From pulses to circuits and back again: A quantum optimal control perspective on variational quantum algorithms

I. INTRODUCTION

• NISQ devices are being explored for classically difficult problems because current devices cannot yet implement landmark fault-tolerant algorithms at scale [1, 2].The paper presents VQAs as candidates for applications including chemistry, optimization [6], and machine learning.Source: §I · p. 1
• VQA implementations have not yet improved on classical capabilities, motivating this Perspective’s search for progress through a QOC lens.The paper focuses on VQA challenges involving ansatz selection, optimization landscapes, noise, and robustness.Source: §I · pp. 1–2
• QOC optimizes control pulses by iteratively minimizing an objective that measures how well a desired control task is achieved.Early qubit-control experiments used electromagnetic pulses, including to implement high-fidelity entangling gates.Source: §I · p. 1
• As devices scaled, research broadened from pulse optimization to circuit compilation, and the authors propose returning to pulse-level ideas to inform VQAs.The Perspective aims to leverage QOC results and tools for VQA challenges.Source: §I · p. 2
• Both VQAs and QOC use quantum-classical optimization loops, but VQAs optimize circuit elements while conventional QOC optimizes continuous pulses.VQA users typically need not know the underlying pulse-level physics.Source: §I · p. 2, Figure 1 · p. 2

VQA formulation

• VQAs minimize an objective over circuit parameters by repeatedly preparing and measuring a parametrized state, then updating parameters with classical optimization.For a linear problem Hamiltonian Hp, exact minimization corresponds to preparing its ground state.Source: §II · p. 2
• Estimating the objective to precision ϵ requires Ns ∝ λ^2/ϵ^2 state re-preparations and measurements, while grouping terms can reduce naive measurement scaling by orders of magnitude.Here λ = Σj |αj| for the Pauli expansion of Hp.Source: §II · p. 3
• VQA objectives are estimated from the quantum system rather than requiring exact parameter labels, giving some robustness to drifts, crosstalk, and other systematic errors.The section also describes QAOA as alternating problem- and driver-Hamiltonian operations, with the original optimum attainable as p → ∞ [6].Source: §II · pp. 3–4

Ansatz selection

• A scalable ansatz should have circuit depth and parameter count that each scale as O(poly(n)); VQE’s unitary coupled-cluster ansatz carries fermionic structure into its gate representation.The unitary ansatz preserves physical properties but is not efficient to evaluate classically.Source: §II · p. 3

Ground-state chemistry

• NISQ noise limits VQE accuracy: devices with ⪆ 50 qubits contrasted with experiments using ≲ 10, while a 12-qubit, approximately 200-gate hydrogen-chain calculation reached chemical accuracy.The reported 12-atom hydrogen-chain calculation used error mitigation from symmetries in a reduced subspace.Source: §II · p. 3

Combinatorial optimization

• Higher-depth QAOA experiments at p = 2 or p = 3 improved solution quality beyond p = 1, although deeper circuits can incur noise and decoherence.Most earlier experiments used p = 1; its two parameters could be optimized analytically, so no variational loop was needed.Source: §II · p. 4

III. QUANTUM OPTIMAL CONTROL

• QOC designs electromagnetic fields or pulses to reach a target state, expectation value, or evolution operator at terminal time T.The objective can incorporate physical constraints, laboratory resources, and robustness to errors or uncertainties.Source: §III · p. 4
• QOC models controls as fields coupled to a drift Hamiltonian, typically under semiclassical and dipole approximations.The resulting bilinear control system follows the Schrödinger equation and is generally not analytically tractable.Source: §III · p. 4
• The QOC problem seeks fields that minimize the terminal-time expectation of H_p, subject to Schrödinger dynamics.Solutions can be constructed iteratively using Euler–Lagrange equations and methods including GRAPE, Krotov, TBQCP, D-MORPH, and SCP.Source: §III · p. 4
• In practice, QOC parametrizes fields with variables c_i and iteratively updates them to minimize the control objective.Learning control evaluates the objective on the quantum system without requiring an underlying system model, inherently accounting for parameter uncertainties and systematic errors.Source: §III · pp. 4–5
• VQAs are digital QOC experiments: both use quantum-classical optimization loops, but VQAs optimize a parametrized circuit ansatz.The circuit ansatz can have tunable depth, dimension, structure, and continuous and discrete parameters.Source: §III · p. 5

IV. PRIOR WORK CONNECTING QOC WITH VQAS

• Continuous-time reformulations of variational quantum optimization can remove discrete optimization and enable optimal-control methods, subject to hardware capabilities.Natural controls may address individual qubits or couplings between two qubits.Source: §IV · p. 5
• Under amplitude-constrained fields and finite solution time, Yang et al. found bang-bang control optimal, while Lin et al. found a bang-singular-bang structure for Grover search.Source: §IV · p. 5
• Controllability analyses link QOC to computational universality and have examined QAOA universality, angle requirements, and strategies to improve reachability [87] [89].The Lie rank criterion connects the dynamical Lie algebra to whether arbitrary unitaries can be generated by shaping unconstrained fields.Source: §IV · p. 5
• Studies of bang-bang protocols examine robustness and reachability across ansatz depths, and report exponential speedups over quasistatic scheduling on certain instances [91] [92] [93].Source: §IV · p. 5
• For fixed-time ground-state preparation, Brady et al. argue the optimal QAOA variational schedule can be bang-anneal-bang rather than bang-bang [95].Other connections map ring-of-disagrees QAOA to independent-spin control and formulate machine-learning tasks as QOC problems on NISQ devices [97].Source: §IV · p. 6
• QOC-inspired ansatz methods adapt circuit structure using derivative information or driving Hamiltonians, including approaches aimed at reducing convergence depth [99] [100] [101] [102].One proposed VQE approach designs ansätze through device-level QOC rather than parametrized quantum circuits [102].Source: §IV · p. 6

V. NEW DIRECTIONS FOR VQAS INFORMED BY QOC

• A shared objective function can be optimized through different natural control parametrizations at pulse, circuit, and logical-circuit levels.The hierarchy links pulse parameters c_i, circuit parameters θ_i, and logical-circuit structure and encoding and decoding maps E_φ and D_φ.Source: §V · p. 6, Figure 2 · p. 7
• At the pulse level, control is parametrized through fields or control Hamiltonians; at the circuit level, it is parametrized through gate parameters.QOC typically operates at the pulse level and VQAs at the circuit level.Source: §V · p. 6
• At the logical-circuit level, physical circuits are converted using error-correcting codes and logical gates, with rotation angles synthesized to a specified precision.Error-corrected experiments operate at this level and benefit from reduced hardware-noise susceptibility through encoding and fault-tolerant operations.Source: §V · p. 6
• Error-correction procedures and decoding parameters could become variational parameters, making QEC part of the learning-control environment.Although maintaining fault tolerance under such variation can be challenging, optimizing small encodings could be useful when only limited error correction is available before fully fault-tolerant operation.Source: §V · pp. 6–7

A. Ansatz selection

• Paths through the abstraction hierarchy can produce a family of richer variational ansätze by connecting parametrizations across levels.The framework allows choices of parametrization to be carried forward, changed, and iterated.Source: A. Ansatz selection · p. 7
• Hamiltonian simulation maps parametrized control evolution into a circuit whose parameters remain determined by the control-field parametrization.This retains pulse-level optimization while enabling conversion to an error-correctable logical circuit through discretization.Source: A. Ansatz selection · p. 7
• Mapping a circuit ansatz back to control parameters is generally non-unique, but gate-wise mappings can yield hybrid control ansätze with added free parameters.Freeing control parameters, including for fixed gates, can systematically increase the resulting circuit’s expressiveness.Source: A. Ansatz selection · pp. 7–8
• For the H2 example, adding controls for the controlled-phase evolution creates a hybrid ansatz that may improve results when gate errors occur.The example also re-digitizes the analog evolution into a Trotter factorization while retaining the original θ parameter.Source: A. Ansatz selection · p. 8
• Iterating mappings between control and circuit descriptions can combine their tools and relate controllability to circuit expressiveness.The proposed cross-connections may also let digital-simulation concepts inform control theory and vice versa.Source: A. Ansatz selection · p. 8

B. Optimization landscapes

• Circuit parameters make the VQA objective non-convex, so local optima can appear despite convexity when optimizing over eigenstate probabilities.The difficulty of finding a global minimum depends on the landscape and available prior information about good optima.Source: B. Optimization landscapes · p. 8
• With a full dynamical Lie algebra, unconstrained controls, and local surjectivity, common QOC landscapes have no local extrema.The paper notes that sufficient control resources are application-dependent and remain an open challenge to assess systematically.Source: B. Optimization landscapes · p. 9
• QOC landscape theory and tools could help analyze VQA ansätze, but practical scaling is difficult when control parameters are insufficient or constrained.Further work is needed when resources scale as O(poly(log(d))) rather than with system dimension; restricted symmetry subspaces are proposed as a possible direction.Source: B. Optimization landscapes · p. 9
• Barren plateaus affect both VQA and QOC landscapes, where gradients can become exponentially small as qubit number increases.The passage links avoidance strategies to careful ansatz, initialization, and training design.Source: B. Optimization landscapes · p. 9

C. Noise and time-optimal control

• QOC robust-control strategies seek pulses that suppress random control errors, and the Hessian of J can help assess robustness.The authors suggest translating these tools to improve VQAs against general errors.Source: C. Noise and time-optimal control · p. 9
• When coherence times are limited and stochastic gate errors dominate, minimum-depth circuits are desirable, linking VQA design to time-optimal control.Time-optimal fields correspond to geodesics on the unitary group; exact results are known only for certain 1–3 qubit systems.Source: C. Noise and time-optimal control · p. 10
• For n-qubit networks, available bounds characterize transformations achievable in polynomial time with 2n local fields, but reachability with fewer controls remains open.Additional control resources may also enable decoupling schemes that suppress interactions with the environment.Source: C. Noise and time-optimal control · pp. 9–10

D. Robust control through digitization

• Controls for unknown quantum states can be mapped to parametrized circuits and then error-corrected circuits for more accurate application.The input state may also need encoding, and effective digital encoding of states from nature has not yet been realized.Source: D. Robust control through digitization · p. 10
• For an N-body state of dimension d ∼ exp(N), maintaining constant fidelity in the stated analog-control model requires precision δ ∝ d^-1/2.The passage frames this as an exponential-in-N control-precision requirement.Source: D. Robust control through digitization · p. 10
• If the state is transduced into n = log2(d) qubits, a VQA circuit with sufficiently high-fidelity gates and quantum error correction can bring control fidelity arbitrarily close to 1 with logarithmic physical-resource overhead.This is conditional on transduction into qubits and sufficiently high gate fidelity.Source: D. Robust control through digitization · p. 10
• Manipulating unknown quantum states in a VQA manner includes quantum machine learning with quantum-state training data and connects to query-based proofs.The passage notes that stronger proofs are possible when only a limited number of quantum states are available.Source: D. Robust control through digitization · p. 10

• QOC concepts and the broader framework integrating traditional QOC models with VQAs are presented as critical to bridging theoretical VQAs and NISQ hardware.Source: §VI · p. 10
• For VQAs, QOC tools can provide richer variational structures and a deeper understanding of variational experiments.Source: §VI · p. 11
• For QOC, VQAs offer an emerging frontier in many-body quantum control that can push the field in new directions.Source: §VI · p. 11