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Comparing, Optimising and Benchmarking Quantum Control Algorithms in a Unifying Programming Framework

S. Machnes, U. Sander, S. J. Glaser, P. de Fouquieres, A. Gruslys, S. Schirmer, T. Schulte-Herbrueggen

arXiv:1011.4874v3quant-phmath.OC

TL;DR

Quantum control requires accurate, efficient steering of increasingly large quantum systems, but evidence comparing commonly used optimisation algorithms across finite-dimensional applications has been limited. The paper benchmarks sequential KROTOV-type and concurrent GRAPE algorithms and introduces the modular DYNAMO framework. Across most test problems, both methods reach similar final fidelities, while concurrent updates are usually faster and scheme handover can exploit their differing convergence behaviour.

  • Problem

    The paper addresses the need to compare sequential and concurrent quantum-control algorithms across a wide range of finite-dimensional applications.

  • Method

    The authors benchmark KROTOV-type and GRAPE algorithms using DYNAMO, a modular framework combining update, gradient, step-size, and subspace-selection methods.

  • Results

    In most test problems, sequential and concurrent methods reach similar final fidelities, while concurrent updates are faster in most cases and overtake sequential updates between fidelity 0.9 and 0.99.

  • Takeaways & Limitations

    Sequential updates can be advantageous for robustness or early optimisation, whereas concurrent updates are recommended for reaching high fidelities in gate synthesis through dynamic handover.

Abstract

from arXiv · show

For paving the way to novel applications in quantum simulation, computation, and technology, increasingly large quantum systems have to be steered with high precision. It is a typical task amenable to numerical optimal control to turn the time course of pulses, i.e. piecewise constant control amplitudes, iteratively into an optimised shape. Here, we present the first comparative study of optimal control algorithms for a wide range of finite-dimensional applications. We focus on the most commonly used algorithms: GRAPE methods which update all controls concurrently, and KROTOV-type methods which do so sequentially. Guidelines for their use are given and open research questions are pointed out. --- Moreover we introduce a novel unifying algorithmic framework, DYNAMO (dynamic optimisation platform) designed to provide the quantum-technology community with a convenient MATLAB-based toolset for optimal control. In addition, it gives researchers in optimal-control techniques a framework for benchmarking and comparing new proposed algorithms to the state-of-the-art. It allows for a mix-and-match approach with various types of gradients, update and step-size methods as well as subspace choices. Open-source code including examples is made available at http://qlib.info.

I. INTRODUCTION

Quantum control uses numerical optimal-control algorithms to steer increasingly complex systems accurately, efficiently, and robustly. The paper compares sequential and concurrent updates and introduces DYNAMO as a modular benchmarking framework.

  • Motivation: Quantum control is increasingly important for steering dynamical systems in applications including quantum simulation, computation, and technology.The systems considered include cold atoms, trapped ions, superconducting qubits, open systems, and decoherence-reduction settings.
  • Algorithms: The study focuses on piecewise-constant time-domain controls and compares sequential-update algorithms with concurrent-update GRAPE methods in finite-dimensional systems.Sequential methods update one or a few time slices before proceeding, whereas GRAPE updates all controls concurrently.
  • Algorithms: Algorithm performance depends on matching update schemes with gradient and step-selection methods, including first-order, second-order, conjugate-gradient, Newton, and quasi-Newton approaches.Exact inverse-Hessian calculations become costly as system dimension increases, motivating iterative approximations.
  • Framework: DYNAMO combines subspace selection, gradient calculation, control updates, and step-size methods in a modular framework for benchmarking quantum-control algorithms.The platform also targets fewer matrix operations and provides MATLAB code and examples for the community.
  • Evaluation: The paper benchmarks sequential and concurrent algorithms across typical finite-dimensional problems and develops guidelines for selecting algorithm variants.It also identifies open research questions for future work with DYNAMO.
  • Unified setting: The framework unifies closed and open quantum-control problems through bilinear system dynamics and supports state-transfer and gate-synthesis applications.The paper treats fully controllable systems and, unless specified otherwise, unbounded control amplitudes.

C. Figures of Merit

The paper formulates quantum-control objectives through final-state quality functions and discretises controlled dynamics into piecewise-constant time slices. These definitions support state-transfer, gate-synthesis, and open-system optimisation.

  • Quality functions: The quality functions studied depend only on the system’s final state X(T), without including running costs.The authors state that omitting running costs is not a principal limitation.
  • Quality functions: A natural figure of merit is the overlap-based projection of the controlled final state onto the target state.The overlap is used for both state and gate control settings.
  • Quality functions: The paper distinguishes fSU := Re g, which respects global phase, from fPSU := |g|, which applies when global phase is immaterial.The projective formulation relates the latter setting to unitary conjugation in Liouville space.
  • Discretisation: Controlled dynamics are discretised into M equal time slices with piecewise-constant amplitudes, fixed final time T = M∆t, and generators comprising drift and control terms.The resulting propagator governs evolution on each interval (t_k−1, t_k].
  • Discretisation: Boundary conditions define the optimisation target: gate synthesis uses X0 ≡ 1l and Xtarget ≡ Utarget, while pure-state transfer uses an initial and target state.The discretised evolution determines the system state at each time slice.
  • Optimisation procedure: The algorithm updates selected time-slice subspaces, with Figure 1 contrasting concurrent, sequential, and hybrid update schemes.The adjoint system is propagated backward to evaluate the figure of merit.

Algorithmic Steps

The generalised DYNAMO search alternates between selecting control-time subspaces and taking gradient-based update steps, with GRAPE and Krotov represented as limiting cases. Hybrid schemes vary subspace size and inner-loop steps dynamically.

  • The algorithm begins with initial control amplitudes, counters, an iteration limit, and a target quality threshold.
  • The outer loop selects the time-slice subspace whose control amplitudes will be updated next.
  • The inner loop performs one or more gradient-based steps within the selected subspace, potentially reducing matrix operations.
  • Forward propagation, gradient-based updates, iteration counters, subspace selection, and final diagnostics form the main computational flow.
  • Krotov uses sequential subspace updates with a degenerate inner loop, whereas GRAPE updates all controls concurrently with a degenerate outer loop.
  • Hybrid algorithms allow arbitrary subspace sizes and dynamically vary subspace updates and inner-loop steps according to gradient magnitude or distance from target fidelity.

E. Overview of the dynamo Package and Its Programming Modules

DYNAMO is modularized around interchangeable update, gradient, step-size, and subspace modules so researchers can adapt algorithms to different tasks and computational regimes. Its design supports both practical optimization and systematic algorithm development.

  • DYNAMO provides a flexible framework for optimizing specific problems and developing new optimal-control algorithms.
  • The MATLAB implementation lets users configure Hamiltonians, select GRAPE, Krotov, or hybrid methods, and impose computational constraints.
  • The framework supports benchmarking new algorithms against established methods while handling shared computational tasks.
  • The study finds no single optimal-control algorithm serves all task types, because accuracy and robustness requirements differ across applications.
  • A universal framework therefore needs modular sequential, concurrent, and hybrid updates with task-adapted parameters.
  • Higher-order gradient choices should match update schemes, while exact Newton inverses become costly as system dimension increases.
  • BFGS reuses previous iterations and suits concurrent updates, whereas it is poorly matched to sequentially changing control subsets.
  • Sequential gradient updates with suitable step-size control can approach sequential Newton performance while avoiding its computational overhead.

2. Gradient Modules

DYNAMO offers exact, approximate, and finite-difference gradient modules alongside computational strategies for reusing propagators and minimizing matrix operations. Approximate gradients trade exactness for efficiency under small time-step conditions.

  • Exact gradients evaluate derivatives of exponential maps for piecewise-constant controls using the Hamiltonian eigendecomposition.
  • Approximate gradients are valid when each digitization interval is sufficiently short relative to the Hamiltonian norm.
  • The approximation replaces a time-integral average with the integrand at the interval’s right endpoint.
  • Approximate gradients cease to be exact when time evolution fails to commute with the control Hamiltonian, with error scaling as |λ_l−λ_m|∆t.
  • Finite differences provide another gradient option, particularly for pure-state transfer.
  • With eigendecompositions available, matrix exponentials can be evaluated cheaply, and sequential and concurrent schemes require basically the same total number of exponentials.
  • The two-level search uses an outer loop for time-slice selection and an inner loop for gradients and search advancement.
  • Hybrid propagation collapses consecutive non-updating slices and caches reusable quantities to reduce matrix multiplications and exponentiations.

A. Closed Quantum Systems

Closed-system control tasks are expressed through common propagators and boundary conditions, allowing gate synthesis and state transfer to share an algorithmic treatment. The initial and target data types determine the evolving state representation.

  • The standard closed-system tasks include phase-independent and phase-dependent gate synthesis, pure-state transfer, and density-operator transfer.
  • Common propagators handle these tasks, while their specific differences enter through initial and target boundary conditions.
  • The evolving state may be represented as a vector or matrix in Hilbert space, or formally in Liouville space.
  • Task 1: unitary gate synthesis up to a global phase: Phase-independent gate synthesis uses a projective superoperator representation of the target gate.
  • Task 2: unitary gate synthesis with fixed global phase: Phase-dependent gate synthesis places the target gate directly in the quality function and differentiates fidelity with respect to control amplitudes.
  • Task 3: state transfer among pure-state vectors: Pure-state transfer uses the target state and propagated initial state in the quality function.
  • Task 4: state transfer among density operators: Density-operator transfer normalizes its quality function by the squared norm of the target state and differentiates the resulting expression.

B. Open Quantum Systems

The framework extends quantum-control optimisation to open Markovian systems and addresses gradient evaluation when drift and relaxation do not commute.

  • Open Markovian tasks: For Markovian dynamics, the Hamiltonian generator is augmented by a relaxation operator Γ to obtain the quantum-map generator.The resulting propagators support computation of trace-fidelity derivatives with respect to control amplitudes.
  • Gradient evaluation: Because Γ and i bHu generally do not commute, exact spectral gradient evaluation is unavailable for the resulting non-normal generator.The paper therefore describes small-time-step approximations and finite differences as alternatives.
  • Scope: The standard Markovian task can represent non-Markovian systems when they embed into a numerically manageable larger Markovian system coupled to an environment.This is an explicit modelling condition rather than a general treatment of non-Markovian dynamics.
  • Open Markovian tasks: Open-system state-transfer optimisation is formulated as a vectorised special case of quantum-map synthesis.The initial and target density matrices are represented using vec-notation, producing propagated and back-propagated vectors.
  • Gradient evaluation: The same gradient approximations used for quantum-map synthesis apply to open-system state transfer.This connects the vector formulation of state transfer directly to the broader open-system optimisation framework.

A. Specification of Test Cases

The benchmark covers 23 finite-dimensional quantum-control systems spanning spin chains, cluster-state systems, NV centres, and driven spin-j systems, with varied couplings and controls.

  • Test-suite scope: The test suite contains 23 systems spanning spin chains, a C4 cluster-state system, an NV-centre system, and driven spin-j systems with j = 3, 6.The cases vary system size, coupling topology, and control scheme.
  • Spin-chain cases: Explorative problems 1–12 use Ising-ZZ spin chains with individually addressable x- and y-controls.The chains vary in length, with parameters including J = 1 and n = 1, . . . , 5.
  • Spin-chain cases: Problem 1 additionally models linear crosstalk through control-Hamiltonian coefficients αk and βk.The selected coefficients are α1 = β2 = 1 and α2 = β1 = 0.1.
  • Cluster-state cases: Problems 13 and 14 represent a C4 graph of Ising-ZZ-coupled qubits intended for cluster-state preparation.The underlying physical system is a completely Ising-coupled set of four locally addressable ions.
  • NV-centre cases: Problems 15 and 16 optimise a CNOT gate for two strongly coupled nuclear spins at a diamond NV centre.The model is specified in the coupled-system eigenbasis and rotating frame using energy levels, a carrier frequency, and relative transition dipoles.

4. Special Applications of Spin Chains

The benchmark includes harder spin-chain and higher-spin cases, and finds that concurrent updates usually reach comparable fidelities faster, while hybrid handovers can combine early sequential and late concurrent advantages.

  • Spin-chain applications: Problems 17–19 extend Ising or Heisenberg spin chains with Stark-shift or global-field terms and restricted control structures.These cases test optimisation under modified drift Hamiltonians and simultaneous or local controls.
  • Spin-chain applications: Problems 20 and 21 use end-only x- and y-controls on isotropic Heisenberg chains, making the systems harder to steer.The chains have n = 3, 4 and model quantum-wire-like control through one or two spins at one end.
  • Higher-spin applications: Problems 22 and 23 study spin-j Hamiltonians with the z term as drift and the other angular-momentum terms as controls, using j = 6 and j = 3.These cases extend the benchmark beyond spin-1/2 systems.
  • Benchmark results: Across 23 test problems, sequential and concurrent schemes usually reach similar final fidelities, but concurrent updates are faster in most cases.Only problem 23 shows sequential average residual errors up to two orders of magnitude higher; final wall times are similar in problems 3, 4, 15, and 16.
  • Benchmark results: In many problems, concurrent optimisation overtakes sequential optimisation between quality 0.9 and 0.99.This supports switching from sequential to concurrent updates at medium fidelity; the example handover uses quality 0.93.
  • Hybrid strategies: Sequential optimisation is typically faster initially, whereas concurrent optimisation overtakes it near the target fidelity.The modular framework supports this handover strategy, while initial-control scaling can alter runtimes and crossing points.
  • Second-order methods: Sequential first-order and trust-region Newton methods provide similar gains per iteration, especially in the long run.The comparison averages trajectories from 100 random initial conditions.
  • Constraint handling: Concurrent constrained optimisation differs between fminunc and the slower fmincon, while the sequential cut-off constraint method has almost no performance effect.The comparison concerns unconstrained versus loosely constrained optimisation.

D. Preliminaries on Trust-Region Newton Methods for Sequential-Update Algorithms

The paper compares sequential and concurrent quantum-control updates across problem settings, showing that performance depends on update order, method order, fidelity target, and initial controls. It introduces dynamo as a modular framework for benchmarking these choices and identifies open directions including hybrid schemes and expensive objectives.

  • Performance comparisons: Sequential first-order updates can approach trust-region Newton quality per iteration, but their higher computational cost can make them slower overall.The comparison motivates using first-order gradients for sequential-update benchmarking.
  • Gradient methods: Gradient computation by diagonalisation is fastest for unitary optimisations, whereas standard approximations remain competitive for pure-state transfer.The comparison covers standard, exact, Taylor-series, Hausdorff-series, and finite-difference procedures.
  • Hybrid schemes: For second-order bfgs methods, fully concurrent grape is an optimum within similarly performing block schemes and cannot be accelerated by shrinking update blocks.The paper characterises bfgs with sequential updates as incompatible with the method’s update structure.
  • Framework and outlook: The study’s conclusions remain exploratory: initial-control selection lacks systematic guidelines, hybrid methods may improve results, and super-expensive objectives require new search algorithms.The authors also identify piecewise-constant time-domain parametrisation as a scope boundary for future benchmarking.
  • Framework and outlook: Dynamo combines subspace selection, gradients, updates, and step sizes while reducing matrix operations and enabling direct benchmarking of new algorithms.The framework also tracks updated controls and supports efficient comparison with existing schemes.
  • Practical recommendations: At high fidelities, grape-bfgs overtakes Krotov, while lower-quality optimisation gives sequential updates a small typical advantage.For unitary gate synthesis at fidelity 1 − 10^-4, grape-bfgs is sometimes more than one order of magnitude faster on average.

A. Exact Gradients (Eqn. (24))

The exact-gradient derivation uses spectral decomposition and substitutions for the propagator Hamiltonian. This approach relies on orthogonality properties that do not provide a comparably simple extension to generic non-normal Lindbladians.

  • Derivation: The derivation of Eqn. (24) applies the spectral theorem to calculate matrix functions through eigendecomposition.The construction uses eigenvectors and eigenvalues of Hermitian matrices.
  • Derivation: For unequal eigenvalues, the derivation uses the corresponding eigenvalue-dependent series identities, while the equal-eigenvalue case is handled separately.The passage explicitly distinguishes λ_l = λ_m from λ_l ≠ λ_m.
  • Derivation: Substituting the time-step Hamiltonian and control-Hamiltonian terms recovers Eqn. (24) for skew-Hermitian generators.The substitution maps A, B, and eigenvalues to the propagator quantities while retaining the eigenvectors.
  • Scope boundary: The exact-gradient construction depends on eigenvector orthogonality, so no equally simple extension is available for generic open systems with non-normal Lindbladians.The limitation concerns the direct spectral approach to exact gradients.

B. Standard Settings in a Nutshell

The paper formulates standard quantum-control tasks within a common iterative framework covering closed and open systems, unitary gates, pure states, density operators, and quantum maps. The algorithms use forward and backward propagation, fidelity evaluation, and iterative control updates.

  • Closed-system tasks: The standard tasks include approximate phase-insensitive and phase-sensitive unitary-gate synthesis in closed systems.Phase-insensitive synthesis absorbs global phase, whereas phase-sensitive synthesis fixes a determinant-compatible phase in SU(N).
  • Closed-system tasks: Pure-state and density-operator transfer tasks define initial and target boundary conditions, then optimise controls over a fixed, digitised final time.The procedures use forward and backward propagation followed by fidelity-based stopping criteria.
  • Algorithmic procedure: Across tasks, the iterative procedure sets controls, propagates forward and backward, evaluates fidelity, and repeats until a threshold or iteration limit is reached.The task descriptions reuse these steps across closed and open settings.
  • Open-system tasks: Open-system tasks extend the framework to Markovian quantum maps and density-operator transfer, with density operators represented in Liouville space.Task 6 is described as the rank-1 version of the quantum-map formulation in Task 5.
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