Steering Quantum Dynamics via Bang-Bang Control:Implementing optimal fixed point quantum search algorithm

Gaurav Bhole, Anjusha V. S., and T. S. Mahesh∗

Department of Physics and NMR Research Center,Indian Institute of Science Education and Research, Pune 411008, India

A robust control over quantum dynamics is of paramount importance for quantum technologies.Many of the existing control techniques are based on smooth Hamiltonian modulations involvingrepeated calculations of basic unitaries resulting in time complexities scaling rapidly with the lengthof the control sequence. On the other hand, the bang-bang controls need one-time calculation of basicunitaries and hence scale much more efficiently. By employing a global optimization routine such asthe genetic algorithm, it is possible to synthesize not only highly intricate unitaries, but also certainnonunitary operations. Here we demonstrate the unitary control through the first implementationof the optimal fixed-point quantum search algorithm in a three-qubit NMR system. More over,by combining the bang-bang pulses with the twirling process, we also demonstrate a nonunitarytransformation of the thermal equilibrium state into an effective pure state in a five-qubit NMRsystem.

PACS numbers: 03.67.Lx, 03.67.Ac, 03.65.Wj, 03.65.TaKeywords: Grover’s algorithm, Quantum control, Genetic algorithm, State preparation

Introduction: Quantum physics offers an exciting plat-form for the next generation devices. While quantumcomputers promise a tremendous computational capabil-ity, other quantum devices have opened up a variety ofinteresting possibilities such as quantum imaging [1, 2],supersensitive field sensors [3–5], and the generation ofrandom numbers [6, 7]. Robust and efficient quantumcontrol is indispensable not only for quantum computers[8], but also for other quantum simulators [9].

Over the past several years a remarkable progresshas been achieved in steering quantum dynamics (eg.[10–14]). Typically one considers a constant internalHamiltonian Hint and a set of M control operators hjsuch that the total Hamiltonian is of the form H(t) =

Hint +∑M

j=1 uj(t)hj , where uj(t) form a set of time-dependent parameter profiles. The overall unitary of du-

ration T is U = D exp(−i∫ T

0H(t)dt

), where D is the

Dyson time ordering operator. Given a target unitaryUT of dimension N , the goal is to optimize uj(t), by

maximizing the unitary fidelity Fu = |Tr(U†TU)/N |2.Generally smooth modulations (SM) of the parameters

uj(t) is assumed (see Fig. 1). The overall quantumevolution is then evaluated by time discretization andsubsequent calculation of instantaneous propagators foreach of the time steps. Here we propose a bang-bang(BB) control scheme which scales much more efficientlywith the size of the control sequence. Combined witha global optimization routine it can generate complexunitaries and even certain nonunitary operations withhigh fidelities.

In the following we first briefly explain the BB schemeand then describe its applications in unitary and nonuni-tary controls. We then illustrate the first experimen-

∗ mahesh.ts@iiserpune.ac.in

tal demonstration of optimal fixed-point quantum search(OFPQS) algorithm on a three-qubit NMR quantum pro-cessor. Finally we also describe a nonunitary BB controlfor initializing an effective pure state in a five-qubit NMRsystem.Bang-bang approach: Although the following methods

can be generalized to other quantum architectures, forthe sake of clarity, we shall use the rotating frame pic-ture of an NMR spin system. Denoting ωr, Jrs, Drs

respectively for resonance off-sets, indirect (scalar), anddirect coupling constants, and Ir for spin operators, thesecular part of the internal Hamiltonian can be writtenin the form

Hint = −∑r

ωrIrz + 2π∑r<s

(Jrs + 2Drs)IrzIsz

+2π∑r<s

(Jrs −Drs)(IrxIsx + IryIsy), (1)

wherein the third term vanishes for weakly interacting

SM

/ B

B

duty cycle

1

10

100

1000

0.001 0.01 0.1 1

T

t

jk

j

kth segment

SM

BB

FIG. 1. Smooth modulation (SM) vs bang-bang (BB) se-quence. Ωj , φjk, ∆t are the amplitude, phase, and duration ofthe segments, and T is the total duration of the sequence. Thehelix represents the twirling operation required for nonunitarygates. Performance of BB improves over SM for lower dutycycle as illustrated in the graph. Here τSM and τBB indicaterespectively SM and BB computational times for calculating10-qubit propagators of T = 0.5 ms duration.

arX

iv:1

512.

0838

5v1

[qu

ant-

ph]

28

Dec

201

5

2

spins, particularly for spins belonging to different nuclearspecies [15].

The BB approach relies on intermittent bursts of fullcontrol power instead of its smooth modulation. For gen-erality we consider a spin-system having several nuclearspecies, and let the jth species be irradiated by an inde-pendent RF source with an amplitude switching betweeneither 0 or a maximum value Ωj , but with a variableRF phase φj ∈ [0, 2π] (see Fig. 1). The propagatorfor the BB sequence can be easily setup by discretizingthe total control period T into short segments of dura-

tion ∆t. Let Sj =∑

m I(j)m be the collective spin oper-

ator of m spins belonging to the jth species. We callXj = exp−i(H0 + ΩjSjx)∆t as the basic propagatorand it needs to be computed only once. During the kthsegment there can be a delay or a pulse. If the kth seg-ment is a delay, then Ujk = exp(−iH0∆t) is a constantoperator and therefore needs to be evaluated only once.On the other hand, if the kth segment is a pulse witha phase φjk, then the propagator is simply obtained by

rotating Xj about the z-axis, i.e., Ujk = ZjkXjZ†jk. Here

Zjk = exp(−iφjkSjz) is a diagonal operator in the Zee-man product basis and is therefore efficiently evaluatedin the run-time of the iterations. The net propagatorfor the entire control sequence is simply the cumulativeproduct U =

∏jk Ujk over all the species j and then

over all the segments k.Typically complex unitaries involving non-local quan-

tum operations require long evolutions under spin-spininteractions, and have low duty cycles. In such cases,the BB approach is orders of magnitude faster thanthe conventional methods which require repeated ma-trix exponentiations to evaluate the segment unitaries(Fig. 1). If [H0, Zjk] = 0 (eg. weak-coupling case:|ωr −ωs| 2π|Jrs−Drs| for all r and s), then a furtherspeed-up ensues, since the delay propagators become di-agonal.

Given a target unitary UT, we optimize the BB param-eters Ωjk, φjk using the genetic algorithm and max-imize the unitary fidelity Fu. If the goal is to pre-pare a specific quantum state ρT starting from an ini-tial state ρin, then we need to calculate the output stateρout = UρinU

† and the state fidelity

Fs =|Tr(ρT ρout)|√Tr(ρ2T )Tr(ρ2out)

(2)

is to be maximized.

In this work, we also describe preparing a target statewith the help of the twirling operation which is essen-tially a nonunitary operator that attenuates all the co-herences, ultimately retaining only the diagonal elementsof the density matrix in the computational basis [16, 17].Although it is possible to incorporate the effects of de-coherence, in our present scheme we ignore such effectsby assuming that the control sequences are much shorterthan the decoherence time-scales. In the following webriefly describe the application of BB sequence in imple-

PPS

000

H

H

0

UG

Z

UG

H

H Z/2 Z

H

H

1H

19F

13C H

HF0

State preparation Generalized Grover iteration Measurement

Z/2

Z/2

eq

FIG. 2. Quantum circuit for OFPQS algorithm.

menting a quantum algorithm.

OFPQS algorithm: Classical search algorithms canfind one or more ‘marked’ items among an unsorteddatabase of Q items in O(Q) steps. On the other hand,Grover’s quantum search algorithm achieves the sametask in O(

√Q) steps, thereby providing a quadratic

speedup over the classical counterpart [18]. Grover’s al-gorithm identifies one of the R marked items among Qunsorted items with the help of a given oracle functionthat can recognize the marked items. It can also be in-terpreted as a rotation in the 2D space spanned by thesuperposition of Q − R non-solution states |ψQ−R〉 andthe superposition of R solution states |ψR〉 [19]. Theapplication of the Grover iterate can thus be visualizedas the rotation of the initial state |ψQ〉 towards |ψR〉 in

O(√Q/R) steps. However, if we do not know the num-

ber of marked items R beforehand, we cannot predict thenumber of iterations which would land the initial stateclosest to the marked state. Too few iterations give us astate comprising of mostly non-solution states, whereas,too many iterations can surpass the solution states andwe may end up getting non-solution states, yet again.In order to overcome this problem, attempts have beenmade to develop fixed point quantum search (FPQS) al-gorithms, which monotonically amplify the probability ofobtaining the marked states [20, 21]. While these FPQSalgorithms lacked the quadratic speedup, a recent opti-mal FPQS (OFPQS) algorithm proposed by Yoder et. al.achieves this speedup while maintaining the fixed pointbehavior [22]. In the following we outline the varioussteps involved in the OFPQS algorithm.

The quantum circuit for the OFPQS algorithm isshown in Fig. 2 (for 3-qubits). We need to extract |ψR〉from |ψQ〉 with a success probability PL with a predefinedlower bound. The algorithm needs a quantum registerwith a total number of qubits n ≥ log2(Q) + 1 includingan ancilla qubit, all initialized in the ground state |0〉⊗n.The system qubits are then transformed into a uniformsuperposition by applying n−1 Hadamard gates. We areprovided with an oracle UG which when acted upon themarked state, flips the ancilla qubit. Thus, UG|ψR〉|a〉 =|ψR〉|a⊕ 1〉 and UG|ψQ−R〉|a〉 = |ψQ−R〉|a〉, where |a〉represents the ancilla qubit. Various gates in the general-ized Grover iteration are also shown in Fig. 2. Let us con-sider l generalized Grover iterations. Defining L = 2l+ 1and γ−1 = T1/L(1/δ) where TL(x) = cos(L cos−1 x) is

the Lth Chebyshev polynomial of the first kind, the phase

3

00

0

00

1

01

0

01

1

10

0

10

1

11

0

11

1

(f)

11 01 10 00 10 00 11 01 10 00 11 01

H /2 (Hz) F /2 (Hz) C /2 (Hz)

(c) (d) (e)

1H

19F

13C

Br

Br (a) (b) H F C

0 49.7 224.5 H 100.0 -310.9 F 100.0 C

0.3 0.2 0.5

13.7 5.2 1.9

T2* T1

FIG. 3. (a) Molecular structure of dibromo fluoromethane,(b) the Hamiltonian parameters and relaxation time con-stants, and (c-e) PPS spectra (upper trace), equilibrium spec-tra (middle trace), and PPS pulse-sequences (bottom trace)for the 1H, 19F and 13C qubits, (f) bar diagram representingtheoretical (red) and experimental (blue) diagonal elements ofthe traceless deviation density matrix corresponding to |000〉PPS. In (b), the diagonal and off-diagonal elements are re-spectively chemical shifts and J-coupling constants in Hz.

rotations are given by

αj = −βl−j+1 = 2 cot−1(

tan(2πj/L)√

1− γ2)

(3)

for all j = 1, 2, . . . l [22]. As each Grover iterate containstwo applications of the oracle UG, the query complexityis L− 1.

NMR Implementation : We implement the OFPQSalgorithm on a quantum register involving two systemqubits 1H and 19F, along with an ancilla qubit 13C ofdibromofluoromethane [23] dissolved in acetone-D6 (seeFig. 3). All the experiments were carried out on a Bruker500 MHz NMR spectrometer at an ambient temperatureof 300 K. Here, we demonstrate the optimal search algo-rithm for searching one and two marked items among fourunsorted items. As described in the circuit in Fig. 2, theexperiment mainly involves three stages: (i) preparationof |000〉 state, (ii) iterations of the generalized Grover op-eration, and (iii) measurement of the final probabilitiesusing the ancilla qubit. In the following each of the abovestages is discussed in detail.

Preparation of the initial state by nonunitary control:The equilibrium state of the 3-qubit register is repre-sented by the density matrix ρeq = (1+ εHIzH + εFIzF +εCIzC)/8, where εi ∼ 10−5 are called the purity factors.

In NMR quantum information processing, one typicallyrealizes a pseudopure state (PPS) that is isomorphic toa pure state using a combination of unitary and nonuni-tary processes [24]. Fig. 3 shows the BB-sequence con-sisting of both RF pulses and twirling operations auto-matically generated for the 3-qubit initialization by max-imizing the state-to-state fidelity Fs. In the experiment,the twirling process is easily achieved using three pulsed-field-gradients (PFGs) whose time instants are obtainedby BB optimization scheme. Fidelity of the experimentalresults of |000〉 PPS preparation shown in Fig. 3 (c-e) isestimated to be 0.998 ± 0.001. Thus by assigning genesfor the locations of the twirling operations along withother genes representing the RF pulses and delays, onecan realize a nonunitary operation.

As shown in the circuit of Fig. 2, we then pre-pare the uniform superposition state |ψQ〉 by applyinga Hadamard gate on each of the system qubits.Generalized Grover iteration by unitary control: The

circuit for the generalized Grover iteration in Fig. 2 isrealized by a single BB sequence with an average fidelityof about 0.98 over 10% RF inhomogeneity. The durationsof the BB sequences for l = 1, 2, . . . , 10 varied in therange 40 ms to 220 ms, and consisted of 8,000 to 44,000segments, each segment being 5µs long. While such longsequences implementing multiple control operations arehard to realize by other control methods, the present BBscheme exploits the very low duty-cycle and the one-timeevaluation of basis propagators to efficiently compute theoverall unitaries. We performed the experiments for oneand two marked items by systematically increasing thenumber of iterations (l) in each case.

Extracting the solutions by measuring ancilla: In order

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10

PL

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 910

PL

= 0

= 5

=10

= 0

= 5

=10

(a) (b)

FIG. 4. The theoretical and experimental results for the prob-ability PL of finding (a) one (|10〉) and (b) two (|10〉 and |11〉)marked states among four items versus the number of itera-tions l. The red and blue points respectively represent thetheoretically predicted and experimentally obtained probabil-ities measured directly from the ancilla (13C) spectra (shownin insets).

4

to measure the final probabilities of finding the markedstates, we first destroy the coherences by applying atwirling operation with the help of a strong PFG (Fig.2). The relative probabilities of the system qubits beingin various eigenstates are encoded in the correspondingtransitions of the ancilla qubit. Therefore we finally mea-sure the ancilla after applying a Hadamard operator, asshown in Fig. 2.

The experimental results of the probabilities versus it-eration number for one and two marked states are shownin Fig. 4(a) and (b) respectively. While the theoreticallower bounds for the probabilities of finding the markedstates are 0.8, the experimental lower bounds were 0.69and 0.64 for one and two marked states respectively. Thelower values of experimental probabilities are mainly dueto pulse imperfections and decoherence. In spite of thelower probabilities, the marked states can be clearly iden-tified from the spectra. It is clear that as the number ofiterations increases, each final state remains close to thesolution state, thus exhibiting the fixed point behavior.

Preparation of PPS in a 5-qubit system: To demon-strate the potential of the BB scheme in a larger quan-tum register, we carried out preparation of pseudopurestate in a five-qubit system consisting of three 19F andtwo 1H spins of 1-bromo-2,4,5-trifluorobenzene (see inset

H4 |0000

H5 |0000

F1 |0000

F2 |0000

F3 |0000

6047 276 115 54 1545 -3680 -25 105 1261 -6744 1522 55 50 -8 29

F1 F2 F3 H4 H5

F1

F2

F3

H4

H5

F1 H5

F2

F3 H4

Br

Frequency (Hz) Frequency (Hz)

00

00

0

00

10

0

01

00

1

01

11

0

10

01

1

11

00

0

11

10

1

(a) (b)

(c)

(d)

(e)

(f)

(g)

FIG. 5. (a) Molecular structure of the five-qubit system 1-bromo-2,4,5-trifluorobenzene and its Hamiltonian parameterswherein diagonal and off-diagonal numbers represent chem-ical shifts and J-couplings (in Hz), (b-f) the spectra corre-sponding to the thermal equilibrium state (blue) and |00000〉PPS prepared from the nonunitary BB sequence (red), and(g) bar diagram representing theoretical (red) and experimen-tal (blue) diagonal elements of the traceless deviation densitymatrix corresponding to |00000〉 PPS.

of Fig. 5) partially oriented in a liquid crystal [25]. Thenonunitary transformation was realized using a single BBsequence of 89.1 ms duration involving four twirling op-erations. The normal and PPS spectra of each spin arealso shown in Fig. 5. The overall fidelity of the PPSpreparation was estimated to be 0.96 ± 0.02. The lowerfidelity compared to the 3-qubit register is not only due tothe increased complexity, but also due to the temporalfluctuations in the dipolar interaction strengths of the5-qubit system. Nevertheless, the automatic procedurefor generating the nonunitary transformation promisesapplications in a variety of physical implementations in-volving quantum control.

Conclusions: Bang-bang pulses are efficient to com-pute, easier to implement, and are robust against pulseerrors. Using a nonlocal optimization algorithm, suchas the genetic algorithm, it is possible to efficiently op-timize the BB sequence to generate any desired unitarytransformation or to prepare a quantum state. For ex-ample, in the case of NMR, long control sequences areoften needed to apply nonlocal transformations. It is un-necessary to apply RF pulses throughout the control se-quence, which is not only hard to compute but also proneto RF inhomogeneity errors. The BB control techniqueallowed us to carry-out the first experimental demonstra-tion of optimal fixed point quantum search algorithm ona three-qubit NMR register. The experimental spectraeasily identified the marked states out of a database offour unsorted items. The entire generalized Grover iter-ate, encompassing several local and nonlocal gates, hasbeen realized using a single BB-sequence lasting morethan 200 ms, involving over 40,000 segments. Such longcontrol sequences are hard to compute by other optimalcontrol techniques. Combined with twirling operationsit is also possible to realize nonunitary transformations.Using this scheme, we have demonstrated the synthesisand experimental implementation of pseudopure statesin 3- as well as 5-qubit NMR quantum registers. BBsequences can be realized by on and off switching of thecontrol field and therefore it is also applicable to a varietyof other architectures where smooth amplitude modula-tion of control fields is difficult. It may also be possibleto realize a hybrid control sequence by combining the BBscheme with other optimal control techniques. Althoughthe present experimental demonstration is in an NMRsystem, the simplicity of the BB scheme allows its ap-plication in other architectures such as SQUID or NV−

center-based quantum registers.

Acknowledgments: We acknowledge useful discussionswith Dr. Abhishek Shukla, Swathi Hegde, Sudheer Ku-mar, and Deepak Khurana. We also thank Prof. AnilKumar, IISc, Bangalore for providing us the three-qubitNMR register.

5

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