Optimal Qubit Control Using Single-Flux Quantum Pulses
OOptimal Qubit Control Using Single-Flux Quantum Pulses
Per J. Liebermann ∗ and Frank K. Wilhelm Theoretical Physics, Saarland University, Campus E2 6, 66123 Saarbrücken, Germany
Single flux quantum pulses are a natural candidate for on-chip control of superconducting qubits.We show that they can drive high-fidelity single-qubit rotations—even in leaky transmon qubits—if the pulse sequence is suitably optimized. We achieve this objective by showing that, for theserestricted all-digital pulses, genetic algorithms can be made to converge to arbitrarily low error,verified up to a reduction in gate error by 2 orders of magnitude compared to an evenly spacedpulse train. Timing jitter of the pulses is taken into account, exploring the robustness of ouroptimized sequence. This approach takes us one step further towards on-chip qubit controls.
I. INTRODUCTION
Rapid single flux quantum (RSFQ) technology hasbeen and is originally pursued as an ultra-high-speedclassical computing platform [1, 2]. The ability to gener-ate reproducible identical pulses at a high clock rate hasbeen demonstrated in integrated circuits [3]. Next to itsoriginal motivation of ultrafast digital circuits, this abil-ity makes RSFQ technology a highly viable candidate forthe on-chip generation of control pulses and readout forquantum computers based on Josephson devices [4–9].The switching time lies in the picosecond range, lead-ing to fast quantum gates [10], but the timing of thesingle-voltage pulses to control the devices is a majorchallenge [11]. The integration capabilities of RSFQ tech-nology are compatible with scaling quantum processors[12, 13]; for example, one can load pulse sequences intoshift registers [14]. Then again, the present-day controlscheme is based on room-temperature electronics, whosesignals are transmitted as analog signals through filtersinto a cryostat, which creates large physical overhead.Besides these engineering considerations, both tech-niques present distinct paradigms for control design.Room-temperature generators have high-amplitude res-olution (currently, 12 bits are typical) at limited speed;thus, methods of analog pulse shaping can be applied [15]to approximating the experiment with high accuracy.This approximation has been done in superconductingqubits, e.g., within the derivative removal by adiabaticgate (DRAG) and wah-wah method [16, 17], that sup-press leakage into higher energy levels. These pulsescan be readily calibrated using a protocol named Ad-HOC [18].On the other hand, SFQ pulses only have a single bitof amplitude resolution: in a given time interval, thereis either a pulse or not. The RSFQ sequence proposedin Ref. [12] again reveals the challenge of balancing gatespeed with suppression of leakage in transmon qubits sim-ilar to DRAG, demonstrating the need for advanced pulsedesign methods which cannot be solved by evenly spacedpulses alone. While their amplitude resolution is mini-mal, an RSFQ sequence has a very short time constant of ∗ [email protected] A m p li t ud e u [ r a d / n s ] t [ps] 2200 FIG. 1. (Color online) Time slicing of the evolution: in eachframe, the control amplitude u ( t ) either has a Gaussian shape(1) or vanishes (0). Therefore, only two unitary operators U i need to be calculated, each with a duration of t c (here, 10 ps).The gap between two consecutive pulses is always an integermultiple of t c . the pulse train—much shorter than the intrinsic frequen-cies of the qubit system—that can be expected to simu-late analog control. To explore this matter, one needs todepart from the conventional optimal-control paradigmsas reviewed in Ref. [15], which often involve gradients,and focus instead on digital control and algorithms thatcan optimize these fully discrete controls.In this paper we show that optimal-control methods,based on genetic algorithms adapted to digital control,can be used to improve gate operations with trains ofSFQ pulses. The single bit of amplitude resolution is en-coded in a discretized time evolution, which also limitspossible clock frequencies. We show that uneven and rel-atively densely populated bit-string pulse sequences canbe used to suppress leakage as well as to drastically re-duce gate times up to the speed limit. We also verifythat optimized pulse sequences are robust against timingjitter, the degree of which depends on the clock integra-tion. II. SFQ CONTROL
As a simple model, we use a qubit with a leakage levelsuch as the transmon qubit [19, 20] that can be con-trolled with voltage pulses. Our single-qubit target gateis a rotation around the y axis, i.e., a Pauli- Y gate withan arbitrary global phase. The time-dependent system a r X i v : . [ qu a n t - ph ] S e p t gate [ns] FIG. 2. Bit-string representation of the pulse sequence before (top) and after (bottom) the optimization for a gate time of20 ns. A vertical black line indicates the time when a pulse is applied, and white space the duration of the free evolution.
Hamiltonian in units of (cid:126) reads H ( t ) = H + u ( t ) H , (1)with drift and control terms H = ω
00 0 2 ω + ∆ , H = − i i − i √ i √ , (2)where ω is the qubit’s angular frequency and ∆ the an-harmonicity of the third level. Gate operations are per-formed by changing the control amplitude u ( t ) . However,instead of finding a proper pulse shape with a duration ofseveral nanoseconds, we apply a single-picosecond pulserepeatedly. This pulse is switched on and off as stored ina shift register and controlled by a clock. The pulse shapewe use is a truncated Gaussian with standard deviation τ and total duration t c u ( t ) = δθ √ πτ e − t / τ (cid:90) t c − t c u ( t ) dt ≈ δθ . (3)The area δθ is approximated by the numerical integrationof the pulse shape. It matches the rotation angle on theBloch sphere for an infinitely narrow δ pulse. To flip theBloch vector around the y axis, we can thus set a lowerbound of n = π/δθ pulses. Note that the three-levelmodel outlined in Eq. (2) is sufficient to describe thequbit dynamics — we perform a posteriori verificationswith more levels and find no discernible difference.We simulate our system by slicing the total gate timeinto N time steps of length t c , i.e., the clock period.Choosing τ accordingly ensures that the applied voltagepulse shape u ( t ) vanishes at the beginning and the end ofa the time interval. We interpret the free evolution of thesystem in a time interval [ t i , t i + 2 t c ] as a pulse with zeroamplitude. Figure 1 shows an example of two consec-utive pulses followed by an interval without an appliedpulse. The time difference between two applied pulsesis always a positive integer multiple of the pixel length t c . For each interval [ t i , t i + 2 t c ] , the time evolutionis captured in a unitary matrix U ( t i ) ≡ U ( t i + 2 t c , t i ) .Since we apply a single pulse only if it is necessary, eachunitary is chosen out of a given database containing justtwo unitary operators, U and U . For both our workand future practical applications, this can be done effi-ciently by storing all of the relevant system parameters in this database. Therefore, the pulse sequence can berepresented as bit string, where zeros represent a free-evolution interval U and ones an applied voltage pulse U ; see Fig. 2. The total time evolution reads U ( t gate ) = (cid:89) i = N − U ( t i ) = U ( t N − ) · · · U ( t ) . (4)The database that the unitaries are chosen from consistsof U = exp ( − it c H ) = U (2 t c ) (5) U = U ( t c ) (cid:20) T exp (cid:18) − i (cid:90) t c − t c H ( t ) dt (cid:19)(cid:21) U ( − t c ) . (6)Any adjustment in the experiment can be captured inthe database and all characterization of the pulses has tobe done once in order to find these two database entries.The target gate is a Pauli- Y gate, where we allow anarbitrary phase for the leakage level, and the global phaseis neglected when using an appropriate fidelity function[Eq. (8)], U target = − e iϕ . (7)We optimize our time evolution within the computationalsubspace of the qubit. The average fidelity functiontherefore reads [21] Φ = 14 (cid:12)(cid:12)(cid:12) tr (cid:110) U † target P Q U ( t gate ) P Q (cid:111)(cid:12)(cid:12)(cid:12) , (8)with the projector onto the qubit subspace P Q .Typical values for a transmon qubit [19] are for thequbit transition frequency ω/ π = 5 GHz and its an-harmonicity ∆ / π = − MHz. The pulse area andduration are set to δθ = π/ and t c = 10 ps, respec-tively, limiting the clock frequency to 100 GHz. The gatetime chosen is t g = 20 ns, leading to a total number of N = 2000 pixels.As a starting point, we use the sequence presented inRefs. [12, 22], so we have exactly n = 100 pulses in thebeginning. After each pulse we wait t wait = 2 π/ω − t c for the qubit to complete a full precession before applyinganother pulse. This scheme increases the gate fidelities P o pu l a t i o n Time t [ns] P ( t ) P ( t ) P ( t ) P ( t )10 × P ( t )10 × P ( t ) P o pu l a t i o n Time t [ns] P ( t ) P ( t ) P ( t ) P ( t )10 × P ( t )10 × P ( t ) FIG. 3. Populations of a qutrit starting in the ground (toppanel) and excited (bottom panel) state for the optimizedsequence (solid lines) and the initial sequence (dashed lines).The population of the leakage level is enhanced by a factor of10 for better legibility. The algorithm suppresses leakage tothe ending gate time. with an increasing number of applied pulses, while de-creasing the area underneath the pulse shape with thesame rate. It also leads to longer gate times, however,because every additional pulse increases the gate time by π/ω due to the waiting time. III. GENETIC ALGORITHMS
As gradient-based algorithms are not straightforwardhere, we use a genetic algorithm [23, 24] to optimize thepulse sequence. This versatile tool for global optimiza-tion is a natural candidate for the problem at hand, sinceour pulse sequence is already encoded in a binary string.Other genetic algorithms have been applied successfullyin analog quantum optimal control, e.g., to optimize laserpulses to control molecules [25].Here, we search for a local minimum in the con-trol landscape starting with the sequence described ear-lier [12, 22], and stop as soon as we reach the targetfidelity. Within the genetic algorithm framework, ev-ery solution for the variational parameters of the con-trol problem is encoded in a genome. At each iteration,a selection of genomes will be merged by a crossover ofgenome pairs. The new and old genomes are mutatedand the genomes with the best fitness make it to the nextgeneration. Finding the right parameters for the geneticalgorithm can be difficult, but it is common practice for − − − G a t ee rr o r − Φ Iteration numberBest fitness errorAverage fitness error
FIG. 4. Starting with a sequence shown in Ref. [12, 22], ge-netic algorithms are used for optimization. At each iteration,a new population of different pulse sequences is created fromtheir parents, while the best solution of a generation with thelowest fidelity error is kept until either the algorithm finds abetter one or a given threshold is exceeded. most optimization problems to choose a high crossoverand a low mutation probability [26]. The parameters ofour optimization are shown in Table I.Using that algorithm and setting our gate fidelity as afitness measure we found the solutions for the sequenceshown in Fig. 2 and for the populations shown in Fig. 3.The algorithm mainly corrects for leakage into the thirdenergy level, which always leads to an increase in thenumber of pulses—in the solutions presented here, from n = 100 to n = 301 . The time taken for a run is about150 s and the improvement is shown in Fig. 4. We en-counter a broad variation over different runs indicatingthe presence of an abundance of local traps. This isconsistent with the common observation that, in princi-ple, trap-free control landscape [27] develops traps whenthe space of the available pulse shapes is strongly con-strained. Population size 70Mutation probability 0.001Crossover probability 0.9Number of genomes to select for mating 64Maximum allowed iterations 200 000Target fitness 0.9999Elitism 1TABLE I. Parameters used in the genetic algorithm. SeeRef. [23] for background. − − − − G a t ee rr o r − Φ Gate time t gate [ns] t gate = 6 ns FIG. 5. The gate errors of the optimizations for different gatetimes. The widths of the pulses have been kept constant;i.e., the number of pixels changes with the gate time linearly.The horizontal red dash-dotted line is the fidelity error of theinitial sequence for 20 ns, and the horizontal black dashedline indicates the stopping condition of the algorithm when afidelity > . has been reached. IV. QUANTUM SPEED LIMIT
With the genetic algorithm at hand, we can searchfor shorter gate times t gate for the Pauli- Y gate. Wekeep the widths of the pulses constant, which decreasesthe number of pixels with a decreasing gate time. Thereachable fidelities are shown in Fig. 5. The shortestpossible gate time we could find within the genetic algo-rithm is t gate = 6 ns. Each optimization stops if a fidelity > . is reached or the maximum number of iterations(200 000) is exceeded. We point out that, for short gatetimes, the evenly spaced pulse sequence is no longer aviable solution if we do not have any control of the pulseamplitude. V. TIMING ERRORS
So far the clock has been assumed to be a perfect one.Here, we take inevitable timing errors into account, thatlead to small pulse delays [28] and, therefore, to devia-tions from the optimal fidelity. We simulate timing errorsby multiplying every applied pulse U in the optimizedsequence with a free-evolution operator of a time interval δt from the right, and its adjoint from the left. There-fore, a positive δt indicates a pulse which arrives with adelay, and a negative δt indicates that the pulse arrivesearlier: U (cid:48) = U ( − δt ) U U ( δt ) . (9) δt is a normal distributed random time with standarddeviation σ for external jitter and √ kσ for internal jitter,where k is the applied pulse number. For each value of σ , − − − − − − G a t ee rr o r − Φ Standard deviation σ [ps]External clockInternal clock FIG. 6. The gate error of the optimized sequence for tim-ing jitters with constant variance (external clock) and lineargrowing variance (internal clock). The gate errors have beenaveraged over 1000 runs of the time evolution for each valueof σ . the fidelity of the time evolution has been averaged over1000 runs for the optimized sequence. As can be obtainedfrom Fig. 6, the external clocking scheme is more robustby an order of magnitude of the standard deviation. It isstill within the target fidelity when the jitter time scaleis 10 % of the pulse width, while, for an internal clock itis around 1 %. If the jitter time reaches the pulse dura-tion, the gate error is still on the same scale as where westarted our optimization. We therefore conclude that anexternal clock should be used in favor of an internal onefor future devices. VI. CONCLUSION
We successfully develop and apply an optimal-controlmethod for pulses with only a single bit of amplituderesolution. Finding the right binary string leads tominimization of the leakage error in the transmonsystem, and thus gate-control precision compatible withthe requirements of fault-tolerant quantum computing.The results presented here show a fidelity improvementof several orders of magnitude over equal pulse-spacingsequences while being robust under external timingjitter. RSFQ shift registers are needed to performthe optimized sequence and are an essential part ofon-chip SFQ-qubit control. This makes the underlyingSFQ-pulse platform together with the single-bit optimal-control theory a possible and attractive candidate for anintegrated control layer in a quantum processor.
ACKNOWLEDGMENTS
We thank Daniel J. Egger and Oleg A. Mukhanovfor the fruitful discussions. Thiswork is supported byU.S. Army Research Office Grant No. W911NF-15-1-0248. This work is also supported by the EU throughSCALEQIT. [1] K. K. Likharev and V. K. Semenov, “RSFQlogic/memory family: a new Josephson-junctiontechnology for sub-terahertz-clock-frequency digitalsystems,” IEEE Trans. Appl. Supercond. , 3–28 (1991).[2] K. K. Likharev, “Superconductor digital electronics,”Physica C Supercond. , 6–18 (2012).[3] Maria Gabriella Castellano, Fabio Chiarello, RobertoLeoni, Guido Torrioli, Pasquale Carelli, Carlo Cosmelli,Marat Khabipov, Alexander B. Zorin, and Dmitri Bal-ashov, “Rapid single-flux quantum control of the energypotential in a double SQUID qubit circuit,” Supercond.Sci. Technol. , 500 (2007).[4] T. P. Orlando, S. Lloyd, L. S. Levitov, K. K. Berggren,M. J. Feldman, M. F. Bocko, J. E. Mooij, C. J. P. Har-mans, and C. H. van der Wal, “Flux-based superconduct-ing qubits for quantum computation,” Physica C Super-cond. , 194–200 (2002).[5] Marc J. Feldman and Mark F. Bocko, “A realistic exper-iment to demonstrate macroscopic quantum coherence,”Physica C Supercond. , 171–176 (2001).[6] Thomas A. Ohki, Michael Wulf, and Marc J. Feldman,“Low-Jc rapid single flux quantum (RSFQ) qubit con-trol circuit,” IEEE Trans. Appl. Supercond. , 154–157(2007).[7] Vasili K. Semenov and Dmitri V. Averin, “SFQ con-trol circuits for josephson junction qubits,” IEEE Trans.Appl. Supercond. , 960–965 (2003).[8] Arkady Fedorov, Alexander Shnirman, Gerd Schön, andAnna Kidiyarova-Shevchenko, “Reading out the state of aflux qubit by josephson transmission line solitons,” Phys.Rev. B , 224504 (2007).[9] Kirill G. Fedorov, Anastasia V. Shcherbakova, Michael J.Wolf, Detlef Beckmann, and Alexey V. Ustinov, “Fluxonreadout of a superconducting qubit,” Phys. Rev. Lett. , 160502 (2014).[10] Thomas A. Ohki, Michael Wulf, and Mark F. Bocko,“Picosecond on-chip qubit control circuitry,” IEEE Trans.Appl. Supercond. , 837–840 (2005).[11] Kris Gaj, Eby G. Friedman, and Marc J. Feldman, “Tim-ing of multi-gigahertz rapid single flux quantum digi-tal circuits,” J. VLSI Signal Process. Syst. Signal ImageVideo Technol. , 247–276 (1997).[12] R. McDermott and M. G. Vavilov, “Accurate qubit con-trol with single flux quantum pulses,” Phys. Rev. Applied , 014007 (2014).[13] Austin G. Fowler, Matteo Mariantoni, John M. Martinis,and Andrew N. Cleland, “Surface codes: Towards practi-cal large-scale quantum computation,” Phys. Rev. A ,032324 (2012).[14] Oleg A. Mukhanov, “Rapid single flux quantum (RSFQ)shift register family,” IEEE Trans. Appl. Supercond. , 2578–2581 (1993).[15] Steffen J. Glaser, Ugo Boscain, Tommaso Calarco, Chris-tiane P. Koch, Walter Köckenberger, Ronnie Kosloff, IlyaKuprov, Burkard Luy, Sophie Schirmer, Thomas Schulte-Herbrüggen, D. Sugny, and Frank K. Wilhelm, “TrainingSchrödinger’s cat: quantum optimal control,” Eur. Phys.J. D , 279 (2015).[16] F. Motzoi, J. M. Gambetta, P. Rebentrost, and F. K.Wilhelm, “Simple pulses for elimination of leakage inweakly nonlinear qubits,” Phys. Rev. Lett. , 110501(2009).[17] R. Schutjens, F. A. Dagga, D. J. Egger, and F. K. Wil-helm, “Single-qubit gates in frequency-crowded transmonsystems,” Phys. Rev. A , 052330 (2013).[18] D. J. Egger and F. K. Wilhelm, “Adaptive hybrid optimalquantum control for imprecisely characterized systems,”Phys. Rev. Lett. , 240503 (2014).[19] Jens Koch, Terri M. Yu, Jay Gambetta, A. A. Houck,D. I. Schuster, J. Majer, Alexandre Blais, M. H. Devoret,S. M. Girvin, and R. J. Schoelkopf, “Charge-insensitivequbit design derived from the Cooper pair box,” Phys.Rev. A , 042319 (2007).[20] Steven M. Girvin, “Circuit QED: Superconducting qubitscoupled to microwave photons,” in Quantum Machines:Measurement and Control of Engineered Quantum Sys-tems (Oxford University Press, 2011).[21] P. Rebentrost and F. K. Wilhelm, “Optimal control of aleaking qubit,” Phys. Rev. B , 060507 (2009).[22] Geoffrey Bodenhausen, Ray Freeman, and Gareth A.Morris, “A simple pulse sequence for selective excitationin Fourier transform NMR,” J. Magn. Reson. , 171–175 (1976).[23] Darrell Whitley, “A genetic algorithm tutorial,” Stat.Comput. , 65–85 (1994).[24] Patrick Sutton and Sheri Boyden, “Genetic algorithms:A general search procedure,” Am. J. Phys. , 549–552(1994).[25] R. S. Judson and H. Rabitz, “Teaching lasers to controlmolecules,” Phys. Rev. Lett. , 1500–1503 (1992).[26] Alexander K. Hartmann and Heiko Rieger, OptimizationAlgorithms in Physics (Wiley-VCH, Weinheim, 2002).[27] Herschel A. Rabitz, Michael M. Hsieh, and Carey M.Rosenthal, “Quantum optimally controlled transitionlandscapes,” Science , 1998–2001 (2004).[28] A. V. Rylyakov and K. K. Likharev, “Pulse jitter andtiming errors in RSFQ circuits,” IEEE Trans. Appl. Su-percond.9