Complete physical characterization of QND measurements via tomography

L. Pereira,1, ∗ J.J. García-Ripoll,1 and T. Ramos11Instituto de Física Fundamental IFF-CSIC, Calle Serrano 113b, Madrid 28006, Spain

We introduce a self-consistent tomography for arbitrary quantum non-demolition (QND) detec-tors. Based on this, we build a complete physical characterization of the detector, including themeasurement processes and a quantification of the fidelity, ideality, and back-action of the mea-surement. This framework is a diagnosis tool for the dynamics of QND detectors, allowing us toidentify errors, and to improve their calibration and design. We illustrate this on a realistic Jaynes-Cummings simulation of superconducting qubit readout. We characterize non-dispersive errors,quantify the back-action introduced by the readout cavity, and calibrate the optimal measurementpoint.

Introduction.- Quantum non-demolition (QND) de-tectors measure an observable preserving its expectationvalue [1, 2]. This property is essential in back-action-free quantum metrology [3–7], and in quantum protocolswith feedback, e.g. fault tolerant computation [8–12].QND measurements are typically implemented indirectlyby monitoring a coupled subsystem, as demonstrated invarious AMO [13–17] and solid-state [18–23] platforms.In superconducting circuits, the standard qubit measure-ment is a dispersive readout [24] mediated by frequencyshifts in an off-resonant cavity. This is a near-QND pro-cess that approximately preserves the qubit’s polariza-tion [cf. Fig 1(b)], and enables rapid high-fidelity single-shot measurements [25, 26], protection by Purcell fil-ters [27–30], fast reset [31], and simultaneous readoutthrough frequency multiplexing [28, 30, 32].

State-of-the-art QND detectors still face experimentalchallenges. A critical problem is the exponential accumu-lation of back-action errors from repeated applications ofthe detector, which limits the scaling of quantum tech-nologies. In superconducting qubit readout, such errorsoriginate in deviations from the dispersive limit in practi-cal devices [33]. This has motivated more complex QNDmeasurement schemes [34–42], which also introduce theirown sources of imperfection.

In order to make progress in the design and opera-tion of QND measurements, we need a complete andself-consistent diagnosis tool, which helps both with thecalibration of the detector and with describing its realdynamics. Many experiments have focused on optimiz-ing simple quantities such as the readout fidelity andthe QND-ness [38, 41]. However, these fidelities do notquantify the QND nature of a measurement, but ratherits projectivity and ideality as shown below. Anotherstandard approach is detector tomography [43–47]. Thismethod characterizes destructive measurements via pos-itive operator-valued measurements (POVMs), but ig-nores the post-measurement state, and therefore a de-scription of the measurement back-action.

In this work, we develop a complete physical charac-terization of QND measurements and their back-actionvia quantum tomography. The protocol, without pre-

FIG. 1. (a) QND detector tomography for generic measure-ments. A self-consistent calibration requires sampling overinput states ρk and two consecutive QND measurements, in-terleaved by a unitary operation Uj . This allows us to gener-ate the measurement processes En, for each possible outcomen = 1, . . . , N , and to reconstruct them tomographically fromthe conditional probabilities p(n|k) and p(mn|jk). (b) Setupfor tomographic characterization of QND qubit readout. Theprotocol requires pulse control on qubit Ωq(t) and cavityΩc(t), as well as continuous homodyne detection 〈a + a†〉c.The detector can have arbitrary qubit-cavity coupling g andany imperfection such as qubit decay γ and dephasing γφ.

calibration of the QND detector, estimates both thePOVM elements and the quantum process operators as-sociated to each measurement outcome. As seen inFig. 1(a), this requires two consecutive applications ofthe detector, interleaved by unitary operations, and re-peated over a set of input states. The information con-tained in the process operators can be used to identifyerrors, calibrate, and optimize the design of QND de-tectors. This can be done either directly, or throughthe analysis of simple metrics such as readout fidelity,QND-ness, and destructiveness, a precise bound of themeasurement back-action that we introduce below. Themethod can be applied to any detector, but we illustrateits power simulating realistically the calibration of super-conducting qubit readout beyond the dispersive approxi-mation. Our study shows that state-of-the-art dispersivereadout is a near-ideal measurement at the optimum of

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QND-ness, but there are other regimes where it becomesmaximally QND with minimal back-action error, as re-vealed by the destructiveness. Other tomographic ap-proaches to non-destructive detectors focus on near-idealmeasurements only [48, 49], and thus do not provide acharacterization of the QND nature and back-action ofthe measurement.General description of QND measurements.- A non-

destructive quantum measurement with N outcomes isrepresented by N completely positive maps En, whichadd up to a trace-preserving map E =

∑n En. The

n-th measurement outcome is obtained with probabil-ity p(n) = TrEn(ρ), leaving the system in the post-measurement state ρn = En(ρ)/p(n). Each En is unam-biguously represented by the Choi matrix Υn, whose N4

elements Υijkln read [50]

Υijkln = 〈ij|Υn |kl〉 = 〈i| En(|k〉〈l|) |j〉 . (1)

The Choi matrices give the post-measurement statesEn(ρ) =

∑ijkl Υ

ijkln ρkl |i〉〈j| and also the measurement

statistics p(n) = TrΠnρ, with the POVM elements

Πn =∑ijk

Υkjkin |i〉〈j|. (2)

Conservation of probability requires the completeness re-lation,

∑n Πn = 11, which imposes

∑nk Υkjki

n = δij onthe Choi components.

A QND measurement of observable O [2, 51, 52] is onewhere the unconditional process E =

∑n En conserves

the probability distribution p(n) over repeated measure-ments. Equivalently, E preserves the average of all com-patible observables Oc

〈Oc〉 = TrOcρ = TrOcE(ρ), ∀[Oc, O] = 0. (3)

Ideal measurements are well known QND measurementswhere consecutive detections project the system onto thesame eigenstate of O. This requires all Choi matrices tobe projectors Υn = (Υn)2, a sufficient condition to sat-isfy Eq. (3). However, as shown below, general QNDmeasurements are not ideal, and allow the state on con-secutive detections to change, provided the averages 〈Oc〉remain constant.Tomographic reconstruction of measurement

processes.- We have developed a self-consistentcharacterization of QND measurements, based on twoconsecutive applications of the detector, interleavedwith unitary operations from a universal set of gatesUj [cf. Fig.1(a)]. The first measurement induces theprocesses En, conditioned on the detected outcome,while the unitary Uj and the second measurement areused for a process tomography of the detector itself. Byrepeating the protocol over a set of input states ρk, weobtain the conditional probabilities p(n|k) = Tr(Πnρk)

and p(mn|jk) = Tr(ΠmUjEn(ρk)U†j ) after the first

and second measurements, respectively. From thesedistributions we reconstruct the POVMs Πn and Choimatrices Υn that best approximate the measurement,without a pre-calibration of the detector.

To recover matrices Πn and Υn that are meaningfuland satisfy all the physical constraints of a measurement,we use maximum likelihood estimation (MLE) [46, 53, 54]in a two-step strategy. First, we reconstruct the POVMsby minimizing the log-likelihood function f(Πj) =∑n,k p(n|k) log[Tr(Πnρk)], which compares the experi-

mental probabilities p(n|k) to the set of feasible matri-ces Πn satisfying Πn ≥ 0 and

∑n Πn = 11. Finally,

we estimate the Choi matrices Υn, minimizing a func-tion fn(Υn) =

∑mjk p(mn|jk) log Tr[(U†jΠmUj⊗ρTk )Υn]

which compares the experimental probabilities p(mn|jk)to a parametrization of the Choi matrix Υn satisfyingΥn ≥ 0 (Υikjl

n = Υijkln ) and the POVM constraint (2).

In total, QND detector tomography solves N + 1 opti-mization problems: one for POVMs of size N2, and Nfor Chois of size N4 [55].QND measurement quantifiers via tomography.- We

use the reconstructed Choi matrices Υn to quantify theperformance and QND nature of a measurement. Stan-dard benchmarks for QND detectors are readout fidelityF =

∑n p(n|n)/N and QND-ness Q =

∑n p(nn|n)/N ,

defined as the average probability that an observable’seigenstate |n〉 remains unchanged after one or two mea-surements, respectively. These quantities are related totomography via

F =1

N

∑n

〈n|Πn|n〉 =1

N

∑nj

Υnjnjn , (4)

Q =1

N

∑n

〈nn|Υn|nn〉 =1

N

∑n

Υnnnnn . (5)

The readout fidelity F quantifies how close the measure-ment is to a projective one Πn = (Πn)2. QND-ness Qand other similar fidelities [49] quantify the overlap withan ideal measurement, satisfying Υn = (Υn)2. Both areimportant measurement properties, but none of them as-sess the QND nature of the detector and its back-actionon the observables (3). For instance, a maximum valueQ = 1 characterizes an ideal measurement, but there arenon-ideal measurements Q 6= 1 which are close to QND.

We introduce the destructiveness D as a precise quan-tifier of the QND nature of a detector, regardless of howideal it is. This new quantifier bounds the change orback-action suffered by any observable compatible withthe measurement of O, in accordance to property (3):

D =1

2max||Oc||=1

||Oc − E†(Oc)||, [O,Oc] = 0. (6)

Evaluating D requires the tomographic reconstruc-tion of the complete measurement process, E†(Oc) =∑ijkln

[Υklijn

]∗Oklc |i〉〈j|, and a maximization over all

3

compatible operators of unit norm ||Oc|| = 1, with||O|| =

√Tr(O†O). In practice, this maximization is

done by finding the maximum eigenvalue of a positivematrix [56]. For instance, in the case of the qubit ob-servable O = σz, the destructiveness reduces to D =||σz − E†(σz)||/

√8. Since D verifies the general QND

condition (3), it also bounds the change of the probabil-ity distribution p(n) over repeated measurements.

As shown below, the three quantities F , Q, and Ddescribe the most important aspects of QND detectors,but there is further information to extract from Υn,which are the most general objects. For instance, de-termining the success probability after M 1 con-secutive QND detections is a demanding experiment,whose outcome may be inferred from our tomographyas p(n . . . n|n) =

∑n〈nn|(Υn)M−1|nn〉.

Calibration of qubit readout beyond dispersiveapproximation.- In standard superconducting qubitreadout [24], the qubit |g〉 , |e〉 couples to an off-resonant cavity mode a with detuning ∆ and couplingg [cf. Fig. 1(b)]. This interaction is described by aJaynes-Cummings (JC) Hamiltonian [57],

HJC =∆

2σz + g(σ+a+ a†σ−) + Ωc(t)(a+ a†), (7)

with Pauli operators σz = |e〉〈e| − |g〉〈g|, σ− = σ†+ =|g〉〈e|, and a resonant drive Ωc(t) on the cavity. In thedispersive limit ∆ g, HJC approximates a dispersivemodel Hd = 1

2 (∆ + χ)σz + χσza†a + Ωc(t)(a + a†) that

predicts a qubit-dependent displacement χ = g2/∆ onthe cavity resonance. In theory, by continuous homodynedetection 〈a+ a†〉c on the cavity, we can discriminatethe qubit state without destroying it. In practice, thenon-dispersive corrections implicit in HJC can slightlydegrade the QND nature of the measurement [33, 58].

To realistically quantify the performance and mea-surement back-action of dispersive readout, we describethe dynamics with the full HJC and a stochastic mas-ter equation (SME) [59–62]. This formalism accountsfor the back-action of the continuous homodyne detec-tion onto the qubit state, as well as cavity decay κ, qubitdecay γ, and qubit dephasing γφ [56]. We simulate nu-merically the tomographic procedure, solving the SMEover many realizations of the experiment. On each tra-jectory, the qubit is prepared in one of the six statesρk ∈

|g〉 , |e〉 , (|g〉 ± |e〉)/

√2, (|g〉 ± i |e〉)/

√2. We per-

form two single-shot measurements, interleaved by a cav-ity reset time, and one of the three qubit gates Uj ∈11, exp(−iπσy/2), exp(−iπσx/2). In Fig. 2 we showa representative trajectory of the protocol, where theoutcome of each single-shot measurement is discrimi-nated as 〈σz〉c = −sign(J) with J =

√κ∫ T

0dt〈a + a†〉c

the homodyne current integrated over the duration Tof the readout pulse Ωc(t) [56]. For simplicity, we ne-glect imperfections in the qubit state preparation and

FIG. 2. Simulation of QND detector tomography for disper-sive qubit readout. (a) Scheme of pulses on qubit (blue) andcavity (red) to implement state preparation, gates, and ho-modyne measurements. (b) Cavity quadrature 〈a+ a†〉c con-ditioned on a single trajectory. (c) Average of Pauli operators〈σz〉c (blue) and 〈σx〉c (light blue) conditioned on the sametrajectory. This realization corresponds to an initial state|+〉 = (|g〉 + |e〉)/

√2 on the qubit, a first measurement with

outcome |e〉, a cavity reset time, the use of gate exp(−iπσy/2),and a second measurement with outcome |g〉. Repeating thisprocedure over many trajectories with different inputs andgates allows us to reconstruct the Choi matrices Υg and Υe.

gates, performed with a local control Ωq(t). Simula-tions consider state-of-the-art parameters of supercon-ducting qubit readout [26]: g/2π = 200MHz, κ = 0.2g,γ = γφ = 10−4g, T = 8/κ ≈ 32ns, and |Ωc| = 0.173g,corresponding to 〈a†a〉 ∼ 1.5 photons on cavity.

We calibrate the measurement by tuning ∆/g and com-puting via tomography the quantifiers 1−F , 1−Q, andD [cf. Fig. 3(a)]. We show predictions using the realisticHJC interaction (solid), as well as the dispersive modelHd (dashed) to benchmark the results. We identify threequalitatively different points of operation (i)-(iii) as indi-cated by vertical lines in Fig. 3(a). For each of them, wedisplay in Figs. 3(b)-(d) the Choi matrices |Υn| for bothmeasurement outcomes n = e, g, where blue (orange)columns correspond to the JC (dispersive) predictionsand the upper color corresponds to the higher values.

(i) Around ∆/g = 7.7, the dispersive model reachesits optimum in fidelity and QND-ness, and the mea-surement is nearly ideal with projective Choi matricesΥ

(d)n ≈ |nn〉〈nn| [cf. orange columns in Fig. 3(d)]. This

occurs near 2χ/κ ∼ 1 as expected by theory [24]. Incontrast, the Choi matrices of the JC model show strongdeviations from an ideal measurement [cf. blue columnsin Fig. 3(b)]. The population transfer |e〉 → |g〉 dur-ing measurement—mainly due to cavity-induced Purcelldecay [33]—reduces Υeeee

e and increases Υggeen . In ad-

dition, the increase of Υgennn and Υegnn

n corresponds tothe growth of qubit coherences during measurement—e.g. due to cavity-mediated qubit driving [58]. Thesenon-dispersive effects increase the infidelity and destruc-tiveness [cf. Fig. 3(a)] and shift the optimal working pointto larger ∆/g.

(ii) Around ∆/g = 19.2, the non-dispersive correc-

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FIG. 3. Measurement quantifiers and reconstructed Choi matrices for QND qubit readout. (a) Readout infidelity 1−F (greencircles), QND-ness infidelity 1−Q (orange crosses), and Destructiveness D (blue squares) as a function of ∆/g. Solid (dashed)lines correspond to predictions for JC (dispersive) models. (b)-(d) Choi matrices |Υn| for n = e, g measurement outcomesat the three representative values ∆/g = [7.7, 19.2, 40], indicated by vertical lines in (a). Blue (orange) bars correspond topredictions from JC (dispersive) models, with the upper color representing the higher values. The probabilities to reconstructΥn are estimated from 2 × 104 trajectories for each initial state ρk and gate Uj (indicated in text). Error bars correspond toone standard deviation, obtained from 103 bootstrap simulations. Simulation parameters are state-of-the-art [26] (see text).

tions decrease and the QND-ness reaches an optimumQ ≈ 0.97 where the actual measurement is most ideal.From Υn in Fig. 3(c), we observe that non-dispersive ef-fects are present but strongly suppressed. In contrast tothe dispersive model, this optimum of QND-ness does notcoincide with the optimum of fidelity—where the mea-surement is most projective [cf. Fig. 3(a)]. This is ex-plained by the different evolution of the destructiveness,and noting that ideality is a compromise between fidelityand measurement back-action. Thus, in the JC model Ddecreases monotonically with ∆/g, pushing the conditionof highest ideality to larger ∆/g, where the back-actionis lower. In contrast, the back-action D of the dispersivemodel is nearly constant with ∆/g, and the optima of Qand F coincide.

(iii) Around ∆/g = 40 the system is deep in the dis-persive limit. The predictions for both models almost co-incide, up to residual cavity-mediated qubit driving [58].Here, 1−F and 1−Q get worse, but D reaches its min-imum, meaning that the realistic measurement is maxi-mally QND, but less ideal than in (ii). The loss of ide-ality is clearly manifested by the large diagonal termsΥmmlln [cf. Fig. 3(d)]. This behavior is explained theo-

retically [56] by the low distinguishability between out-comes n = g, e when the cavity displacement ∼ g2/∆ istoo small compared to the measurement uncertainty ∼ κ.The minimum value of D is attributed to how decoher-ence γ = γφ = 10−4g breaks the QND condition (3) evenwhen non-dispersive effects are suppressed [56].

We see that a tomography-based calibration allowsus to characterize different regimes of the detector, and

to identify error sources via the process matrices Υn.When simulating the measurement dynamics with HJC,we account for all non-dispersive effects and the back-action appearing in realistic superconducting circuit ex-periments, in a unified way. We also consider imperfec-tions due to larger intrinsic qubit decay and dephasingon the operation points (ii) and (iii). The effect of in-trinsic decay γ is qualitatively similar to Purcell decay,whereas pure dephasing γφ has a negligible effect on theChoi matrices for long readout pulses T 1/κ [56]. Fi-nally, our physical analysis can be used to choose an op-timal working point for the detector. If one is interestedin a near-ideal QND measurement, optimizing Q gives agood compromise between fidelity F and back-action D,as illustrated in (ii). This requires a larger detuning ∆/gthan expected by the standard dispersive prediction [24].However, if one is interested in a maximally QND mea-surement with minimal back-action, D is a more suitablequantity to optimize. This may require loosing idealityof the measurement as shown in (iii).

Conclusions and Outlook.- We developed a tomo-graphic procedure to characterize and calibrate arbitraryQND detectors, quantifying fidelity F , ideality Q, andback-action D via Choi matrices Υn. We applied themethod to a realistic simulation of superconducting qubitreadout using the full JC interaction. We identified non-dispersive errors and showed that the standard operationpoint with optimal QND-ness Q does not correspond tothe regime with minimal back-action. The tomographycan be immediately implemented in experiments as itonly requires standard control and measurement tools,

5

whose imperfections can be self-consistently included viagate set tomography [63–65]. This opens the door to ex-perimentally analyze unexplored effects on qubit readoutsuch as strong cavity driving [66], leakage to higher levels[67, 68], or cross-talk [69, 70]. This understanding mayhelp improve the QND measurement performance andguide the design of alternative schemes [34–42]. More-over, the method can be directly applied to other plat-forms such as QND detectors of microwave [71–76] or op-tical photons [77–80]. Finally, identifying regimes of non-ideal QND measurements with minimal back-action maybe also exploited for quantum information tasks since thechange in post-measurement states may be corrected viaerror mitigation strategies [52, 81, 82].

This work has been supported by funding from Spanishproject PGC2018-094792-B-I00 (MCIU/AEI/FEDER,UE) and CAM/FEDER Project No. S2018/TCS-4342(QUITEMAD-CM). L.P. was supported by CONICYT-PFCHA/DOCTORADO-BECAS-CHILE/2019-772200275. T.R. further acknowledges support from theEU Horizon 2020 program under the Marie Skłodowska-Curie grant agreement No. 798397, and from the Juande la Cierva fellowship IJC2019-040260-I.

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Information Dynamics 24 (2017).

7

Supplemental Material for:Complete physical characterization of QND measurements via tomography

Luciano Pereira1, Juan José García-Ripoll1, and Tomás Ramos11 Instituto de Física Fundamental IFF-CSIC, Calle Serrano 113b, Madrid 28006, Spain

CONTENTS

• I.— Additional properties of Destructiveness

• II.— Stochastic master equation for qubit readoutvia homodyne detection on cavity

• III.— Choi matrices in the dispersive model

• IV.— Effect of l.qubit decoherence on the Choi ma-trices for qubit readout

I.- ADDITIONAL PROPERTIES OFDESTRUCTIVENESS

In this section, we give a practical recipe to calculatethe Destructiveness D via the maximum eigenvalue ofa specific positive semi-definite matrix [cf. Sec. I.A], aderivation of a close formula for D in the case of a qubit[cf. Sec. I.A], and a numerical check that D vanishes forqubit readout deep in the dispersive limit and for zeroqubit decoherence [cf. Sec. I.B].

I.A.- Practical calculation of Destructiveness

We define the Destructiveness as a precise measure ofthe back-action and QND nature of a detector, given byEq. (6) of the main text as,

D =1

2max||Oc||=1

||Oc − E†(Oc)||, [O,Oc] = 0. (8)

This involves a maximization over the set of all normal-ized and compatible operators with O, and here we pro-vide a simple recipe to calculate this quantity. Let usconsider the eigenvector decomposition of Oc =

∑j ojPj ,

with P 2j = Pj and oj ∈ R. Using this decomposition, the

destructiveness can be rewritten as

D2 =1

4max||o||=1

∑jk

ojBjkok, (9)

with Bjk = Tr([Pj−E†(Pj)][Pk−E†(Pk)]). From Eq. (9),we see that D can also be calculated as the largest eigen-value of the positive semi-definite matrix Bjk.

In addition, it is possible to find a close expression ofD in the qubit case O = σz, which reads

D = ||σz − E†(σz)||/√

8. (10)

FIG. 4. (a) Destructiveness D obtained from QND tomogra-phy of dispersive readout, for different values of qubit decayγ. From top to bottom, γ = 10−2 (blue), 5 × 10−3 (orange),10−3 (green), 5× 10−4 (red), 10−4 (purple). (b) Average De-structiveness for each value of decay. The solid orange line isa linear fit, D = αγ+β, with α = 27.7/g and β = 8.5×10−3.

This expression is obtaining by noticing that the ma-trix Bjk has at least one null eigenvalue with eigen-vector o = (1/

√2, 1/√

2) since the identity Oc = Iis compatible with any observable and this does notchange with any quantum process. Using this observa-tion in the qubit case, we find that the no-null eigen-value of Bjk is simply Tr(B), and that its eigenvector iso = (1/

√2,−1/

√2). This eigenvector defines the com-

patible observable Oc = σz/√

2, which solves the opti-mization problem (9). Evaluating, we obtain Eq. (10).

I.B.- Destructiveness in the low back-action limit ofdispersive qubit readout

In Fig. 3(a) of the main text we showed that the de-structiveness D does not vanish deep in the dispersivelimit of qubit readout, ∆ g, despite non-dispersiveeffects are strongly suppressed. Here, we show that thisbehavior comes from the presence of qubit decoherence inour realistic model, which breaks the exact QND condi-tion. This occurs even in the dispersive model Hd, whichis exempt of non-dispersive effects.

To do so, we numerically compute D by simulating thetomography via the stochastic master equation with dis-persive Hamiltonian Hd, and for five values of the qubitdecay γ/g = [10−4, 5 × 10−4, 10−3, 5 × 10−3, 10−2]. Forsimplicity, we consider no qubit dephasing, γφ = 0, butthe results are qualitatively similar when varying dephas-ing. We also consider a weak driving case, Ωc = g/2,

8

since here we are mainly interested in the effect of qubitdecay on D. Other parameters of the simulations areκ = g/5, and T = 10/κ, and we vary the detuning ∆/g.Fig. 4(a) shows D as a function of ∆/g, for the five valuesof decay γ. These results are averaged over 103 trajecto-ries for each initial state and measurement, indicated inthe main text, and the error bars correspond to 3 stan-dard deviations obtained from 103 bootstrappings. Wecan see that D is almost constant in ∆/g for all the de-cays studied. This is consistent with the dispersive modelas there are no terms depending on ∆/g that may spoilthe QND condition. More importantly, we see that Ddecreases with γ, showing that γ is responsible for atleast part of the finite back-action quantified by D. Inthe dispersive model, decoherence γ is the only quantitythat breaks the QND condition, and therefore, it shouldbe the only quantity that contributes to D as well. Toshow this, we plot the average of D as function of γ inFig. 4(b), and we perform a linear fit D(γ) = αγ + β,with α = 27.7/g and β = 8.5× 10−3. We observe a goodagremeent between data and fit within the statistical er-ror, which suggests that there exists a linear relationshipbetween qubit decay and Destructiveness. Extrapolat-ing D to the decoherence-free limit, γ → 0, we obtainthat the destructiveness is lower than the statistical noiseD ≤ O(1/

√Nt), where Nt = 103 is the number of tra-

jectories used for each input state and gate in the QNDdetector tomography. Therefore, within the statisticalprecision of the tomographic reconstruction, we concludethe method predicts a vanishing destructiveness D → 0,deep in the dispersive limit, and for zero decoherence.

II.- STOCHASTIC MASTER EQUATION FORQUBIT READOUT VIA HOMODYNE

DETECTION ON CAVITY

In this section, we explain how to model the dynamicsof a dispersive qubit readout within the formalism of astochastic Master equation (SME) [1-3].

A single-shot measurement is carried out by applyinga resonant pulse Ωc(t) on the cavity along a time T . Thecoherent dynamics of the measurement is given eitherby the Jaynes-Cummings H = HJC or the dispersiveH = Hd Hamiltonians, given in the main text. The cav-ity is continuously monitored via homodyne detection oftransmitted photons, which introduces a stochastic dy-namics on the qubit-cavity system. This dynamics isdescribe by the following SME [1-3]

dρ =− i[H, ρ]dt+ κD[a]ρdt+√κM[a]ρdW

+ γD[σ−]ρdt+γφ2D[σz]ρdt, (11)

where the continuous homodyne measurement is de-scribed by the the Wiener process dW and the super-operator M[A]ρ = (A − 〈A〉)ρ + H.c.. Markovian decay

of the cavity κ, decay of the qubit γ, and dephasing ofthe qubit γφ are described by standard Lindblad super-operators, D[A]ρ = AρA† −A†A, ρ/2. We numericallyintegrate this equation using an implicit Euler algorithm[4]. The continuous measurement gives us access to thehomodyne current over a single trajectory,

J(t) =√κ 〈a+ a†〉c (t) + ξ(t), (12)

where 〈a+ a†〉c is the intracavity quadrature conditionedto the trajectory, and ξ(t) = dW/dt is the vacuum shot-noise 〈〈ξ(t)ξ(t′)〉〉 = δ(t− t′). The direction of the qubit-dependent phase shift of the resonator can be determinedfrom the homodyne current, which allows us to discrim-inate the state of the qubit without destroying it. Toreduce the noise of the signal, we integrated J(t) overthe measurement time T , obtaining, J =

∫ T0dtJ(t) =√

κ∫ T

0dt 〈a+ a†〉c. Since the qubit-dependent cavity dis-

placements are in opposite directions, the measurementoutcome can be discriminated from the sign of the inte-grated signal J as 〈σz〉c = −sign(J).

In order to benchmark our code that solves Eq. (11),we numerically calculate the readout fidelity F via twomethods. The first method consists in computing theprobabilities p(g|g) and p(e|e) by a direct simulation ofthe experiment with a single qubit readout. The secondmethod is applying our tomographic protocol involvingtwo measurements and then obtaining F via Eq. (4) ofthe main text. In addition, we compare these two numer-ical methods with the analytical prediction in the case ofthe dispersive model, given in Ref. [5] by

F = 1− 1

2erfc(SNR/2), (13)

where erfc is the complementary error function and SNRis the signal-to-noise ratio,

SNR =4Ω sin(ϕ)T√

2κT

[1− 4

κT

(cos(ϕ/2)2

− sin(χT + ϕ)

sin(ϕ)e−κT/2

)]. (14)

Here, ϕ = 2 arctan(2χ/κ). For the simulations, we setthe parameters on g units as κ/g = 1/5, γ = γϕ = 0,Ωc/g = 1/10, and T = 10/κ. The average number ofphotons in the resonator is 〈a†a〉 = 0.5. We considerhere a weak driving case since it requires a low numberof trajectories to achieve a good precision in the numeri-cal calculations. For each initial state and gate needed forthe tomography (indicated in the main text), we perform103 trajectories. The error bars are 5 times the standarddeviation obtained with 103 bootstrap simulations. Theresults are shown in Fig 5. We see that the three in-dependent estimations of F —direct, tomography, andanalytical— agree well within an error of 5 standard de-viations.

9

6 8 10 20 30 40 50

∆/g

10−2

10−1

1−F

Analytic

Direct

Tomography

FIG. 5. Comparison of three independent methods to esti-mate the readout infidelity 1−F of dispersive qubit readout.Blue corresponds to the analytical formula (14), orange to thedirect definition via probablities F = [p(g|g) + p(e|e)]/2, andgreen to the tomographic reconstruct and the use of Eq. (4)of the main text. The error bars correspond to 5 standarddeviations obtained with 103 bootstrap simulations.

III.- CHOI MATRICES IN THE DISPERSIVEMODEL

In this section, we study the general structure of Choimatrices Υn predicted by the dispersive model. Withthis we show that the diagonal form of Υn in Fig. 3(d) ofthe main text is characteristic from a low measurementindistinguishably occurring deep in the dispersive limit.

To do so, let us consider that the qubit is initially in thestate |Ψ〉 = ψe |e〉 + ψg |g〉 and the cavity is empty |0〉r.Modeling the system with the dispersive HamiltonianHd,the coherent evolution of the joint system |Ψ〉 = |Ψ〉q |0〉rafter a time t is given by [6]

|Ψ(t)〉 = ψg |g〉 |αg(t)〉+ e−i∆tψe |e〉 |αe(t)〉 . (15)

Here, |αj(t)〉 is the resonator state whose phase is shiftedconditioned on the qubit state j ∈ g, e. Therefore, wecan infer the qubit state by measuring the cavity field.This evolution is described by the unitary operation,

U(t) = |g〉〈g| ⊗D(αg(t)) + e−i∆t |e〉〈e| ⊗D(αe(t)),(16)

where D(α) is the cavity displacement operator. Read-ing the qubit state requires continuous homodyne detec-tion during a time T of the cavity quadrature operatorQ =

√κ(a+ a†

). The outcome of the qubit measure-

ment is discriminated form the integrated homodyne cur-rent, J = 1

T

∫ T0dt〈Ψ(t)|Q |Ψ(t)〉. This can be obtained

as the expected value of Q in the time average densitymatrix, Eqr(|Ψ〉〈Ψ|) = 1

T

∫ T0dt |Ψ(t)〉〈Ψ(t)|. In addition,

the reduced evolution for the qubit can be obtained bytracing over the cavity Eq(|Ψ〉〈Ψ|) = Trr[Eqr(|Ψ〉〈Ψ|)].Considering the eigenvector basis of Q, |q〉, we have

that the Kraus representation of the qubit dynamics is

Eq(|Ψ〉〈Ψ|) =

∫ T

0

dt

∫ ∞−∞

dqKq(t) |Ψ〉〈Ψ|Kq(t)†, (17)

with Kraus operators Kq(t) = r〈q|U(t)|0〉r/√T [7]. Con-

sidering the wave function of the cavity states φj(q, t) =〈q|αj(t)〉, the Kraus operators can be expressed as

Kq(t) =1√T

(φg(q, t) |g〉〈g|+ φe(q, t)e−i∆t |e〉〈e|). (18)

We can divide the trajectories in order to determinethe outcome of the measurement. If the homodyne cur-rent is larger than a predefined value δ, we assign theresult e, and otherwise we assign g. For instance, in thespecial case of a dispersive model without decoherenceboth states displace the cavity by the same amount, butin opposite directions, having δ = 0. Using this sepa-ration in the general case, the components of the Choimatrices are given by

Υijklg =

∫ T

0

dt

∫ δ

−∞dq 〈i|Kq(t) |k〉〈l|Kq(t)

† |j〉 , (19)

Υijkle =

∫ T

0

dt

∫ ∞δ

dq 〈i|Kq(t) |k〉〈l|Kq(t)† |j〉 . (20)

We can further simplify the Choi matrices as,

Υg =εg |gg〉〈gg|+ (1− εe) |ee〉〈ee|+ ζg |ge〉〈ge|+ ζ∗g |eg〉〈eg| , (21)

Υe =(1− εg) |gg〉〈gg|+ εe |ee〉〈ee|+ ζ∗e |ge〉〈ge|+ ζe |eg〉〈ge| , (22)

where we have defined the four auxiliary quantities εg =1T

∫ T0dt∫ δ−∞ dq|φe(q, t)|2, εe = 1

T

∫ T0dt∫∞δdq|φg(q, t)|2,

ζg = 1T

∫ T0dt∫ δ−∞ dqei∆tφg(q, t)φe(q, t)

∗, and ζe =1T

∫ T0dt∫∞δdqe−i∆tφe(q, t)φg(q, t)

∗.The general form of Υn in Eqs. (21)-(22) deviate from

projectors and thus we conclude that the measurementis not ideal in general. In the case without decoherence,the wave functions correspond to coherent states. In thiscase, the outcomes of the measurement cannot be per-fectly discriminated since the coherent states are not or-thogonal. Therefore, we have that ε < 1 and |ζj | > 0, andthat the dispersive readout cannot be a perfectly idealmeasurement. When the cavity displacement ∼ g2/∆ istoo small compared to the measurement uncertainty ∼ κ,the overlap between the wave functions increase, imply-ing that εj decrease and |ζj | increase. We can see thiseffect in the Choi matrices of Fig. 3(d) of the main text,which exhibits large diagonal terms Υgege

n and Υegegn .

10

FIG. 6. Choi matrices of dispersive readout with JC modelin presence of decay γ for (a) ∆/g = 19.2 and (b) ∆/g =40. Columns of different colors represent different values ofdecay, which increase in the direction indicated by the arrowas γ/g = 10−4 (blue), 5× 10−3 (green), 10−2(yellow).

FIG. 7. Choi matrices of dispersive readout with JC model inpresence of dephasing γφ for (a) ∆/g = 19.2 and (b) ∆/g =40. Columns of different colors represent different values ofdephasing, which increase in the direction indicated by thearrow as γφ/g = 10−4 (blue), 5× 10−3 (green), 10−2(yellow).

IV.- EFFECT OF QUBIT DECOHERENCE ONTHE CHOI MATRICES FOR QUBIT READOUT

In this section, we simulate the tomographic character-ization of dispersive readout as a function of qubit decayγ and dephasing γφ to study its effect on the Choi matrixcomponents Υg and Υe.

The simulations of QND detector tomography are done

using the JC Hamiltonian, and with Nt = 2 × 103 tra-jectories for each initial state and gate (indicated in themain text). Parameters are κ/g = 1/5, T = 6/κ, andΩc/g =

√3/10. For ∆/g, we consider the operating

points (ii) and (iii) of Fig. 3(a) of the main text, thatis a near-ideal measurement ∆/g = 19.2, and the deepdispersive limit ∆/g = 40, respectively. For both cases,we compute the Choi matrices |Υn| for three values of de-cay γ/g as shown by columns of different colors in Fig. 6.Similarly, we consider three values of dephasing γφ/g,and the corresponding |Υn| are shown in Fig. 7. Bothoperating points (ii) and (iii) show similar behavior whenincreasing qubit and dephasing, indicating that decoher-ence manifests in Υn independently of ∆/g. From Fig. 6we see that the main effect of decay γ is to increase theChoi components Υggee

n and decrease Υeeeen . This behav-

ior is consistent with the effect of cavity-induced Purcelldecay discussed in the main text. From Fig. 7 we seethat the dephasing has marginal effect on the Choi ma-trix because all the deviations in its components are onthe order of magnitude or lower than the statistical errorO(1/

√Nt). This is reasonable since dephasing cannot

appreciably affect the dynamics of the qubit after it hasbeen projected to |g〉 or |e〉 by the measurement. De-phasing may be relevant for short measurement times T ,when the projection has not yet fully happened, but inall our simulations we have considered T & 6/κ.

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