From pulses to circuits and back again: A quantum optimal control perspective on variational quantum algorithms
Alicia B. Magann, Christian Arenz, Matthew D. Grace, Tak-San Ho, Robert L. Kosut, Jarrod R. McClean, Herschel A. Rabitz, Mohan Sarovar
FFrom pulses to circuits and back again:A quantum optimal control perspective on variational quantum algorithms
Alicia B. Magann,
1, 2, ∗ Christian Arenz, ∗ Matthew D. Grace, Tak-San Ho, Robert L. Kosut,
4, 3
Jarrod R. McClean, Herschel A. Rabitz, and Mohan Sarovar Department of Chemical & Biological Engineering,Princeton University, Princeton, New Jersey 08544, USA Extreme-Scale Data Science & Analytics, Sandia National Laboratories, Livermore, California 94550, USA Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA SC Solutions, Sunnyvale CA, 94085 Google Research, 340 Main Street, Venice, CA 90291, USA (Dated: September 16, 2020)The last decade has witnessed remarkable progress in the development of quantum technologies.Although fault-tolerant devices likely remain years away, the noisy intermediate-scale quantumdevices of today may be leveraged for other purposes. Leading candidates are variational quantumalgorithms (VQAs), which have been developed for applications including chemistry, optimization,and machine learning, but whose implementations on quantum devices have yet to demonstrateimprovements over classical capabilities. In this Perspective, we propose a variety of ways thatprogress toward this potential crossover point could be informed by quantum optimal control theory.To set the stage, we identify VQAs and quantum optimal control as formulations of variationaloptimization at the circuit level and pulse level, respectively, where these represent just two levels ina broader hierarchy of abstractions that we consider. In this unified picture, we suggest several waysthat the different levels of abstraction may be connected, in order to facilitate the application ofquantum optimal control theory to VQA challenges associated with ansatz selection, optimizationlandscapes, noise, and robustness. A major theme throughout is the need for sufficient controlresources in VQA implementations; we discuss different ways this need can manifest, outline avariety of open questions, and conclude with a look to the future.
I. INTRODUCTION
The development of large scale, fault-tolerant quan-tum computers would enable diverse and disruptive ap-plications, such as the ability to break RSA encryptionprotocols using Shor’s factoring algorithm [1] and to ef-ficiently simulate the dynamics of complex quantum sys-tems [2]. Although significant progress has been made,the noise levels present in current quantum devices meansthey cannot yet serve as platforms for implementing theselandmark algorithms at scale. As such, a major goal is toidentify classically difficult problems that could be solvedwith these noisy intermediate-scale quantum (NISQ) [3]devices.This goal has motivated the development of variationalquantum algorithms (VQAs) for a variety of applicationsincluding ground state chemistry [4], optimization [5],and machine learning [6]. In VQAs, the problem to besolved is reformulated as an optimization problem, whosesolution is sought using quantum hardware and classicaloptimization in concert [7]. The quantum device is usedto evaluate the objective function, which is accomplishedvia a relatively shallow, parametrized quantum circuitapplied to an appropriately initialized register of qubits,after which the value of the objective function can be de-termined by measuring the register. Meanwhile, the clas-sical co-processor iteratively optimizes the parameters of ∗ These two authors contributed equally. the shallow quantum circuit. To-date, hardware imple-mentations of VQAs have not yet demonstrated improve-ments over the capabilities of classical computers, andthe aim of this Perspective is to examine how progresscan be made towards meeting this milestone in the fu-ture. In particular, this Perspective will consider VQAs,their associated challenges, and potential paths forward,through the lens of quantum optimal control. To mo-tivate this choice, we first look back and review certainaspects of the research efforts that have led us to theNISQ era of today.We begin by recalling early efforts to create quantumcomputers, which focused on developing methods to con-trol their components, and involved one or a few qubits[8, 9]. In these experiments, control was typically realizedusing electromagnetic fields or “pulses” designed to drivethe dynamics of the qubits in a desired fashion. Tech-niques for qubit control were studied extensively, espe-cially in the context of implementing high-fidelity entan-gling gates between qubits, as this is a necessary ingre-dient for quantum computation in the gate model. Onemethod, quantum optimal control (QOC), stands out forits ability to improve gate fidelities beyond what othertechniques could offer [10, 11]. In QOC, the control goalis formulated as an objective functional; then, the pulsesto minimize the objective functional are sought using it-erative optimization methods.Following these early demonstrations of qubit control,devices began to scale up to higher qubit counts, whichhas led to the advent of the NISQ era today, and in tan- a r X i v : . [ qu a n t - ph ] S e p Classical Computer
Create Experiment
Quantum
Device
MeasureUpdate
Cost functional optimizationCost functional evaluation(a) Pulse level (b) Circuit level
QOC VQA U t [ { f k ( t ) } ] = T e i R t dt H [ { f k ( t ) } ] U ( { ✓ i } ) = Y i U i ( { ✓ i } ) Figure 1. The quantum-classical optimization loop used inVQA and QOC experiments is shown. The quantum device(green) evaluates the objective functional, whose value is de-termined via measurements and input into a classical com-puter (yellow), which updates the optimization parameters.In conventional QOC experiments, these parameters are de-fined at the pulse level (a), and serve to parametrize a setof control fields { f k ( t ) } entering in a Hamiltonian describingthe physics of the system. In VQAs, the parameters typicallyenter at the circuit level (b), via a set of gates parametrizedthrough a set of e.g., gate angles { θ i } . In this Perspective, weexplore how VQAs may be informed by returning to a morephysical, pulse-level description inspired by QOC. dem, to the development of VQAs. These concurrent de-velopments have inspired significant research on circuitcompilation and optimization, similar in spirit to manyof the earlier efforts that studied pulse optimization inthe context of QOC. Thus, the reach of technology andthe focus of the community have broadened “from pulsesto circuits” . In this Perspective, we explore how the de-velopment of VQAs can be informed by going from thecircuit level “back again” to the pulse level to strengthenties to QOC and leverage results and tools from this well-developed field.As illustrated in Fig. 1, we explore this prospect byframing VQA implementations and QOC experiments asquantum-classical optimization loops; in the former (b),the optimization is done over quantum circuit elements,in contrast to a conventional QOC experiment (a), wherethe optimization is performed over a set of continuouspulses. We remark now that although the parametrizedquantum circuits in (b) are formed by gates, which arein turn implemented using pulses, a user seeking to im-plement a VQA typically has no need for any knowledgeof what happens at the pulse level, which offers a layerof abstraction separating the user from the underlyinghardware physics.With this unifying picture in mind, we now describethe remainder the this article. We begin by introduc-ing the concept of VQAs and discussing their currentstate in Section II. We then discuss QOC theory in Sec-tion III and review connections that have been made to VQAs to-date in Section IV. In Section V we presenta hierarchy of abstractions in variational optimizationthat serves to provide a common framework for VQAand QOC experiments. This is followed by an in-depthdiscussion of QOC-motivated future research directionsaimed at addressing four important challenges associatedwith VQAs: ansatz selection, optimization landscapes,noise, and robustness. In each of these cases, we empha-size the need for appropriate control resources to enhancethe performance of VQAs. Finally, we conclude in Sec-tion VI with a look ahead. II. VARIATIONAL QUANTUM ALGORITHMS
Variational quantum algorithms seek to solve problemsby leveraging the dynamical and representational powerof quantum computers in conjunction with classical com-puters. They do so by reformulating problems of interestas the minimization of some objective or loss function J [ { θ i } ] over a set of parameters { θ i } , asmin { θ i } J [ { θ i } ] , (1)where { θ i } parametrizes a quantum circuit U ( { θ i } ) = (cid:89) i U i ( θ i ) , (2)on n qubits. The objective function can be nonlinear inthe general case, such as cross-entropy for machine learn-ing. However, for reasons of simplicity, it is often formu-lated as the minimization of the expectation of a linearoperator H p , referred to as the “problem” Hamiltonian[12]: J [ { θ i } ] = (cid:104) ψ ( { θ i } ) | H p | ψ ( { θ i } ) (cid:105) , (3)where the state of the qubits at the culmination of the cir-cuit is | ψ ( { θ i } ) (cid:105) = U ( { θ i } ) | ψ (cid:105) , with | ψ (cid:105) denoting theirfixed initial state. Exact minimization of J correspondsto the preparation of the ground state of H p .The set of variational parameters { θ i } that mini-mize J [ { θ i } ] are sought iteratively, where at every it-eration, J [ { θ i } ] is evaluated by preparing the qubits inthe state | ψ (cid:105) , applying a circuit U ( { θ i } ) with a partic-ular parametrization { θ i } , and then measuring a set ofqubit observables to estimate J [ { θ i } ]. This can be ac-complished by expanding H p in the Pauli operator basisas per H p = (cid:80) Nj =1 α j P j , where α j are scalar coefficients, P j are tensor products of Pauli operators that are easyto measure on a quantum device, and N = O (poly( n )),and then measuring the expectations of each of the N Pauli basis operators in the expansion. Due to thestochastic nature of measurement in quantum mechan-ics, repeated measurements on an identically preparedstate | ψ ( { θ i } ) (cid:105) are needed to estimate these expectations.Then, the expectation value of H p can be computed todetermine J by classically evaluating the weighted sum (cid:104) ψ ( { θ i } ) | H p | ψ ( { θ j } ) (cid:105) = (cid:80) Nj =1 α j (cid:104) ψ ( { θ j } ) | P j | ψ ( { θ i } ) (cid:105) .After the value of J [ { θ j } ] has been evaluated in this man-ner, a classical optimization routine is then used to iter-atively update the values of { θ i } to convergence. Usingthis method to estimate the value of J [ { θ j } ] to a specifiedprecision (cid:15) requires a number of re-preparations and mea-surements of the state that scales as N s ∝ λ /(cid:15) , where λ = (cid:80) j | α j | [13]. Recent work has shown that clevergrouping of terms and other techniques can be used toreduce the naive scaling of these measurements by ordersof magnitude, even with techniques in the near-term [14].As quantum computers advance, it is possible to improvethe scaling of this estimation even further using tech-niques that leverage phase estimation, but the increasedresource costs for such approaches can be prohibitive.For practical reasons, the circuit represented by U ( { θ i } ) is usually assumed to be formed by a sequenceof elementary one- and two-qubit gates drawn from aspecified gate set, which is constructed and parametrizedaccording to a particular ansatz. However, we remarkhere that knowledge of how the parameters { θ i } explic-itly enter into the circuit is not always required, as J [ { θ i } ]is evaluated via measurements using the quantum de-vice. That is, the method is robust to many types oflabeling errors, since properties of the quantum systemrather than specific parameter values are of interest. Thismeans that VQAs possess some degree of robustness todrifts, crosstalk, and other systematic errors that can oc-cur during the implementation of the circuit [15], whichis a primary reason why VQAs are believed to be a wayto derive practical algorithmic use from NISQ devices.Furthermore, we note that the choice of ansatz for anyVQA is a crucial step. Although there is no general ap-proach for developing good ans¨atze, they are often de-rived from physical intuition (e.g., the QAOA ansatz,see below), knowledge of states generated by a particu-lar ansatz (e.g., the coupled-cluster ansatz, see below),or practical convenience (e.g., hardware-efficient ans¨atze[4, 16]). A critical feature of an ansatz is that it shouldbe scalable, i.e., having a circuit depth that scales as O (poly( n )) for an n -qubit quantum computer. Further-more, the number of variational parameters { θ i } shouldalso scale as O (poly( n )).VQAs have been developed for numerous applicationareas including machine learning [6], linear systems [17],and the compilation of quantum circuits [18]. However,the first application area for VQAs was the ground stateproblem in quantum chemistry [7, 19]. In this context,the variational quantum eigensolver (VQE) was devel-oped as a VQA for seeking the electronic ground state ofa chemical system in a field of fixed nuclear charges. Thesolution of this chemistry problem has a variety of ap-plications, including in chemical reaction prediction, thedetermination of molecular properties, etc. One commonansatz is the unitary coupled-cluster ansatz [4, 20], whichis a norm-preserving variant of the common coupled-cluster ansatz used in quantum chemistry, which con-structs a size-extensive ansatz through an exponential parametrization. Due to its unitary formulation, it nat-urally preserves physical properties of the state, but itis not efficient to evaluate classically. It represents anexample of a structured ansatz that carries the fermionicstructure into its translation into gates after the use of aJordan-Wigner transformation and Suzuki-Trotter split-ting. Strictly speaking, this exponentially splitting andthe choice of ordering represents a slight deviation fromthe formal construction [21], but the construction re-mains unitary, independent of the choice of parameters,and upon repetition it can be used to express arbitrarystates within the manifold of fixed particle number.With the evidence that NISQ devices can achieve clas-sically intractable, but perhaps not useful, tasks [22],there is a belief that variational preparation of groundstates of correlated systems may represent one of thefirst classically intractable and useful roles for NISQ de-vices. As such, the study of correlated quantum sys-tems remains a major objective for NISQ devices, and isa focus of many experimental and theoretical efforts atpresent. VQE demonstrations have been shown experi-mentally on a variety of photonic, ion trap, and super-conducting qubit setups in combination with error mit-igation techniques [16, 23–28]. Indeed the discrepancybetween available qubits in current quantum devices ( (cid:39)
50 qubits) and the number used in VQE experiments ( (cid:46)
10 qubits) is that the impact of noise makes the experi-ments incompatible with the high accuracy necessary toclaim an application advantage. With the help of errormitigation from symmetries in the reduced subspace, thelargest variational calculation performed to chemical ac-curacy on a quantum computer utilized 12 qubits andapproximately 200 quantum gates to simulate a 12-atomhydrogen chain [29]. Using more qubits will require ad-vances in both hardware and error reduction techniques.Another major application of VQAs is combinatorialoptimization. Here, the quantum approximate optimiza-tion algorithm (QAOA) was developed as a variationalmethod for determining approximate solutions to com-binatorial optimization problems, by encoding them intodiagonal Ising Hamiltonians, such that the solution of theproblem is encoded in the ground state of the Hamilto-nian [5]. QAOA seeks to find the solution by variationallyminimizing the expectation value of the Ising Hamilto-nian. Unlike the VQE, the QAOA ansatz is typicallyfixed, and consists of an alternating sequence of uni-tary operations generated by the problem Hamiltonian H p , and a so-called “driver” Hamiltonian, which doesnot commute with the problem Hamiltonian, and is de-noted by H d . Explicitly, the QAOA ansatz is formed by p rounds of alternating applications of these two Hamil-tonians, U ( { θ i } ) = p (cid:89) j =1 exp ( − iβ j H d ) exp ( − iγ j H p ) , (4)where { θ i } is the set of variational parameters { β j , γ j } ,and the true optimum for the original combinatorial op-timization problem can be achieved as p → ∞ .QAOA has been implemented experimentally using su-perconducting circuits for up to 23 qubits [30–34], a pho-tonic system with 2 qubits [35], and trapped ions with upto 40 qubits [36]. Due to limited qubit coherence times,most of these implementations (with some exceptions[32, 34]) considered combinatorial optimization problemsdefined on the connectivity graph of the hardware only,keeping circuit depths at a minimum. Connections witha physical theory of how these algorithms perform mech-anistically in the low depth regime have shown that phys-ical considerations, such as those considered in QOC,could be crucial for performance [37]. Furthermore, al-though in theory, ans¨atze with higher p should improveon the quality of the solutions achievable at lower p , theyalso require deeper circuits. So in practice, the effects ofnoise and decoherence may negate any improvements inquality. For this reason, most experiments implementeda single round of QAOA only, i.e., p = 1. These cases in-volved only two parameters whose optimal values couldbe obtained analytically, meaning that a variational op-timization loop was not necessary. However, in recentsuperconducting circuit experiments it was shown thatgoing to p = 2 [33] or p = 3 [34] and performing theoptimization could improve the solution quality beyond p = 1.Although these recent hardware demonstrations showa promising trend, the ultimate goal of implementingVQAs to solve problems that are intractable classicallyhas yet to be reached. The path towards meeting thisgoal will involve a confluence of theoretical and experi-mental progress. In the following, we propose a few waysthat this progress could be informed by QOC. In par-ticular, we focus in on how methods and results fromQOC could be leveraged to address VQA challenges as-sociated with ansatz selection, optimization landscapes,noise and robustness. Before getting to this, we first in-troduce QOC and discuss its connections to VQAs. III. QUANTUM OPTIMAL CONTROL
The aim of QOC is to design one or more electromag-netic fields or “pulses” to steer the dynamics of a quan-tum system towards a desired control target, which canbe a state, observable expectation value, or evolution op-erator, at some terminal time T . A standard formulationin QOC seeks to minimize a control objective functional J [ { f k ( t ) } ] over { f k ( t ) } , as [38–42]min { f k ( t ) } J [ { f k ( t ) } ] , (5)where J [ { f k ( t ) } ] includes the control target and physicalconstraints, often along with other criteria, which canbe defined to represent available laboratory resources orquantify robustness to errors or uncertainties [43].The set of pulses { f k ( t ) } used in QOC are typicallyconsidered to be classical fields in the semiclassical ap-proximation [44], in contrast to fully quantized fields. In addition, the wavelengths of the fields are typically as-sumed to be much greater than the size of the controlledquantum system in the dipole approximation [44], suchthat any spatial variation in the field amplitudes canbe neglected across the controlled system. In this set-ting, the control fields { f k ( t ) } typically enter the time-dependent Hamiltonian H [ { f k ( t ) } ] in an affine manner[45, 46], as follows: H [ { f k ( t ) } ] = H + (cid:88) k f k ( t ) H k , (6)where H is the drift Hamiltonian describing the time-independent system and { H k } is the set of control Hamil-tonians that couple the fields to the system, e.g., via dipoleinteractions. The dynamical equation for the systemtime-evolution operator U t is given by the Schr¨odingerequation ˙ U t = − iH [ { f k ( t ) } ] U t , with U = . This isa bilinear control system, making an analytical formula-tion of QOC solutions intractable in general [45–47]. Itsformal solution reads U t [ { f k ( t ) } ] = T e − i (cid:82) t dt (cid:48) H [ { f k ( t (cid:48) ) } ] , (7)where T indicates time ordering, such that the systemstate at time t is given by | ψ ( t ) (cid:105) = U t | ψ (cid:105) where | ψ (cid:105) isthe initial state.In analogy to the formulation of VQAs in Eqs. (1) and(3), we now turn our attention to the QOC problem inEq. (5), where we define J [ { f k ( t ) } ] = (cid:104) ψ ( T ) | H p | ψ ( T ) (cid:105) , (8)such that (unconstrained) { f k ( t ) } are sought to mini-mize the expectation value of H p at a designated time T . Solutions of this QOC problem, subject to the dy-namical constraint that | ψ ( t ) (cid:105) evolves according to theSchr¨odinger equation, are required to satisfy the follow-ing first-order necessary conditions: δJδ | ψ ( · ) (cid:105) = δJδ (cid:104) χ ( · ) | = δJδ { f k ( · ) } = 0 , (9)as well as the boundary conditions | ψ (0) (cid:105) = | ψ (cid:105) and (cid:104) χ ( T ) | = ∇ | ψ ( T ) (cid:105) J , where (cid:104) χ ( · ) | is a Lagrange multi-plier introduced to ensure that the dynamical constraintis satisfied [48]. Optimal control fields can then be con-structed to satisfy Eq. (9) via the corresponding Euler-Lagrange equations. To this end, a plethora of methodshave been developed for updating the QOC solutions,including GRAPE [49], Krotov [50–52], TBQCP [53–55],and D-MORPH [56–58] methods, until optimality as perEq. (9) is achieved.In practice, the minimization of J is usually accom-plished by first parametrizing one or more continuouscontrol fields by a set of variables { c i } , such that theHamiltonian describing the controlled system becomes H ( t, { c i } ) = H + (cid:88) k f k ( t, { c i,k } ) H k . (10)Common parametrizations include setting { c i } to be theamplitudes and phases of a set of frequency componentsof the field, or setting { c i } to be piecewise-constant fieldamplitudes in the time domain. Then, the objective is tooptimize { c i } to generate U T ( { c i } ) = T e − i (cid:82) T H ( t, { c i } ) dt such that the control objective functional J [ { c i } ] is min-imized at the terminal time T . In practice, this mini-mization is typically performed in an iterative fashion.If a tractable and accurate model is available, this it-eration can be performed numerically. However, it canalso be carried out experimentally via learning control[59–64], which does not require knowledge of the under-lying system model. Instead, at each iteration of suchQOC experiments, J [ { c i } ] is evaluated by first prepar-ing the system in a specified initial state, then evolvingit in the presence of applied fields with parametrization { c i } , and finally measuring the observable expectationvalue(s) needed to estimate J [ { c i } ]. A classical optimiza-tion routine is used to update the values of { c i } from oneiteration to the next, until J [ { c i } ] converges [59]. In thismanner, objective functional evaluations are performedby the quantum system directly, inherently accountingfor parameter uncertainties and other systematic errors,limitations, etc.This quantum-classical optimization loop associatedwith QOC at the pulse level is directly analogous to theprocedure used in VQA implementations at the circuitlevel, as shown in Fig. 1. In fact, if one has access to thequantum circuit at the pulse level, then Eq. (1) can in-stead be solved by optimizing over the set of continuouspulses { f k ( t ) } that are available. As such, we considerVQA implementations to be a form of digital QOC exper-iments on qubits, where the quantum circuit generatingthe unitary transformation U ( { θ i } ) is designed directly,through the selection and optimization of a parametrizedunitary ansatz. In general, the depth, dimension, andstructure of the ansatz, as well as the continuous and dis-crete parameters within it, are all tunable, giving the re-sulting optimization space both continuous and discretedegrees of freedom.In the past few years, this fundamental relationshipbetween QOC and VQA has been exploited to derivea deeper understanding of VQAs and novel variationalstrategies for quantum computing problems. In the fol-lowing section, we review some of this work at the inter-section of VQAs and QOC. IV. PRIOR WORK CONNECTING QOC WITHVQAS
As argued above, standard parametrized quantum cir-cuit ans¨atze can be viewed as examples of digitized QOCimplementations. There are many benefits to relaxing,or embedding, such digitized ans¨atze into a continuously(in time) parametrized framework, similar to the typi-cal setting in QOC. Such a relaxation often allows oneto eliminate any discrete optimization component of the problem, and more importantly, by formulating VQAswithin a standard control theory setting, this enables oneto apply many powerful methods and results of optimalcontrol theory.An early example of work with this reformulation is byYang et al., [65], who used a continuously-parametrizedformulation of variational quantum optimization to showthat a bang-bang approach (similar to the the alternat-ing structure of QAOA) is optimal for preparing the stateencoding the optimization solution, given amplitude-constrained control fields and a finite time to solution.This is achieved by applying Pontryagin’s minimum prin-ciple [48] to the continuously-parametrized formulation ofthe problem. Similarly, Lin et. al. apply Pontryagin’sminimum principle associated with time-optimal quan-tum control to Grover’s quantum search problem, andfind that the time-optimal control solution has a bang-singular-bang structure [66].Another direction has considered connections betweenthe QOC concept of controllability and the quantum com-puting concept of computational universality [67]. Inbrief, controllability is the study of which control ob-jectives can be realized with a given set of controls andconstraints. For unconstrained control fields, the dynam-ical Lie algebra L , formed by iterated commutators of thedrift and the control Hamiltonians and their real linearcombinations, is a powerful tool for deciding these mat-ters. In particular, the dynamical Lie algebra gives rise tothe Lie rank criterion [45, 46], which states that if L spansthe full space (i.e., the special unitary algebra su (2 n )for a n qubit system), then every unitary transformation V ∈ SU(2 n ) can be created to arbitrary precision in finitetime by shaping the control fields, and the system is saidto be fully operator controllable . A vast literature charac-terizing the controllability of quantum systems has beendeveloped in recent decades [68–80]. More recently, therelationship between controllability and computationaluniversality was utilized in refs. [81, 82] to show thatthe QAOA ansatz is universal for quantum computingfor specific choices of the problem Hamiltonian. In addi-tion, Mbeng et. al. made connections between digitizedquantum annealing, QAOA, and QOC, e.g., analyzingthe number of angles that are needed for controllability[83], while Akshay et. al. examined reachability deficitsin QAOA, providing strategies for improving reachability[84].Some groups have investigated using the QAOAans¨atze for bang-bang control of state transitions inquantum spin systems, e.g., Refs. [85–87], exploring ro-bustness and reachability as a function of the ansatzdepth. In addition, Bapat and Jordan analyzed the per-formance of bang-bang control protocols for optimizationalgorithms, showing that on certain problem instances,these protocols can yield an exponential speedup for bothclassical and quantum optimization, compared with qua-sistatic scheduling [88]. Using QOC, Brady et. al. showthat in a fixed amount of time, the optimal QAOA proce-dure has a pulsed bang-anneal-bang structure [89]. Witha fermionic representation, Wang et. al. [90] show thatthe evolution of a quantum system implementing QAOAon the so-called “ring of disagrees” problem translatesinto QOC of an ensemble of independent spins, therebysimplifying the determination of the optimal angle vec-tors. On a related note, Wu et. al. propose a schemeto machine learning tasks into corresponding QOC prob-lems on NISQ devices [91].Recently, several groups have proposed adaptive, vari-ational, or QOC-inspired approaches to design improvedVQA ans¨atze [92–97]. For example, the approach de-veloped in Refs. [92–94] uses derivative information toadaptively modify the circuit depth and ansatz structureusing a “pool” of predetermined single- or multi-qubitHamiltonian operators. Whereas, in the context of quan-tum chemistry, the approach presented in Ref. [95] usesa set of QOC-informed driving Hamiltonians to gener-ate VQA ans¨atze with symmetry-breaking features thatcan decrease the circuit depth required for convergence.Similarly, ref. [96] proposes ans¨atze determined by QOCat the device level, rather than parametrized quantumcircuits, to perform VQE simulations. These last worksespecially strengthen the connections between QOC andVQAs, and provide a natural segue to the next section,where we outline some promising new directions of re-search at this intersection.
V. NEW DIRECTIONS FOR VQAS INFORMEDBY QOC
A unifying view of VQAs and QOC can be obtained byviewing both as formulations of variational optimizationat different levels within an abstraction hierarchy. Thisis illustrated in Fig. 2, which we now discuss. We firstassume that the objective function J , determined by theapplication, is shared between the two approaches. Then,one can think of experimental ways to evaluate J withina hierarchy of abstractions modeling the experimentalhardware. At each level (i)-(iii) of this hierarchy, there isa natural parametrization of the control one has over thehardware and this defines a natural variational ansatz atthat level.At the bottom of this hierarchy is a pulse-level abstrac-tion (i), where we are closest to a first-principles modelof the hardware and think in terms of Hamiltonians forthe localized computing elements (e.g., qubits, qudits)and fields coupling them. At this level, the parametriza-tion of control is in terms of a continuously-parametrizedcontrol field or control Hamiltonian (with parameters c i )that is often realized by a set of electromagnetic fields,which are coupled to the computing elements. QOC typ-ically operates at this level of the modeling hierarchy.In the middle of the hierarchy (ii), we abstract awayfirst-principles descriptions of the hardware and think interms of universal circuit elements, or gates, that (ide-ally) perform well-defined maps on the computing ele-ments. Although there might be a discrete set of types Figure 2. A representation of the hierarchy of abstractionsused to model quantum hardware, and how variational op-timization enters into each level of the hierarchy. Outputproduced by the hardware is the desired objective function, J , which is parametrized in different ways by each level of thehierarchy – by control field parameters c i at the pulse level(i), by circuit parameters (usually angles θ i ) at the quantumcircuit level (ii), and by the structure of the circuit and theencoding and decoding maps ( E and D ) at the logical circuitlevel (iii). By understanding the relationship between the nat-ural variational parametrizations at each of the levels, we candevelop a rich family of variational ans¨atze. of gates, they can be continuously-parametrized by someset of gate parameters { θ i } . VQAs typically operate atthis level of the modeling hierarchy.Finally, at the highest level of the modeling hierar-chy, the logical circuit level (iii), we think in terms ofcircuits operating on quantum states encoded within anerror correction code. Any physical circuit at the circuitlevel of abstraction can be converted into a logical circuitgiven an error correction code and its associated logicalgates, with an error rate that is determined by the codeand the hardware. Strictly speaking, only discrete gatesare thought to be error correctable to arbitrary precision,thus, arbitrary rotation gates depending on θ i are syn-thesized as a sequence of discrete gates, which performsthis rotation to a specified precision. Hence, the rotationangle θ i is still present in the logical circuit, but only upto the precision that is given by the synthesis and codeprocedure. In practice, one may optimize the angle asif it is continuous, so long as the synthesis map is per-formed after, and the precision is great enough to impactthe optimization in practice. Error-corrected quantumcomputing experiments operate at this level of the mod-eling hierarchy, and benefit from decreased susceptibilityto hardware noise due to the encoding and careful imple-mentation of fault-tolerant operations. A. Ansatz selection
With this hierarchy of modeling abstractions and a de-scription of variational optimization at each level, we canexploit the connections between the levels in the hierar-chy to define new and richer variational ans¨atze, and infact, a family of ans¨atze, built from paths on this dia-gram. To see this, we explore a few sample paths acrossthis diagram and understand the implications of follow-ing particular trajectories.As a first example, we consider a path from the pulselevel (i) to the circuit level (ii). We begin at thepulse level (i), with a continuous control perspective offields acting on n qubits, which is then discretized as aparametrized control. We now have a time-dependentHamiltonian H ( t, { c i } ) acting on a system of n qubits.The field of quantum algorithms of simulation of time-dependent Hamiltonians is well developed, and a rangeof methods exist for simulating the time evolution gen-erated by H ( t ) over some desired time interval [0 , T ]to arbitrary accuracy. Among the simplest is an op-erator splitting, also known as “Trotter factorization”[98], but a host of methods with more accurate imple-mentations without direct classical simulation analogs,including quantum walk and so-called linear combina-tions of unitary approaches, exist that may also be used[99–101]. Hence, we may understand this step as imply-ing that the parametrized control can be combined withHamiltonian simulation to yield a quantum circuit, whoseparametrization is naturally understood at the level of H ( t ). As such, optimizations proceed on a landscapedetermined by a parametrization at the pulse level (i),despite being implemented at the circuit level (ii). Thatis, while arbitrary rotations depending on some θ i exist inthe circuit, they are entirely determined by the composi-tion θ i ( { c i } ) in conjunction with the chosen Hamiltoniansimulation map. This has the advantage that this circuitcan be converted to a logical circuit by discretization intoerror-correctable gates, mitigating the effects of decoher-ence while retaining the essential prescription of control.In this case, we retain all of the power of QOC machin-ery in manipulating and understanding the ansatz, whileleveraging the power of quantum algorithms and errorcorrection to ensure theoretical guarantees of implemen-tation.We can take this same path farther, by appending toit additional gate parametrizations. That is, if Hamil-tonian simulation also maps parametrized control to acircuit at the quantum circuit level (ii), e.g., ref. [102],additional parameters may also be added to the gener-ated circuits, creating a hybrid ansatz. To explore thefull power of these connections though, let us explore theother direction that one can take in this perspective.Consider starting at the quantum circuit level (ii),with descriptions of hardware in terms of quantum cir-cuits. To make these circuits realizable, a structure andparametrization is selected. Once this circuit has beendetermined, one can, in principle, map this back to aQOC problem at the pulse level (i) in a number of ways,where now the control parametrization depends on theangles from the circuit such that c i ( { θ i } ) is determinedthrough a map that we denote as “Hamiltonian genera-tion”. This mapping is typically non-unique in a more se- vere way than in the other direction, especially if one con-siders mapping the entire circuit to a generating Hamilto-nian, whether it be time independent or time dependent.However, much better formed mappings can be used andare related to existing strategies for gate design. For ex-ample, one map can perform a gate-wise mapping foreach parametrized gate back to the Hamiltonian controlparameters c i ( { θ i } ) that are used to accurately imple-ment this gate. Similar to before, a hybrid control ansatzcan then be created by dropping the dependence on θ i and allowing free variation of the parameter c i . This canalso be done for fixed gates without parametrizations inthe gate model, increasing the overall expressiveness ofthe resulting circuit in a systematic and new way. We seethen, that the gate formulation also naturally provides afamily of control ans¨atze.To give a specific example, we consider a system andansatz that has received considerable attention in theVQA literature, which is the preparation of the H ground state in a minimal, molecular orbital basis. With-out symmetry reduction, the ground state of this problemis given by | Ψ GS (cid:105) = cos θ | (cid:105) + sin θ | (cid:105) . A simpli-fied version of a circuit that contains this state within itsparametrization is given by | (cid:105) R y ( θ ) •| (cid:105) H • H •| (cid:105) X H • H •| (cid:105) X H • H Where, in this diagram, H is the Hadamard gate, R y is a rotation about the Pauli Y axis, and the con-necting line is a two-qubit controlled-Z gate, CZ =diag(1 , , , − θ provides the variational freedom required to pre-pare the ground state. In addition, as outlined above,we can map back to control implementations of gates,including those that don’t yet have parameters. For ex-ample, a controlled-phase gates may be written as a time-independent evolution under a Hamiltonian of the form H z = c Z Z + c Z + c Z for a time τ , up to anunimportant global phase. If this is how the gate is im-plemented in our physical system, then the parameter τ can be added to the list of parameters to create a hybridcontrol ansatz. The ansatz in this case includes the be-ginning of the circuit, as well as analog evolution under H z which may be implemented by physical means.Finally, taking the mapping in the other di-rection, the evolution under the Hamiltonian H for a time τ in this case, could be re-digitizedinto a Trotter factorization of exp( − iH z τ ) =exp( − iτ c Z Z ) exp( − iτ c Z ) exp( − iτ c Z ), whereeach term in the product can be translated back intoa digital sequence, and now we can vary the set ofparameters { θ, c , c , c } , where we have retained the θ parametrization and the rest of the circuit, but addedmore controls. Note that for this simple problem, if theimplementation is error-free, this additional freedommay be unnecessary. However, if some errors occurin the gates, this freedom can be used to improve theresult. Moreover, problems that are similar, but notidentical to the hydrogen molecule, can then leveragethis base ansatz and natural extensions from it. Thissimple example is designed to illustrate the ways thatone can move from the circuit level (ii) to a hybridansatz that combines tools from the circuit and pulselevels (ii) and (i) and back again, while changing thenature of the ansatz.Finally, we see that beginning at any level in the hi-erarchy in Fig. 2, it is possible to iterate on the connec-tions between these approaches, choosing to carry for-ward parametrizations or develop new ones, and mapback to the other corresponding formalism. For exam-ple, if one takes a variational parametrization at thecontrol-field level, maps it to a quantum circuit witha Hamiltonian simulation map, then selects a partic-ular gate and maps it back to a control parametriza-tion, and uses the combined circuit, we get a new hy-brid parametrization that can be manipulated closer toa device-level description. It is easy to see that thesechoices can be mixed, matched, and iterated on to createan entire family of parametrizations. Here, we envisionthat the tools of QOC are bolstered by concepts from dig-ital simulation, including error correction and advancedtime-propagation algorithms, and vice versa.In addition to expanding the set of options and tools ofboth areas, we hope that this formulation and perspec-tive of connections allows more direct cross-pollination ofideas. For example, by formulating a circuit ansatz basedon control theory, we more directly understand the con-sequences of controllability in a circuit model. In turn,by developing a control-theoretic ansatz from a circuitfamily known to be universal, we may find new insightsin relation to entanglement-generating power or expres-siveness of quantum circuits. B. Optimization landscapes
A critical consideration in VQAs is the difficulty of op-timizing over the circuit parameters. The ease of findingthe global minimum of J during this optimization pro-cess is dictated by the structure of the underlying opti-mization landscape as well as available prior informationabout the location of good optima. A central theme ofQOC is the study of such landscapes, and translatingthese tools into the context of VQAs will allow for newstrategies in ansatz and problem design.At their core, VQAs represent the mapping of a con-vex optimization problem in an exponentially-large lin-ear space, to a non-convex optimization problem overparametrized quantum circuits. This can be seen by ex-panding the variational state | ψ ( { θ i } ) (cid:105) in the eigenbasis {| n (cid:105)} of H p . The objective functional then takes the form J [ { θ i } ] = (cid:80) dn =1 λ n ( { θ i } ) E n , where E n are the eigenval-ues of H p and λ n = |(cid:104) n | ψ ( { θ i } ) (cid:105) | . When J is optimizedover the set { λ n } , the optimization problem is convex,which implies that the corresponding optimization land-scape is free from local optima. In contrast, the parame-ters θ i typically enter in a non-linear fashion in λ n , oftenvia an exponential map in U i ( θ i ). As such, when J isoptimized over the set { θ i } the optimization problem isnon-convex and in general, local optima can appear.In the context of QOC, to address these and otherlandscape considerations, a sizeable body of researchhas analyzed the topological features and properties ofthe (dynamic) QOC landscape, defined by J as a func-tional of the control fields [103–106]. To characterizethese landscapes, recall that J depends on the time-evolution operator up to time T , such that we have J = J ( U T [ f ( t )]). Then, the functional derivative of J with respect to a single field δJδf ( t ) is given by the com-position δJδf ( t ) = ∂J∂U T ◦ δU T δf ( t ) . The first term in the com-position captures the properties of the kinematic controllandscape, i.e., J defined as a function of U T . The com-position above implies that if the variation of U T withrespect to the control field δU T δf ( t ) is assumed to be fullrank, i.e., equal to d , then the dynamic and kinematiccritical points coincide. This result is referred to as localsurjectivity . In this scenario, the topology of the dynamiccontrol landscape is fully characterized by the criticalpoint structure of the kinematic control landscape. Re-sults from QOC theory have shown that the kinematiccontrol landscapes of typical objective functionals consistof global extrema and saddles [105–109]. As such, if weare able to create every unitary transformation such that J as a function of U T can be varied arbitrarily, and if U T can be varied arbitrarily also by varying the control field,then the landscape of J consists of saddles and globalextrema. More precisely, under the assumptions of (1)a full dynamical Lie algebra, (2) unconstrained controlfields of arbitrary length and shape, and (3) local sur-jectivity, the control landscapes for commonly-employedobjective functionals are free from local extrema. Exten-sive QOC simulation and laboratory studies indicate therelative ease of satisfying all three assumptions [44]. Fur-thermore, it has been shown that assumptions (1) and (3)are almost always satisfied in a measure theoretic sense[68, 110, 111], thereby implying that QOC landscapes arealmost always free of local minima under the premise ofsufficient control field resources [112]. While the precisemeaning of “sufficient” is application-dependent and re-mains an open challenge to systematically assess, it hasrecently been shown that local surjectivity is almost al-ways met when the control fields allow for approximatingHaar random evolutions [111, 113] within the time inter-val of interest: [0 , T ]. As such, even though performingthe optimization over a set of parametrized control fieldsdoes not avoid the non-convexity of the problem, pro-vided the assumptions (1)-(3) hold, the interior of theoptimization landscape provably contains saddles only.Looking ahead, we hope that the theory of QOC land-scapes can provide insight into optimization landscapesof VQAs, which could include better tools to understanddifferent ans¨atze. Most directly, the optimization land-scape of a variational ansatz formulated at the control-field level in Fig. 2 could be analyzed in terms of existingQOC landscape theory. Here, in addition to the theo-retical foundations, numerical tools, such as D-MORPH(a flexible continuation-method developed in the contextof QOC) [114, 115] and FLACCO (software for feature-based landscape analysis of continuous and constrainedoptimization problems) [116], could be employed to ex-plore these landscapes. However, there are a numberof caveats to consider for this direct approach; that is,although QOC has many well developed tools for thecharacterization of QOC landscapes, there remain chal-lenges for applying these tools to scalable and practicalimplementations of VQAs. For instance, characterizingthe landscape topology in the presence of constraints ischallenging; although these landscapes are expected tobe free from local extrema when sufficiently many un-constrained variables { c i } are used to parametrize thecontrol fields, there is evidence that an insufficient num-ber of parameters (i.e., less than d ), leading to a viola-tion of the surjectivity assumption (3), and control fieldconstraints are among the reasons for local extrema toappear [117]. In general, further research is needed whenthe control resources do not scale with the dimension d of the quantum system being controlled, but rather onlyscale as O (poly(log( d ))). To this end, it may be inter-esting to study systems that are not fully controllable,but where all states within a certain subspace definedby the symmetries of the (controlled) system are reach-able. One may then ask whether a (significantly) smallernumber of control parameters could be used to obtaina trap-free landscape when moving only in a restrictedsubspace (containing the ground state) whose dimensiondoes not scale exponentially.We also remark that as the dimension d of the quantumsystem increases the landscape flattens out, such that forcertain objective functionals, gradients become exponen-tially small as a function of the number of qubits, makingthese regions difficult to leave with local optimization al-gorithms. These so-called barren plateaus [19] are a con-sequence of a concentration of measure phenomenon, andoccur in both VQA implementations and QOC [111]. Un-derstanding how to avoid them in each context indepen-dently could yield useful and transferable techniques ben-efiting both areas of study, and will likely involve care-ful design of ansatz, initialization, and training methods,e.g., ref. [118]. C. Noise and time-optimal control
Today, errors in NISQ devices severely restrict the cir-cuit depths that are achievable for VQAs. For example,certain errors can arise from stochastic fluctuations in the gate implementations, leading to errors that accumulateas circuit depth increases. In QOC, robust control strate-gies have been developed to suppress random errors likethis, by seeking pulses that are robust in the presenceof finite control noise; the condition of the Hessian of J can be used to determine such properties [115]. We be-lieve more direct translation of robust control tools fromQOC into variational algorithms will help improve theirrobustness against general errors.Other, often more insidious, errors stem from interac-tions of the qubits with their surrounding environmentthat can cause the system to decohere over time. Thus,the timescale associated with performing a quantum cir-cuit should ideally be restricted to the coherence time ofthe system. When coherence times are limited and gatesare dominated by stochastic errors, it is desirable to seekVQA circuits that drive the qubits to the minimum of J as quickly as possible, i.e., using circuits with minimumdepth. In the context of QOC, the theory of time-optimalcontrol, and the associated theory of quantum speed lim-its, provide a powerful framework for considering this is-sue at the pulse level, and may offer valuable tools toenhance VQA performance. In general, identifying time-optimal fields is a challenging task, as it correspondsto finding the associated geodesic on the unitary group[119–124]. Exact results are only known for certain sys-tems consisting of 1-3 qubits [125–129]. However, upperand lower bounds on T can be found [130–133]. In addi-tion, for n qubit networks, the upper-bounds in [132, 133]allow for characterizing the unitary transformations thatcan be created efficiently with 2 n local fields, i.e., where T = O (poly( n )). While some progress has recently beenmade to characterize efficiently-controllable qubit graphs[134], in general, it remains an open challenge to system-atically determine the set of unitary transformations thatare reachable in polynomial time with fewer controls. Fi-nally, we remark that additional control resources notonly offer faster control strategies and in general, richerans¨atze, but can also allow for counteracting decoher-ence by enabling decoupling schemes [135–138] that cansuppress interactions with the environment. D. Robust control through digitization
Until this point, we have considered a setting naturalto VQA’s, where one typically starts from a known ini-tial state, e.g. | .. (cid:105) and builds up the state of interestusing a set of parametrized gates or controls. However,QOC also deals with situations where one may be givenan unknown quantum state | ψ (cid:105) , and wishes to manip-ulate it. For example, this quantum state may be thestate of reactants in a chemical system and the controlgoal is to steer the reaction products. Alternatively, in aquantum sensing context, the quantum state may be thestate of the sensed environment and the control goal isto distinguish features in the environment.If we now consider such a QOC experiment from the0point of view of VQAs, and think of the applied con-trols as a parametrized unitary transformation, one cantranslate these controls into a parametrized circuit ansatz[102] and further, into an error corrected circuit. Thiswould have the effect of ensuring that the controls areapplied to an arbitrarily high degree of accuracy, an ad-vantage of taking the digital point of view. Of course, tobe fully compatible with quantum error correction, theinput quantum state may also need to be encoded, andeffective digital encoding of quantum states from natureis an experimental challenge that has not yet been re-alized, but this perspective has the potential to enableQOC experiments of arbitrary accuracy on many-bodysystems.The increasing difficulty of precisely controlling aquantum system with non-ideal analog controls as its di-mension increases can be understood in various ways.Conceptually, this is a consequence of the orthogonalitycatastrophe – where small perturbations from the idealHamiltonian (in this setting, caused by small errors inthe analog controls) can lead to an exponential decay(with system dimension) of fidelity with the state underideal evolution [139, 140]. More operationally, consideran N -body quantum system with state vector of dimen-sion d ∼ exp( N ). We can estimate the requirements oncontrol precision with a simple model. Assume we wantto prepare the state with 1 in the first entry of the statevector, and 0 on the remaining ( d −
1) entries. Now foreach dimension we are able to achieve a precision δ , andconsider for simplicity the normalized erred state thatis off by δ in each of the entries except the first. Inthis case, the fidelity with the target state is given by1 / [1 + ( d − δ ], which means to maintain a constantfidelity, one must have δ ∝ d − / . This is an exponential(in N ) precision requirement for the control.In contrast, if the state of this N -body system can be transduced into n = log ( d ) qubits, the control of thestate can be encoded in a VQA circuit [102]. Then, as-suming the fidelity of qubit gates are sufficiently high andusing the tools of quantum error correction, the fidelityof controlling this state can be brought arbitrarily closeto 1 with logarithmic overhead in physical resources.At a broader application level, the manipulation of un-known quantum states in a VQA manner encompasseswhat is sometimes called “quantum machine learning”for training data provided in the form of quantum statesrather than classical inputs, or quantum data. This ideaallows for the tools QOC to come to bear in this com-munity. Moreover, this permits a closer connection toquery based proofs in quantum computer science, wherestronger proofs are possible in the setting where only alimited number of quantum states are available [141–145]. VI. OUTLOOK
In this perspective, we have explored the connectionsbetween VQAs and the theory of QOC. While VQA ap-plications constitute some of the most promising appli-cations of near-term quantum computers, more progressbridging the gap between theoretical VQAs and NISQcomputing hardware is necessary to realize this promise.We have argued that concepts from QOC, and the moregeneral framework outlined in Sec. V, which integratestraditional models of QOC with VQAs and extends both,are critical to bridging this gap. Both fields have some-thing to gain from exploring this fertile area of over-lap. From the perspective of VQAs, the mature theoryand powerful tools of QOC can provide richer variationalstructures and offer a deeper understanding of variationalexperiments. Conversely, from the perspective of QOC,VQAs present an exciting frontier of many-body quan-tum control, pushing this established field in new direc-tions. For these reasons, we expect bountiful fruits at theintersection of these two areas of research in the years tocome.
ACKNOWLEDGMENTS
A.B.M., M.D.G., and M.S. were supported by the U.S.Department of Energy, Office of Science, Office of Ad-vanced Scientific Computing Research, under the Quan-tum Computing Application Teams program. A.B.M.also acknowledges support from the U.S. Departmentof Energy Computational Science Graduate Fellowship,grant number DE-FG02-97ER25308. M.D.G. also ac-knowledges support from the U.S. Department of Energy,Office of Advanced Scientific Computing Research, un-der the Quantum Algorithm Teams program. C.A. andH.A.R. were supported by the U.S. Army Research Of-fice, grant numbers W911NF-16-1-0014 and W911NF-19-1-0382, respectively. T.S.H. acknowledges support fromthe U.S. Department of Energy, grant number DE-FG02-02ER15344.Sandia National Laboratories is a multimission labo-ratory managed and operated by National Technology &Engineering Solutions of Sandia, LLC, a wholly ownedsubsidiary of Honeywell International Inc., for the U.S.Department of Energy’s National Nuclear Security Ad-ministration under contract DE-NA0003525. This paperdescribes objective technical results and analysis. Anysubjective views or opinions that might be expressed inthe paper do not necessarily represent the views of theU.S. Department of Energy or the United States Govern-ment. [1] P. W. Shor, Algorithms for quantum computation: Dis-crete logarithms and factoring, in
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