Speedup For Quantum Optimal Control From Automatic Differentiation .
PHYSICAL REVIEW A 95, 042318 (2017)Speedup for quantum optimal control from automatic differentiationbased on graphics processing unitsNelson Leung,1,* Mohamed Abdelhafez,1 Jens Koch,2 and David Schuster11The James Franck Institute and Department of Physics, University of Chicago, Chicago, Illinois 60637, USA2Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA(Received 20 December 2016; published 13 April 2017)We implement a quantum optimal control algorithm based on automatic differentiation and harness theacceleration afforded by graphics processing units (GPUs). Automatic differentiation allows us to specifyadvanced optimization criteria and incorporate them in the optimization process with ease. We show that theuse of GPUs can speedup calculations by more than an order of magnitude. Our strategy facilitates efficientnumerical simulations on affordable desktop computers and exploration of a host of optimization constraints andsystem parameters relevant to real-life experiments. We demonstrate optimization of quantum evolution basedon fine-grained evaluation of performance at each intermediate time step, thus enabling more intricate control onthe evolution path, suppression of departures from the truncated model subspace, as well as minimization of thephysical time needed to perform high-fidelity state preparation and unitary gates.DOI: 10.1103/PhysRevA.95.042318I. INTRODUCTIONThe techniques and algorithms used to optimize the controlof quantum systems [1–17] and those underlying the fieldof deep neural networks [18,19] share a number of commonelements. Both areas heavily use linear algebra operationscombined with gradient descent optimization. Thus, advancedhardware and software technology recently emerging from therapid development of machine learning also paves the way fora significant boost of optimal quantum control techniques.A crucial factor for recent impressive progress in machinelearning has been the leveraging of massive parallelism nativeto graphics processing units (GPUs) [20–24]. Similarly, GPUshave been used to accelerate computations in many areasof quantum physics and chemistry [25–31]. Specifically,GPUs are extremely efficient in multiplying very large matrices [32,33]. Such multiplications also form a central stepin the simulation and optimal control of quantum systems.Exploiting this advantageous feature of GPUs, we achievesignificant speed improvements in optimizing control schemesfor systems at the current frontiers of experimental quantumcomputation. As the number of qubits in these experimentsis increasing [34–36], it becomes increasingly important totake advantage of optimal control techniques. Moreover,recent advances in commercially available electronics—e.g.,arbitrary waveform generators enabling base-band synthesis ofthe entire microwave spectrum [37]—afford new capabilitieswhich quantum optimal control is uniquely well suited toharness.There have been numerous theoretical developments ofnumerical and analytical methods for quantum optimal control(see Ref. [1] for a recent review). The algorithms involved arepredominantly based on gradient methods, such as realizedin gradient ascent pulse engineering (GRAPE) [2,3], Krotovalgorithms [4–10], or rapid monotonically convergent algorithms [11–17], and are available in several open-source packages, including QuTiP [38,39], DYNAMO [40], Spinach [41], 42318(14)SIMPSON [42]. Quantum optimal control has been remarkablysuccessful in determining optimized pulse sequences [43], designing high-fidelity quantum gates [7,44–53], and preparingentangled states [54–58].Optimal control is a versatile concept which can be appliedto a vast variety of quantum systems. Typically, there is aprimary goal (e.g., maximizing fidelity to a target state or unitary), as well as additional constraints and costs associated withspecific experimental systems. Examples of such constraintsinclude fixed maximum amplitudes of control pulses [59,60],maximum overall power of control signals [61], and limitedtime resolution of arbitrary waveform generator [62]. Further,finite coherence of quantum systems motivates minimizingthe overall time needed for reaching the intended state orunitary (time-optimal control) [63]. In certain cases, steeringthe quantum system among an optimal path (time-dependenttarget) may be desired [64]. Incorporating new constraintsin the optimization process often requires the analyticalderivation and implementation of additional contributions tothe gradient calculation, and may necessitate significant effortto deploy on large computer clusters. This issue can greatlyimpede the ability to quickly develop control strategies for newproblems.To overcome these obstacles, we have implemented aquantum optimal control scheme that incorporates constraintsvia automatic differentiation [65,66] and utilizes GPUs forboosting computational efficiency. Specifically, automaticdifferentiation handles the updating of gradient calculations inthe backward-propagation algorithm [19], and thus eliminatesthe need to hard code additional gradient contributions fromconstraints. For the actual optimal control applications wepresent in this paper, we find that the computational speedupfrom utilizing GPUs becomes significant for Hilbert-spacesizes exceeding dimensions of the order of one hundred;see Fig. 3. Together, these features allow a quick turnaroundfor varying optimization constraints and system parameters,rendering this approach invaluable for the study of quantumoptimal control. In this paper, we describe the implementationof automatic differentiation, demonstrate its application toquantum optimal control of example systems relevant to042318-1 2017 American Physical Society
LEUNG, ABDELHAFEZ, KOCH, AND SCHUSTERPHYSICAL REVIEW A 95, 042318 (2017)TABLE I. Relevant contributions to cost functions for quantumoptimal control. Names of contributions indicate the quantity to beminimized.μCost-function contributionCμ (u)1Target-gate infidelity1 tr(KT† KN )/D 22Target-state infidelity1 T N 23Control amplitudes4Control variations5j,k6Evolution time (target gate) 7Evolution time (target state) uk,j uk,j 1 2j1 A. Important types of cost-function contributions u 2 Occupation of forbidden state1N1 1N F j 2†2j tr(KT Kj )/D 2j T j quantum computing and quantum optics, and discuss theperformance gains achieved by utilizing GPUs.II. THEORYWe briefly review the essential idea of quantum optimalcontrol and introduce the notation used throughout our paper.We consider the general setting of a quantum system withintrinsic Hamiltonian H0 and a set of external control fields{u1 (t), . . . ,uM (t)} acting on the system via control operators{H1 , . . . ,HM }. The resulting system Hamiltonian is given by H (t) H0 Mk 1 uk (t)Hk . Optimal control theory aimsto minimize deviations from a target state or target unitaryby appropriate adjustments of the control fields uk (t). Toimplement this optimization, the time interval of interest isdiscretized into a large number N of sufficiently small timesteps δt. Denoting intermediate times by tj t0 j δt, theHamiltonian at time tj takes on the formHj H0 M uk,j Hk .(1)k 1The control fields subject to optimization now form a set {uk,j }of d MN real numbers.The quantum evolution from the initial time t t0 to timetj is described by a propagator Kj , decomposed according toKj Uj Uj 1 Uj 2 . . . U1 U0 ,(2)Uj exp( iHj δt)(3)whereis the propagator for the short time interval [tj ,tj δt]. (Hereand in the following, we set h̄ 1.) Evolution of a select initialstate 0 from t t0 to t tj then takes the usual form, j Kj 0 .cost-function contributions used for quantum optimal control.The total cost functionis a linear combination of these cost functions, C μ αμ Cμ . The weight factors αμ must bedetermined empirically and depend on the specific problemand experimental realization at hand. In the following, wediscuss these relevant cost-function contributions.(4)In the decomposition of Kj , each short-time propagator Ui canbe evaluated exactly by matrix exponentiation or approximatedby an appropriate series expansion. Propagation methodswhich go beyond the piecewise-constant approximation forthe propagation can further improve speed and accuracy [48].Optimization of the discretized control fields u Rd can beformulated as the minimization of a cost function C(u) whereC : Rd R . Table I shows some of the most importantThe first cost contribution C1 (u) is the primary tool forrealizing a target unitary KT , such as a single- or multiqubitgate. Cost is incurred for deviations between the target unitaryand the realized unitary KN at a given final time tN . For asystem with Hilbert-space dimension D, its expression 1 † tr(KT KN )/D 2 [2] represents the infidelity obtained fromthe trace distance between the target unitary and the realizedunitary. Minimizing this cost function is the principle goal ofthe quantum control problem.The second cost function C2 (u) 1 T N 2 measures the distance between a desired target state T andthe state N realized at the final time tN , as obtained fromevolution of a given initial state 0 . In addition, generalizingC2 to multiple initial and target states is useful for performinga unitary KT which is only defined on some subspace HSof the modeled Hilbert space. Such restriction to a selectedsubspace is of practical importance whenever a desired unitaryis to be implemented within some computational subspaceonly, as is common for quantum computation applications.There, evolution of higher excited states or auxiliary systemsoutside the computational subspace is immaterial. Optimalcontrol, then, can be achieved by simultaneous evolution of aset of initial states { 0s } (s 1,2, . . . ,S) that forms a basis ofHS . Optimal control fields are obtained from minimizing thecomposite state infidelity C2 (u) 1 S1 s Ts PS Ns 2relative to the desired target states Ts KT 0s . (Here, PSis the projector onto subspace HS .)This composite state-transfer cost function when used overa complete basis is equivalent to the gate fidelity, but hasseveral advantages. Most importantly it is more memoryefficient requiring only the current state to be stored ratherthan the whole unitary. In addition, it is very amenable todistributed computing approaches. However, when the unitarytransfer matrix can be stored in memory, propagating the fullunitary can take advantage of the parallelism of the GPU forsmaller problems (see Fig. 3).Like many optimization problems, quantum optimal controlis typically underconstrained. In order to obtain control fieldsthat are consistent with specific experimental capabilities andlimitations, it is often crucial to add further constraints onthe optimization. Control fields must be realizable in thelaboratory, should be robust to noise, and avoid large controlamplitudes and rapid variations based on signal output specifications of instruments employed in experiments. Exceedinglystrong control fields may also be problematic due to heatdissipation which may, for instance, raise the temperatureinside a dilution refrigerator. These points motivate theconsideration of additional cost-function contributions in thefollowing.One such contribution C3 (u) u 2 suppresses largecontrol-field amplitudes globally and is commonly employed042318-2
SPEEDUP FOR QUANTUM OPTIMAL CONTROL FROM . . .PHYSICAL REVIEW A 95, 042318 (2017)in quantum optimal control studies [2,59,60,67]. (The generalization to more fine-grained suppression of individual controlfields is straightforward to implement as well.) Penalizingthe L2 norm of the control fields favors solutions with lowamplitudes. It also tends to spread relevant control fieldsover the entire allowed time window. While C3 constitutes a“soft” penalty on control-field amplitudes, one may also applya trigonometric mapping to the amplitudes to effect a hardconstraint strictly enforcing fixed maximum amplitudes [68]. The fourth type of2contribution to the cost function, C4 (u) j,k uk,j uk,j 1 , penalizes rapid variations of controlfields by suppressing their (discretized) time derivatives [67].The resulting smoothening of signals is of paramount practicalimportance, since any instrument generating a control field hasa finite impulse response. If needed, contributions analogousto C4 which suppress higher derivatives or other aspects ofthe time dependence of fields can be constructed. Together,limiting the control amplitudes and their time variation filtersout high-frequency “noise” from control fields, which is anotherwise common result of less-constrained optimization.Smoother control fields also have the advantage that essentialcontrol patterns can potentially be recognized and given ameaningful interpretation. The contribution C5 (u) j F j 2 to the cost function has the effect of suppressing occupation of a select “forbidden” state F (or a set of such states, upon summation)throughout the evolution. The inclusion of this contributionaddresses an important issue ubiquitous for systems withHilbert spaces of large or infinite dimension. In this situation,truncation of Hilbert space is needed or inevitable due tocomputer memory limitations. (Note that this need even arisesfor a single harmonic oscillator.) Whenever the evolutiongenerated by optimal control algorithms explores highlyexcited states, truncation introduces a false nonlinearity whichcan misguide the optimization. Including additional statescan, in principle, mitigate this problem, but is generallycomputationally very expensive. An independent physicsmotivation for avoiding occupation of highly excited statesconsists of spontaneous relaxation in realistic systems: highenergy states are often more lossy (as is usually the case,e.g., for superconducting qubits) and possibly more difficultto model. Active penalization of such states therefore hasthe twofold benefit of keeping Hilbert-space size at bay andreducing unwanted fidelity loss from increased relaxation. Toaddress these challenges, we employ an intermediate-time costfunction [64,69]: the cost function C5 limits leakage to higherstates during the entire evolution and at the same time preventsoptimization from being misinformed by artificial nonlinearitydue to truncation. We note that the efficacy of this strategy issystem dependent: it works well, for example, for harmonicoscillators or transmon qubits [70] which have strong selectionrules against direct transitions to more distant states, but maybe less effective in systems such as the fluxonium circuit [71]where low-lying states have direct matrix elements to manyhigher states.Customarily,algorithms minimizing the cost function C μ αμ Cμ for a given evolution time interval [t0 ,tN ]aim to match the desired target unitary or target state atthe very end of this time interval. To avoid detrimentaleffects from decoherence processes during the evolution, itis often beneficial to additionally minimize the gate duration(or state preparation) time t tN t0 itself. Instead ofrunning the algorithms multiple times for a set of different t, we employ cost-function contributions of the form †C6 (u) 1 N1 j tr(KT Kj )/D 2 for a target unitary or C7 (u) 1 N1 j T j 2 for a target state, respectively.These expressions penalize deviations from the target gate ortarget state not only at the final time tN , but at every timestep. This contribution to the overall cost function thereforeguides the evolution towards a desired unitary or state in asshort a time as possible under the conditions set by the otherconstraints, and thus results in a time-optimal gate.We will demonstrate the utility of these cost-functioncontributions in the context of quantum information processingin Sec. V. The versatility of automatic differentiation allowsstraightforward extension to other contexts such as optimization of quantum observables.B. Gradient evaluation The weighted sum of cost functions, C μ αμ Cμ , canbe minimized through a variety of gradient-based algorithms.Such algorithms are a very popular means of optimizationthanks to their good performance and effectiveness in findingoptimized solutions for a wide range of problems. At themost basic level, gradient-based algorithms minimize the costfunction C(u) by the method of steepest descent, updating thecontrols u in the opposite direction of the local cost-functiongradient u C(u):u u η u C(u).(5)The choice of the step size η for the control field parametersu plays an important role for the convergence properties ofthe algorithm. A number of schemes exist which adaptivelydetermine an appropriate step size η in each iteration ofthe minimization algorithm. Our implementation supportssecond-order methods such as L-BFGS-B [73] as well asgradient descent methods developed for machine learning suchas ADAM [74].For the evaluation of the gradient u C, we make useof automatic differentiation [65,66] in reverse-accumulationmode. In brief, this algorithm utilizes the decomposition of themultivariable cost function C(u) into its computational graphof elementary operations (addition, matrix multiplications,trace, etc.), each of which has a known derivative. In reverseaccumulation mode, all partial derivatives of C are evaluated ina recursion from the top level (C) back towards the outermostbranches (variables u)—rather similar to the procedure ofobtaining a derivative with pencil and paper. For instance,for the simple function shown in Fig. 1 C(u) sin(u1 ) u1 u2 f [sin(u1 ), f (u1 , u2 )],one obtains all partial derivatives by a recursion starting withthe evaluation of C D1 f sin D2 f f · · · . uj uj ujHere, Dj f stands for the derivative of a multivariablefunction f with respect to its j th argument; square brackets042318-3
LEUNG, ABDELHAFEZ, KOCH, AND SCHUSTERPHYSICAL REVIEW A 95, 042318 (2017)forward (evolution)C(u)u1,1 u2,1 u3,1 sin H1 H0 ·u1 H2 H1 H3FIG. 1. Sample computational graph for automatic differentiation. Automatic differentiation utilizes the decomposition of themultivariable cost function C(u) into its computational graph ofelementary operations, each of which has a known derivative. Inreverse-accumulation mode, all partial derivatives of C are evaluatedin a recursion from the top level (C) back towards the outermostbranches (variables u).denote subsequent numerical evaluation of the enclosed term.(Function arguments are suppressed for brevity.)Automatic differentiation has become a central tool inmachine learning [75] and equally applies to the problem ofoptimal control of quantum systems. In this approach, thegradient of a set of elementary operations is defined and morecomplex functions are built as a graph of these operations. Thevalue of the function is computed by traversing the graphfrom inputs to the output, while the gradient is computedby traversing the graph in reverse via the gradients. Thismethodology gives the same numerical accuracy and stabilityof analytic gradients without requiring one to derive andimplement analytical gradients specific to each new trial costfunction.All cost functions summarized in Table I can be conveniently expressed in terms of common linear-algebra operations. Figure 2 shows the network graph of operations in oursoftware implementation, realizing quantum optimal controlwith reverse-mode automatic differentiation. For simplicity,the graph only shows the calculation of the cost functions C2and C5 . The cost-function contributions C1 ,C6 , and C7 aretreated in a similar manner. The suppression of large controlamplitudes or rapid variations, achieved by C3 and C4 , issimple to include since the calculation of these cost-functioncontributions is based on the control signals themselves anddoes not involve the time-evolved state or unitary. The hostof steps for gradient evaluation is based on basic matrixoperations such as summation and multiplication.Reverse-mode automatic differentiation [19] provides anefficient way to carry out time evolution and cost-functionevaluation by one forward sweep through the computationalgraph, and calculation of the full gradient by one backwardsweep. In contrast to forward accumulation, each derivative isevaluated only once, thus enhancing computational efficiency.The idea of backward propagation is directly related to theGRAPE algorithm for quantum optimal control pioneered byKhaneja and co-workers [2]; see the Appendix. While theoriginal GRAPE algorithm bases minimization exclusively on H2H0 H3ΨT C2e iδtΨ0,Cu1,2 u2,2 u3,2e iδtu2backward (gradient)Ψ1ΨFΨ2C5ΨFC5 0ΨN e iδtAA BABBAIABA e iδtIBAΨFC2 1 iδte iδtA z 2ΨF ΨjΨjC5ΨF2 Re(z) Ψ F 2Re(z) 2C5 2 Im(z)Im(z) 2FIG. 2. Computational network graph for quantum optimal control. Circular nodes in the graph depict elementary operations withknown derivatives (matrix multiplication, addition, matrix exponential, trace, inner product, and squared absolute value). Backwardpropagation for matrices proceeds by matrix multiplication or, wherespecified, by the Hadamard product . In the forward direction,starting from a set of control parameters uk,j , the computationalgraph effects time evolution of a quantum state or unitary, and thesimultaneous computation of the cost function C. The subsequent“backward propagation” extracts the gradient u C(u) with respectto all control fields by reverse-mode automatic differentiation. Thisalgorithm is directly supported by TensorFlow [72], once such acomputational network is specified.the fidelity of the final evolved unitary or state, advanced costfunctions (such as C5 through C7 ) require the summation ofcost contributions from each intermediate step during timeevolution of the system. Such cost functions go beyondthe usual GRAPE algorithm, but can be included in themore general backward propagation scheme described above.[The Appendix shows analytical forms for gradients for costfunctions that are based on time evolution ({C1 ,C2 ,C5 }).]042318-4
SPEEDUP FOR QUANTUM OPTIMAL CONTROL FROM . . .PHYSICAL REVIEW A 95, 042318 (2017)III. IMPLEMENTATIONOur quantum optimal control implementation utilizes thelibrary developed by Google’s machine intelligence research group [72]. This library is open source and isbeing extended and improved upon by an active developmentcommunity. TensorFlow supports GPU and large-scale parallellearning, critical for high-performance optimization. The simple interface to PYTHON allows those who are not software professionals to implement high-performance machine-learningand optimization applications without excessive overhead.Typical machine-learning applications require most of thesame building blocks needed for quantum optimal control.Predefined operations, along with corresponding gradients,include matrix addition and multiplication, matrix traces, andvector dot products. In addition, we have implemented anefficient kernel for the approximate evaluation of the matrixexponential and its gradient. Using these building blocks,we have developed a fast and accurate implementation ofquantum optimal control, well suited to deal with a broadrange of engineered quantum systems and realistic treatmentof capabilities and limitations of control fields.In common applications of quantum optimal control,time evolving the system under the Schrödinger equation—more specifically, approximating the matrix exponential forthe propagators Uj at each time step tj —requires thebiggest chunk of computational time. Within our matrixexponentiation kernel, we approximate e iHj δt by seriesexpansion, taking into account that the order of the expansionplays a crucial role in maintaining accuracy and unitarity. Therequired order of the matrix-exponential expansion generallydepends on the magnitude of the matrix eigenvalues relativeto the size of the time step. General-purpose algorithms suchas expm() in the PYTHON SciPy framework accept arbitrarymatrices M as input, so that the estimation of the spectral radiusor matrix norm of M, needed for choosing the appropriateorder in the expansion, often costs more computational timethan the final evaluation of the series approximation itself.Direct series expansion with only a few terms is sufficient forHj δ with spectral radius smaller than 1. In the presence oflarge eigenvalues, series convergence is slow and it is moreefficient to employ an appropriate form of the “scaling andsquaring” strategy, based on the identity 2nMexp M exp n,(6)2TensorFlowwhich reduces the spectral range by a factor of 2n at the costof recursively squaring the matrix n times [76]. Overall, thisstrategy leads to an approximation of the short-time propagatorof the form p2n ( iHj δt/2n )k,(7)Uj k!k 0based on a Taylor expansion truncated at order p. Computational performance could be further improved by employingmore sophisticated series expansions [77,78] and integrationmethods [79].As opposed to the challenges of general-purpose matrixexponentiation, matrices involved in a specific quantumcontrol application with bounded control field strength (iHj δt)will typically exhibit similar spectral radii. Thus, rather thanattempting to determine individual truncation levels pj andperforming scaling-and-squaring at level nj in each timestep tj , we make a conservative choice for global p and nat the beginning and employ them throughout. This simpleheuristic speeds up matrix exponentiation over the defaultSciPy implementation significantly, primarily due to leavingout the step of spectral radius estimation.By default, automatic differentiation would compute thegradient of the approximated matrix exponential via backpropagation through the series expansion. However, for sufficientlysmall spectral radius of M, we may approximate [2]d M(x)e M (x) eM(x) ,(8)dxneglecting higher-order corrections reflecting that M (x) andM(x) may not commute. (Higher-order schemes taking intoaccount such additional corrections are discussed in Ref. [3].)Equation (8) simplifies automatic differentiation: within thisapproximation, only the same matrix exponential is neededfor the evaluation of the gradient. We make use of this ina custom routine for matrix exponentiation and gradientoperator evaluation, further improving the speed and memoryperformance.The TensorFlow library currently has one limitation relevant to our implementation of a quantum optimal controlalgorithm. Operators and states in Hilbert space have naturalrepresentations as matrices and vectors which are genericallycomplex valued. TensorFlow, designed primarily for neuralnetwork problems, has currently only limited support forcomplex matrices. For now, we circumvent this obstacle bymapping complex-valued matrices to real matrices via the 1 Hre iσy Him , and state vectorsisomorphism H re , im )t . Here, 1 is the 2 2 unit matrix and σy ( is one of the Pauli matrices. Real and imaginary part ofthe matrix H are denoted by Hre Re H and Him Im H ,respectively; similarly, real and imaginary parts of state vectors and im Im . Written out in explicit block re Re are matrix form, this isomorphism results in H Hre HimHimHre re , im (9)rendering all matrices and vectors real valued. For the Hamiltonian matrix, this currently implies a factor two in memory cost(due to redundancy of real- and imaginary-part entries). Thereare promising indications that future TensorFlow releases mayimprove complex-number support and eliminate the need fora mapping to real-valued matrices and vectors.IV. PERFORMANCE BENCHMARKINGObtaining a fair comparison between CPU-based and GPUbased computational performance is notoriously difficult [80].We attempt to provide a specific comparison under a unifiedcomputation framework. TensorFlow allows for straightforward switching from running code on a CPU to a GPU. Foreach operation (matrix multiplication, trace, etc.), we use thedefault CPU and GPU kernel offered by TensorFlow. Note042318-5
LEUNG, ABDELHAFEZ, KOCH, AND SCHUSTERPHYSICAL REVIEW A 95, 042318 (2017)10102101102Hilbert space dimension103(a) target unitaryruntime per iteration (s)runtime per iteration (s)Hilbert space dimension4 1910010-22468101102102103(b) target state 610010-2102No. of qubits4681012No. of qubitsFIG. 3. Benchmarking comparison between GPU and CPU for (a) a unitary gate (Hadamard transform) and (b) state transfer (GHZ statepreparation). Total runtime per iteration scales linearly with the number of time steps. For unitary-gate optimization, the GPU outperformsthe CPU for Hilbert-space dimensions above 100. For state transfer, GPU benefits set in slightly later, outperforming the CPU-basedimplementation for Hilbert-space dimensions above 300. The physical system we consider, in this case, is an open chain of N spin-1/2systems with nearest-neighbor σz σz coupling, and each qubit is controlled via fields x and y .that properly configured, TensorFlow automatically utilizes allthreads available for a given CPU, and GPU utilization is foundto be near 100%. Not surprisingly, we observe that the intrinsicparallelism of GPU-based matrix operations allows much moreefficient computation beyond a certain Hilbert-space size;see Fig. 3.In this example, we specifically inspect how the computational speed scales with the Hilbert-space dimension whenoptimizing an n-spin Hadamard transform gate and n-spinGreenberger-Horne-Zeilinger (GHZ) state preparation for acoupled chain of spin-1/2 systems presented in Sec. V D.(Details of system parameters are described in the samesection.) We benchmark the average runtime for a singleiteration for various spin-chain sizes and, hence, Hilbertspace dimensions. We find that the GPU quickly outperformsthe CPU in the unitary-gate problem, even for a moderatesystem size of
quantum computing and quantum optics, and discuss the performance gains achieved by utilizing GPUs. II. THEORY We briefly review the essential idea of quantum optimal control and introduce the notation used throughout our paper. We consider the general setting of a quantum system with intrinsic Hamiltonian H 0 and a set of external control .
mum speedup can occur at 1 big core, n base cores, or with an intermediate number of middle-sized cores. Consider, n 16. With f 0.5, one core (of cost 16 BCEs) gives the best speedup of 4. With f 0.975, 16 single-BCE cores provide a speedup of 11.6. With n 64, f 0.9, 9 cores of 7.1 BCEs each provides an overall speedup of 13.3. Imp
Performance Recap Latency is the time it takes to perform a single task Performance(X) 1/ Latency(X) Speedup of X over Y: S Perf(X) / Perf(Y) Latency(Y) / Latency(X) Amdahl's Law limits speedup from optimization Corollary 1: Max speedup bounded by time in optimization Corollary 2: Make the "common case" fast Corollary 3: Parallel speedup limited by time in .
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