. 10f, we measure a spatially varying average er-ror rate along a string of 10 trapped ion qubits (shownin Fig. 10f inset). In this system, ions are simultaneouslyaddressed by a global microwave control field to driveglobal single-qubit rotations. However, due to a gradientin the strength of the microwave field, the qubits rotatewith a spatially varying Rabi rate meaning that the con-trol cannot be synchronously calibrated for all 10 qubits.Qubit 1 in the figure is used for calibration in this exper-iment, yielding the lowest error rate.

Error is measured using parallel randomized bench-marking in which all ions are illuminated with mi-crowaves simultaneously and experience the same RBsequence. The general approach to RB sequence con-struction and experimental implementation is describedin detail in the supplemental material of Ref. [38]. Se-quences here are composed of up to 500 operations se-lected from the Clifford set. Measurement is conductedusing a spatially resolved EMCCD camera in order to ex-tract average error rates for each individual qubit. Giventhe relatively low quantum efficiency of this detectionmethod, SPAM errors are in the range of ∼ 3− 5%, ap-proximately an order of magnitude higher than achievedusing single-qubit RB, as measured via an avalanche pho-todetector.

4. Molmer-Sorensen drift measurements

In Fig. 10g-i, we compare two different construc-tions of phase-modulated two-qubit Mølmer-Sørensengates, both designed to produce the entangled Bell state

43

(|00〉 − i |11〉) /√

2. The gates are implemented by il-luminating two ions with a pair of orthogonal beamsfrom a pulsed laser near 355 nm, driving stimulatedRaman transitions as described in [52]. The geome-try of the beams enables coupling to the radial mo-tional modes of the ions, which have approximate fre-quencies ωk/2π = (1.579, 1.498, 1.485, 1.398) MHz andare denoted from highest to lowest frequency by k = 1to k = 4. One of the Raman beams is controlled byan acousto-optic modulator driven by a two-tone radio-frequency signal produced by an arbitrary waveform gen-erator. This results in a bichromatic light field that off-resonantly drives the red and blue sideband transitions,creating the state-dependent force used in the gate. Wemodulate the phase of the driving force φ(t) by adjustingthe phase difference between the red and blue frequencycomponents, φ(t) = [φb(t)− φr(t)] /2.

After an initial calibration of the mode frequencies andgate Rabi frequency, we repeatedly perform the entan-gling operations over a period of several hours withoutfurther calibration, alternating between the two differentphase-modulated gate constructions (panel g vs i) in or-der to mitigate any systematic differences between mea-surements. The ions are optically pumped to |00〉 andthe selected gate is performed by applying the Ramanbeams for a duration of 500 µs. For both constructions,the gate detuning set to -2 kHz from the k = 2 mode.In this configuration, only the k = 2 and k = 3 modesare significantly excited during the operation. Each gateis repeated 500 times and the ion fluorescence measuredafter each repetition. We use a maximum likelihood pro-cedure described in [52] to extract the state populationsPn, the probability of measuring n ions bright, for eachset of repetitions.

The first phase-modulated gate construction consistsof four phase segments and is calculated to ensure modesk = 2 and k = 3 are de-excited at the conclusion of theoperation. The second is calculated to provide additionalrobustness to low frequency noise affecting the closure ofmode k = 2, which necessitates doubling the number ofphase segments as per the analytic procedure outlinedin [52].

The required gate Rabi frequencies are Ω = 2π × 18.3kHz and Ω = 2π × 22.9 kHz, for the first and secondgates, respectively. The laser amplitude required to pro-duce the desired Ω for a particular gate construction iscalibrated by fixing the amplitude of the single-tone Ra-man beam and varying the amplitude of the bichromaticbeam. As the amplitude of the beam is increased, thepopulations P0 and P2 will converge to the point at whichP0 = P2 ≈ 0.5, indicating the creation of the Bell stateand the correct laser amplitude.

Appendix F: Product Features Summary

This appendix provides a concise summary of key prod-uct features in the Q-CTRL product suite. Additional in-

formation is available from https://q-ctrl.com/products.

1. BLACK OPAL

• Evaluate and select error-robust controls from a li-brary in order to address specific noise-mitigationchallenges.

• Use machine-learning optimization packages togenerate tailored error-robust control solutions forcustom hardware.

• Characterize noise in your hardware using qubitsas transducers and custom spectrum reconstructionpackages

• Perform error budgeting for in-built or custom up-loaded controls using the filter-function framework

• Build intuition through advanced visualizationmodules to explore the impact of noise on quantumoperations, and how control can mitigate error

• Access dedicated preconfigured workspaces andcontrol libraries for single-qubit gates, two-qubitgates for superconducting circuits (cross-resonanceand parameteric drive) and trapped ions (Molmer-Sorensen), and noise characterization.

• Output control solutions in machine-ready formatssuch as CSV and JSON for upload to laboratoryinstrumentation or cloud based hardware.

2. BOULDER OPAL

The python package provides (via current and forth-coming features):

• Create numerically optimized error-robust controlsfor arbitrary high-dimensional quantum systemsthat suppress various error channels including de-phasing, crosstalk, and leakage. Incorporate realexperimental constraints on maximum power orcontrol bandwidth.

• Simulate individual gates through to full circuits inthe presence of realistic colored noise spectra (e.g.1/f) to capture drift and non-Markovian noise dy-namics.

• Verify and compare the error-robustness of any con-trol, including optimized controls and controls cre-ated by other means and uploaded for analysis.Probe error robustness through direct calculationof quasistatic-error susceptibility and filter func-tions to illustrate time-varying noise susceptibility.

• Characterize noise in hardware through tailoredqubit measurements to reconstruct noise spectra indifferent channels

44

• Calibrate control hardware to account for nonlin-earities in modulators, finite band limits, and dis-tortions induced by transmission lines.

• Discover and eliminate drifts and other system in-stabilities through time-series analyses of measure-ment records

• Automate control calibration and noise character-ization through scripting routines integrated di-rectly with experimental control software.

3. OpenControls

• Access a detailed open-source library of error-robust control solutions in a simple python module.

• Format controls directly for integration into domi-nant quantum programming frameworks includingQiskit, Cirq, and Pyquil. Modules exist for out-putting controls directly formatted as circuits foreach language.

• Integrate standard controls into programmingworkflows for circuit analysis incorporating Q-CTRL and custom python libraries.

4. Devkit

• Build deterministic robustness to noise into com-piled quantum circuits using the physics of coher-ent averaging. Use optimization tools to find logi-cally identical compilations which improve robust-ness against various error channels.

• Rapidly discriminate between logically identicalcircuit compilations using using high-dimensionalfilter functions to understand relative susceptibilityto error of nominally identical circuit compilations.Error channels include general common-mode de-phasing, local perturbations, crosstalk, etc.

• Create hybrid circuits combining low-level robustcontrols on circuit subspaces with general circuit-level compilation. Optimize low-level circuit blocksat the analog layer and optimally compile forcircuit-level error robustness [76, 77]. Tools arebased on a custom singular-value-decompositionframework which allows efficient separation of opti-mization tasks between sub-blocks within a circuit.

Appendix G: BLACK OPAL Visualizations of noiseand control in quantum circuits

Q-CTRL provides an advanced visual interface thatenables users to compose quantum circuits (Figs. 17(a)and 4(a)) and interactively track state evolution subject

using Bloch spheres and a visual representation of en-tanglement based on correlation tetrahedra. These toolsoffer unique, interactive, 3-dimensional visualizations, as-sisting users to build intuition for key logical operationsperformed in quantum circuits.

In our tools, an arbitrary sequence of single and mul-tiqubit operations may be graphically composed or se-quenced in python. The associated state evolution iscalculated and displayed using interactive three.js ob-jects. These may then be rendered in Q-CTRL products,or directly embedded in Jupyter notebooks or websites.Current visualization packages are limited to two-qubitsubspaces, with forthcoming development expanding tolarger circuits.

As an example consider the time-domain evolution ofthe state of a single qubit subject to control. A series ofgates is described by an ideal target Hamiltonian:

Htot =1

2Ω(cos(φ)σx + sin(φ)σy) +

1

2∆σz, (G1)

where, Ω is the Rabi rate, ∆ the detuning and σx,y,z arethe standard Pauli matrices. Depicted in Fig. 17(b) isa snapshot of the state vector evolution (purple pointer,Fig. 17(b)) along its present trajectory (solid red line,Fig. 17(b)) given by the last Hadamard gate (highlightedthrough a dynamic indicator). The visualization is dy-namic and evolves in time; a user can interact with atime-indication slider in order to move through the timesequence. Moreover the view of the Bloch sphere andits color palette may be adjusted by the user, allow-ing for a user to gain insights that may be challengingin a simple 2D representation. In this specific exam-ple, for instance, it becomes immediately obvious that aHadamard gate has the action not only of transforminga state |+z〉 → |+x〉, but that it does so as a rotationabout an axis tilted out of the equatorial plane of theBloch sphere.

The visualizer module also provides a means by whichone may intuitively explore the effect of noise on Uni-tary operations performed within quantum circuits. Twonoise channels are available: control-amplitude noiseΩ→ Ω(1 + β); and ambient dephasing noise ∆→ ∆ + η,where β is the fractional fluctuation away from the tar-get driving rate Ω and η is a fluctuation away form thetarget detuning ∆. In this circumstance the ideal stateevolution is perturbed by the presence of noise, as il-lustrated by a displaced trajectory on the Bloch sphere.Ultimately, the discrepancy between the final location ofthe state at the end of the circuit and the ideal trans-formation, illustrated by a dotted line, provides a clearvisual representation of how noise reduces the fidelity ofa Unitary transformation.

Producing an intuitive visual representation of entan-glement poses a significant challenge due to the presenceof non-classical correlations between quantum systems.The Q-CTRL visualizer provides an exact representationof a two-qubit system exhibiting entanglement and sub-ject to Unitary controls utilizing two Bloch spheres and

45

Ideal evolution

a

b

c Evolution under noise

Q1

Q1

Q1

FIG. 17. Circuit evolution under noise. (a) The interactiveQ-CTRL quantum circuit interface, paused during the secondHadamard gate (highlighted). (b) Evolution of the state vec-tor (purple pointer) on the Bloch sphere, under ideal, noise-free conditions. The current trajectory on is highlighted inred ( H-gate) which sequentially continues on the prior circuitevolution (purple). (c) Evolution of the state in the presenceof two noise channels: control amplitude (β) and ambient de-phasing (η). The cumulative error in the qubit state (dashedred) due to the presence of these noise channels is indicated bythe dashed line, relative to the ideal target trajectory (greenpointer).

three correlation tetrahedra. The Bloch spheres depictthe standard Pauli observables corresponding to each ofthe qubits (Fig. 18(b)), given for qubits one and two re-spectively by:

X1 = 〈σx ⊗ I〉, X2 = 〈I⊗ σx〉,Y1 = 〈σy ⊗ I〉, Y2 = 〈I⊗ σy〉,Z1 = 〈σz ⊗ I〉, Z2 = 〈I⊗ σz〉. (G2)

The surfaces of the Bloch spheres fully describe the spaceof separable states; the presence of any entanglement ne-cessitates that the Bloch vectors shrink and the states

move off of the Bloch sphere surfaces. For instance, inFig. 18(b), the magnitude of both Bloch vectors shrinksto zero following the first CNOT gate, as the systemevolves into a maximally entangled Bell state as a resultof this gate.

We visually depict entanglement using a set of ob-servables corresponding to correlations between pairs ofCartesian observables for the two qubits. The correlationvalue between the observables is given by:

V (AB) = 〈σA ⊗ σB〉 − 〈σA ⊗ I〉〈I⊗ σB〉,A,B ∈ X,Y, Z. (G3)

The nine correlation-pairs are organized into threecoordinate systems bounded by a tetrahedral geome-try [78] given by the following axial arrangement ofthe observable pairs: (XX,Y Y,ZZ), (XY, Y Z,ZX) and(XZ, Y X,ZY ). For separable states all correlation val-ues are zero and the visual indicator is set to the center ofthe tetrahedra. In the presence of non-zero entanglement,however, these visual indicators emerge from the originof the correlation coordinate systems and grow towardsthe extrema of the convex hull for maximally entangledqubit pairs. The degree of entanglement is also visuallyrepresented using the concurrence C(ψ) defined as:

C(ψ) = 2|〈00|ψ〉〈11|ψ〉 − 〈01|ψ〉〈10|ψ〉|, (G4)

which varies between 0 (separable states) and 1 (maxi-mally entangled) throughout the system evolution, andshown using a horizontal indicator. Maximally entangledstates may traverse the correlation tetrahedra when lo-cal unitaries are applied to the individual qubits, but theBloch vectors remain at the centers of the Bloch spheres.Once again, noise processes may be added to the system’sevolution in order to represent how the presence of noiseperturbs the entangled states.

Appendix H: Details on cloud architecture

Q-CTRLs suite of closed source products is designedand built upon a multitier application architecture. Of-ten referred to as an n-tier architecture or multilayeredarchitecture, a multitier architecture is a clientserver ar-chitecture in which presentation, logic, and data func-tions are physically separated. The most widespreaduse of the multitier architecture, and the form which Q-CTRL employs, is the three-tier architecture.

The three-tier architecture is a software architecturepattern in which the user interface (presentation), func-tional processes or business rules (logic), and data stor-age and access (data) are developed and maintained asindependent layers.

By segregating an application into tiers, we acquire theoption of modifying a specific layer, or even an individualcomponent of each layer, instead of reworking the entireapplication. The three-tier architecture allows any tier

46

Q1 Q2

a

b

c

Q1

Q2

FIG. 18. Visualization of two-qubit entangled states using BLACK OPAL. (a) Evolution of a two-qubit entangling circuitpaused during a pair of gates (circled in red). (b) The trajectories of individual qubit observables depicting separable states arerepresented on an interactive 3D Bloch sphere. The bloch vector goes to zero as qubits become entangled through the actionof the CNOT gate. (c) Three interactive entanglement tetrahedra track correlations between the nine pairs of observablesenabling visual representation of complete two-qubit state evolution. Concurrence indicates the level of entanglement (redmarkers) between the qubits throughout the evolution of the circuit. Control and/or dephasing noise may be added to assistin understanding its impact on the evolution of entangled states. Details of the visualization packages within BLACK OPALare presented in App. G.

to be upgraded or replaced independently in responseto changes in requirements or technology. For example,a change of database technology in the data tier wouldonly affect the Application Programming Interface (API)code as the API is the interface between the data andlogic tiers. Neither the presentation tier, nor any othercomponent of the logic tier, would be aware of the changeand would keep functioning as if there were no change atall. This provides a model by which we are able to createflexible, reusable and modular software with well-definedinterfaces, or ways of communicating, between each tier.

1. Presentation tier

The presentation tier is the topmost level of the Q-CTRL application stack. This tier displays informationrelated to such services as searching and viewing pre-vious optimizations, creating new quantum circuit sim-ulations or even signing up, logging in and managingyour account subscription. It communicates with othertiers via the API component of the logic tier and is con-cerned with presentation logic, e.g. how the results ofa simulation should be displayed, as opposed to busi-

ness logic, e.g. how a simulation is performed. In sim-ple terms, it is the layer which users can access directlyvia, for example, a website such as the Q-CTRL WebApp, or a client software package such as the Q-CTRLPython package. These two examples are part of the Q-CTRL product suite; however, due to the interoperablenature of the architecture it is possible for Q-CTRL cus-tomers to develop their own Custom applications thatlive alongside Q-CTRL-developed software on the pre-sentation tier. Some examples of potential custom appli-cations would be a client software package written in theR programming language or a tablet app used to design,simulate, and visualize quantum circuits.

2. Logic tier

The logic tier controls the applications functionality byperforming detailed processing and consists of three dis-tinct components: API, Queue, and Core. Controlled en-tirely by the API, the logic tier accepts requests from thepresentation tier and routes them to the relevant layersand components within the stack. As mentioned above,the logic tier is concerned with business logic (sometimes

47

referred to as domain logic) and is the part of the ap-plication that encodes the real-world business rules thatdetermine how data can be created, stored, and changed.It is contrasted with the other tiers of the applicationthat are concerned with lower-level details of managinga database or presenting the user interface.

API : As mentioned above, the API provides the inter-face for, and is the entry point to, all requests made tothe logic tier by the presentation tier. Strictly speaking,the API is a server-side web API consisting of a collectionof public endpoints and can be characterized as being:

1. Exposed via the web by means of a Hyper-text Transfer Protocol (HTTP) web serverThe Hypertext Transfer Protocol (HTTP) is anapplication protocol. HTTP functions as a re-questresponse protocol in the clientserver comput-ing model. A web browser running the Q-CTRLWeb App, for example, may be the client andthe Q-CTRL API, in this example, is the server.The client submits an HTTP request message tothe server. The server, which provides resourcessuch as JSON files and other content, or performsother functions (business logic) on behalf of theclient, returns a response message to the client.The response contains completion status informa-tion about the request in the form of a HTTP sta-tus code (e.g. 201 Created) and may also containrequested content in its message body.

2. Implemented in the Representational statetransfer (REST) architectural styleRepresentational state transfer (REST) is a soft-ware architectural style that defines a set of con-straints to be used for creating Web services. Webservices that conform to the REST architecturalstyle, called RESTful Web services, provide inter-operability between computer systems on the In-ternet. RESTful Web services allow the requestingsystems to access and manipulate textual represen-tations of Web resources by using a uniform andpredefined set of stateless operations. By using astateless protocol and standard operations, REST-ful systems aim for fast performance, reliability,and the ability to grow by reusing components thatcan be managed and updated without affecting thesystem as a whole, even while it is running.

3. Consisting of a well-defined re-quest/response message system expressedin JavaScript Object Notation (JSON)JavaScript Object Notation (JSON) is an open-standard file or data interchange format that useshuman-readable text to transmit data objectsconsisting of attribute/value pairs and array datatypes (or any other serializable value). It is a verycommon and language-independent data formatwith a diverse range of applications, including e.g.Openpulse formatting within Qiskit [40]. By using

a language-independent data format, the Q-CTRLAPI affords interoperability between systems,meaning applications can be developed in a widearray of languages and form factors.

4. Formatted as per the JSON:API specifica-tionJSON:API is a specification for building RESTfulAPIs in JSON. The specification consists of a setof shared conventions relating to common require-ments of RESTful APIs - including:

• Content negotiation: how a client should re-quest and how a server should respond, in-cluding specific HTTP headers

• Document structure: how the JSON shouldbe formatted

• Errors: How a server should respond in thecase of an error, including specific HTTP sta-tus codes

By implementing the JSON:API specification, theQ-CTRL API provides client developers with theoption to take advantage of generalized tooling inthe form of the JSON:API compliant client librariesavailable in various languages, including Python,JavaScript, Ruby, R and iOS. Clients built aroundJSON:API are also able to take advantage of fea-tures relating to the efficient caching of responses,sometimes eliminating network requests entirely.

5. Discoverable via an endpoint formatted asper the OpenAPI specificationThe OpenAPI Specification is a broadly adoptedindustry standard for describing modern RESTfulAPIs. Q-CTRL employs the OpenAPI specifica-tion as a way to both document its APIs, and toprovide a means by which its APIs are discoverableby client applications. Q-CTRLs OpenAPI docu-ment is available at https://api.q-ctrl.com/. Thisis the root of Q-CTRLs API and acts as both asource of documentation for client application de-velopers and as a discovery endpoint upon whichclient applications can self-generate client librarycode based on the specification. By adopting theOpenAPI specification, the Q-CTRL API providesclient developers with the option to take advantageof generalized tooling in the form of OpenAPI com-pliant software such as auto document generationand industry standard API productivity and col-laboration tools such as Postman.

Queue: The Queue is a distributed, asynchronous taskscheduler based on distributed message passing. It is fo-cused on real-time operation and also supports schedul-ing of long-running tasks (e.g. optimizations and simu-lations). Scheduling is the method by which work is as-signed to resources that complete the work. In the case ofQ-CTRL, the work is data flows performed by Core (see

48

below), which are in turn scheduled onto either shared ordedicated hardware resources.

The execution of tasks is performed concurrently onworker servers using multiprocessing and parallel pro-cessing to implement a variety of scheduling disciplinesincluding (but not limited to): First come, first served(sometimes referred to as first in, first out or FIFO); Pri-ority scheduling, and; Shortest remaining time first. Theuse of cloud-based worker servers allows the Queue toscale efficiently as the load and number of requests fluc-tuates. Additionally, the option for Q-CTRL customersto opt for dedicated hardware resources for the process-ing of their tasks enables even greater scaling and perfor-mance that would not be possible on personal hardwaresuch as laptop devices.

The Queue is never accessed directly. Tasks are passedto the Queue by the API and execute asynchronously (inthe background) so as not to disrupt or block the opera-tion of client applications in the presentation tier. Thisallows users of the Q-CTRL Web App, for example, torequest a control optimization to be performed and thento carry on using the app to perform other tasks while therequested optimization is performed in the background.Once the task is complete, the user is notified in-app andcan then view the results.

Core: Core is the encapsulation of Q-CTRLs world-leading quantum control engineering expertise in theform of a Python package. However, unlike many Pythonpackages that our customers are used to installing locallyand accessing directly, Core (like the Queue above) isprivate and accessible via the Q-CTRL API. This allowsCore to take advantage of the benefits of its siblings onthe logic tier including: A language-independent inter-face allowing for almost unlimited interoperability and;Parallel processing on dedicated hardware for highly scal-able performance.

3. Data tier

The data tier includes the data persistence mechanisms(database and storage), and the data access layer that en-capsulates the persistence mechanisms and exposes thedata. The data access layer provides an API to the logictier that exposes methods of managing the stored datawithout exposing or creating dependencies on the data-storage mechanisms. As discussed earlier, avoiding de-pendencies on the storage mechanisms allows for updatesor changes without the logic tier being affected by thechange, allowing for easier scalability and maintenance.

Database: The Database is a relational database man-agement system. While the database itself is simply anorganized collection of data, the implementation of therelational model is used in concert with the concepts ofdomain-driven design in order to build a collection of dataobjects, and relationships (or cardinality) between theseobjects, to represent the domain of quantum control. Forexample, consider a series of one-to-many relationships

between data objects in which one system has many con-trols. Further, each object has attributes that furtheridentify their context and purpose: a control can be oneof three types (shift, drift or drive), whereas a simulationcontains information relating to its initial state vector,point times and number of trajectories to be averaged.

As has been the case with all components of the ap-plication stack, the database is never accessed directly,rather, interactions are orchestrated via the API in or-der to maintain the security and integrity of the datacontained within.

Storage: Storage is a file server that acts to provideaccess to static files accessible from disk. This is an alter-native to retrieving information from the database whichcan be slow and unnecessary for requests that are com-mon. For example, when certain types of optimizationsor simulations prove to be commonplace or occurring reg-ularly, the results are saved to storage as files which canbe retrieved quickly without the need to queue the re-quest for Core to reprocess.

4. Infrastructure

Providing the delivery mechanism for Q-CTRLs appli-cations is the physical IT infrastructure comprising bothphysical equipment such as bare-metal servers as well asvirtual machines and associated configuration resources.Provisioned on any cloud - public, private, hybrid or on-premise - and via the use of infrastructure as code (IaC)processes, Q-CTRLs infrastructure provides a platformfor its applications to ensure the following at all times:

1. Performance:

• Utilization of more hardware-on-demand thancould possibly fit into a consumer PC

• Utilization of all CPUs

• GPU acceleration

• Pre-compiled binaries

2. Availability:

• Redundancy in hardware due to infrastructurehosted across multiple availability zones in or-der to survive outage of a given data centre

• Autohealing via service checks with containersautomatically restarted if health checks fail

• Cloud agnostic and code-driven so we canquickly re-deploy to another cloud provider ifrequired

• Monitoring and alerting

3. Portability:

49

• Infrastructure as code (IaC)

• Containerized with Docker

• Orchestrated with Kubernetes

4. Scalability:

• Autoscaling with the ability to add more com-pute resources on demand via public cloudproviders

• Almost infinitely scalable and not limited toresources we need to provision and host our-selves

• Images tuned to start up as quickly as possiblewhen joining the cluster

• Ability to scale down resources when not re-quired (cost effective)

5. Security:

• Nothing can reach production environmentswithout going through a defined pipeline,starting with code. This ensures that we knowthat the state of our environment matches ex-actly what has been declared in code, and thateverything that is deployed has gone through

a quality assurance and security check pro-cess. For example, all code and dependenciesgo through a security vulnerability check

• End-to-end SSL

• Enterprise-grade secrets management, lever-aging HSMs for key management

• All network traffic is via Ingress

• Only select teams have access to virtual ma-chines and containers running in the cluster

• All access is monitored and logged

• Leverages security provided by public cloudproviders, e.g. Google Cloud Platform(GCP), Amazon Web Services (AWS), Mi-crosoft Azure, etc.

• Kubernetes cluster upgrades and database se-curity upgrades are applied automatically

• Zero-trust networking

• Applications run on non-privileged accountsinside Docker containers for greater isolationof running processes

• Surface attack area minimised by only de-ploying required components on minimalistCoreOS images that are constantly updatedwith security patches

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