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Optimizing Variational Quantum Algorithms using Pontryagin's Minimum Principle

Zhi-Cheng Yang, Armin Rahmani, Alireza Shabani, Hartmut Neven, Claudio Chamon

arXiv:1607.06473v3quant-phcs.CC

TL;DR

Variational quantum algorithms need effective finite-time control protocols for optimization problems where slow adiabatic evolution can be impractical. This paper applies Pontryagin’s minimum principle to derive bang-bang protocols for closed and Markovian open systems, finding system-size-independent pulse-duration distributions in the SK model and improved performance over conventional quantum annealing under the weak-noise conditions studied.

  • Problem

    VQAs require effective protocol parameterizations for finite-time quantum optimization, while adiabatic approaches can involve long timescales and weak energy gaps.

  • Method

    The paper applies Pontryagin’s minimum principle to VQA control dynamics, including closed systems and Markovian open systems, and studies the resulting protocols in the SK spin glass.

  • Results

    The optimal protocols have bang-bang form, with system-size-independent pulse-duration distributions in the studied SK model and lower errors than linear quantum-annealing ramps under weak noise and thermal coupling.

  • Takeaways & Limitations

    Bang-bang optimality and a characteristic pulse timescale support efficient protocol parameterization and the square-pulse ansatz used by QAOA.

Abstract

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We use Pontryagin's minimum principle to optimize variational quantum algorithms. We show that for a fixed computation time, the optimal evolution has a bang-bang (square pulse) form, both for closed and open quantum systems with Markovian decoherence. Our findings support the choice of evolution ansatz in the recently proposed Quantum Approximate Optimization Algorithm. Focusing on the Sherrington-Kirkpatrick spin-glass as an example, we find a system-size independent distribution of the duration of pulses, with characteristic time scale set by the inverse of the coupling constants in the Hamiltonian. The optimality of the bang-bang protocols and the characteristic time scale of the pulses provide an efficient parameterization of the protocol and inform the search for effective hybrid (classical and quantum) schemes for tackling combinatorial optimization problems. For the particular systems we study, we find numerically that the optimal nonadiabatic bang-bang protocols outperform conventional quantum annealing in the presence of weak white additive external noise and weak coupling to a thermal bath modeled with the Redfield master equation.

I. INTRODUCTION

Variational quantum algorithms search for nonadiabatic control paths through a classical–quantum feedback loop. The paper connects this framework to optimal control and argues that bounded-time protocols can be represented by bang-bang pulses.

  • Motivation: Quantum annealing suppresses excitations through sufficiently slow Hamiltonian changes, but alternative rapid paths can also reach low-energy states.Variational algorithms search over such paths by optimizing a fixed number of protocol parameters.
  • Reachability: Increasing the total time expands the reachable set of final states, allowing the optimal protocol to approach a low-energy target more closely.Beyond a critical time at which the target becomes reachable, increasing time offers no further advantage.
  • Contribution: The paper connects VQAs to optimal control theory and uses Pontryagin’s minimum principle to identify their optimal protocol structure.This connection provides the basis for the paper’s analysis of bounded-time evolution.
  • Variational quantum algorithms: VQAs use classical optimization and quantum measurements in a closed loop to select time-dependent variational protocols.The quantum system generates final states for candidate protocols, while a classical computer updates the parameters.
  • Contribution: For a fixed total time, the optimal protocol has a bang-bang form, supporting the square-pulse ansatz used by QAOA.The paper contrasts this with continuous-time variational evolution.

A. Bang-bang optimal protocols

Because the dynamical equations and optimal-control Hamiltonian are linear in bounded controls, Pontryagin’s minimum principle selects control values at their permissible extremes. This yields nonadiabatic bang-bang protocols, with a finite pulse timescale set by the Hamiltonian energy scale in the SK model.

  • Control setting: Bounded Hamiltonian controls are assumed to vary independently within prescribed ranges during the fixed evolution interval.For a fixed initial state, these controls determine the final wavefunction and therefore the terminal cost.
  • Pontryagin’s principle: Pontryagin’s minimum principle minimizes the optimal-control Hamiltonian pointwise over the admissible controls.The cost depends on the final dynamical variables, with conjugate variables satisfying backward equations and terminal boundary conditions.
  • Bang-bang optimal protocols: When the optimal-control Hamiltonian is linear in a control, the minimizing value is generically its minimum or maximum permissible value.A finite interval with a vanishing coefficient is the stated caveat to this argument.
  • Bang-bang optimal protocols: Quantum dynamics linear in the controls therefore produces sudden switches between control extrema, yielding nonadiabatic bang-bang protocols.The result applies regardless of the number of variational parameters in the formulation described.
  • Pulse parametrization: For the SK Ising spin-glass model, the finite timescale of fixed couplings is set by the Hamiltonian energy scale and determines how many switching times are needed.As the characteristic timescale grows, fewer parameters are needed to represent the protocol.

B. Presence of decoherence

The Pontryagin framework extends from closed systems to Markovian open-system dynamics. When the controls are controllable, both Hamiltonian controls and controllable decoherence rates take bang-bang form.

  • Open-system extension: Pontryagin’s principle applies to open quantum systems whose Markovian dynamics are described by a Lindblad equation.The extension follows from the linearity of the open-system dynamical equation.
  • Bang-bang controls: For controllable open-system protocols, the Hamiltonian controls and decoherence controls are bang-bang.The decoherence operators can represent noise in Hamiltonian parameters or an engineered bath.

IV. VQA FOR THE SK SPIN-GLASS MODEL

The paper applies VQA to the Sherrington–Kirkpatrick spin glass by optimizing a bounded control that interpolates between a transverse-field Hamiltonian and the problem Hamiltonian. The resulting square-pulse ansatz offers a variational alternative to slow adiabatic evolution.

  • Problem model: The SK Ising spin glass is a combinatorial optimization problem whose objective is to minimize an energy function over 2^n spin configurations.Many practical combinatorial optimization problems map to this model, while classical minimization has exponential cost in n.
  • VQA setup: The VQA uses a parameterized Hamiltonian with one bounded control g(t) and initializes the system in a ground state of the transverse-field Hamiltonian.The control range is 0 ≤ g(t) ≤ 1, and the objective is the final expectation value of the SK cost Hamiltonian.
  • Protocol comparison: Unlike the smooth linear ramp of adiabatic evolution, the variational protocol permits arbitrary time dependence and is optimized at fixed total time.Shorter times are desirable for variational searches because evaluating many candidate protocols requires repeated measurements.
  • Square-pulse ansatz: The QAOA-style ansatz alternates sudden evolutions under B and C, with total time T = Σ_i(γ_i + β_i).B and C are sums of commuting one- and/or two-qubit terms, so the protocol can be interpreted as a sequence of simple gates.
  • Square-pulse ansatz: Pontryagin’s principle identifies this bounded alternating square-pulse ansatz as the optimal VQA choice for the stated SK control setting.The number of variational layers needed depends on the characteristic pulse timescales analyzed in the paper.

V. NUMERICAL STUDIES

Numerical studies find bang-bang protocols through unbiased Monte Carlo searches and faster interior-point optimization, with pulse durations showing weak system-size dependence. At equal total time, the optimized nonadiabatic protocols outperform linear quantum annealing in the studied ideal systems.

  • Protocol optimization: Monte Carlo optimization converged to bang-bang protocols from different initial protocols for n = 5 and T = 0.8.The search used piecewise-constant controls without assuming a bang-bang form.
  • Protocol optimization: Interior-point minimization agreed with Monte Carlo results and ran much faster for optimized protocols.For T = 2, the ansatz used around ∼20 × T variational parameters and converged to fewer bangs than allowed.
  • Pulse time scales: The typical bang duration is determined by a characteristic Hamiltonian energy scale and remains finite rather than vanishing with system size.This distinguishes the protocols from generic Trotterization, whose pulse durations must approach zero.
  • Pulse time scales: The bang-time distributions for n = 6, 7, 8, 9 and 10 collapsed onto nearly the same peak at fixed T = 2.The distributions were averaged over 50 Hamiltonian instances, indicating a typical pulse scale with extremely weak dependence on n.
  • Performance: At equal total time, nonadiabatic bang-bang protocols significantly outperformed linear quantum-annealing ramps in final-state and final-energy errors for the studied systems.The comparison averaged errors over 20 of 50 generated realizations selected for the highest success rates.

VI. EFFECTS OF DISSIPATION AND DEPHASING

The study evaluates bang-bang protocols against quantum annealing under external-control and thermal-environment perturbations. It frames robustness to these effects as necessary for practical applications.

  • Robustness analysis: The authors compare bang-bang and quantum-annealing protocols under noise in external controls and coupling to a thermal environment.The section motivates these two noise sources as inevitable in real-world implementations.

A. Random Dephasing Noise

The random-dephasing study models hardware-like noise with random fields and compares its effect on bang-bang and linear quantum-annealing protocols. In the studied weak-noise regime, bang-bang fidelity remains higher at the strongest tested noise strength, while the white-noise model has a practical high-frequency limitation.

  • Noise model: Random fields in the x and z directions model pure dephasing associated with hardware electronics.For bang-bang evolution, the control takes g(t) = 1 or g(t) = 0 at each time.
  • Noise model: The noise-averaged density matrix evolves through a master equation under independent white noise with zero mean and specified second moments.The noise strengths are taken equal, Wb = Wh = W, for simplicity.
  • Evaluation: The comparison tracks fidelity error and final-energy error for bang-bang and linear quantum-annealing protocols at different noise strengths.The fidelity error is 1 − ⟨ψGS|ρ(T)|ψGS⟩, while the energy error is Tr[ρ(T)C] − EGS.
  • Results: At small W, noise only slightly decreases fidelity, and at W = 0.01 the optimal bang-bang protocol retains higher fidelity than the linear ramp.The linear-ramp error changes from its W = 0 value by an amount of the same order of magnitude.
  • Model limitation: The white-noise model is an idealization because real experiments have a finite correlation time and characteristic high-frequency cutoff.The authors note that the delta-correlated model implies infinitely large random fields, despite their lack of correlation.

B. Weak Thermal Bath

The protocols are evaluated under weak thermal-bath coupling using a Redfield master equation and an Ohmic Markovian bath model. Across coupling strengths, the optimal bang-bang protocol retains higher fidelity than QAA, despite an intermediate regime where QAA is more robust.

  • Bath model: The open-system dynamics is modeled with the Redfield master equation for a weak thermal bath at temperature 1/β.The bath is assumed Ohmic and in thermal equilibrium; an infinite cutoff frequency enforces the Markovian approximation.
  • Bath model: The coupling strength η is a dimensionless parameter controlling the interaction between the system and environment.
  • Fidelity under coupling: 8.5 ≲ T ≲ 11.5: QAA shows remarkable robustness and a much smaller change in η = 0 error than the bang-bang protocol.
  • Fidelity under coupling: As T increases, the errors of the bang-bang and QAA protocols become closer, but the optimal bang-bang protocol retains higher fidelity under open-system dynamics.

VII. PULSE DURATION FROM THE PONTRYAGIN’S MINIMUM PRINCIPLE

Pontryagin’s minimum principle determines bang-switching times through the conjugate-momentum dynamics. In the SK model, the resulting pulse-duration distribution has a finite, system-size-independent characteristic time set by the problem’s energy scale.

  • Control dynamics: The wave function is expanded in the computational basis, with amplitudes initialized by the all-spins-x state before the Schrödinger dynamics is optimized.
  • Switching rule: Pontryagin’s minimum principle determines the optimal bang-bang control, which takes values 0 and 1 and switches when the control-Hamiltonian derivative vanishes.
  • Switching rule: Within a g(t) = 1 interval, the first root of w(t) = 0 after the bang begins determines the pulse duration.
  • Characteristic scale: The pulse-duration scale is governed by the energy difference ΔCz,k, whose time dependence appears through e^-i(Cz−C̄z(k))t.
  • Characteristic scale: The characteristic time scale is finite and system-size independent, unlike generic-protocol Trotterization, where individual pulse durations must approach zero.

VIII. SUMMARY AND OUTLOOK

The paper establishes bang-bang protocols as optimal for bounded linear controls and verifies their performance numerically in the SK spin glass. These protocols reduce error relative to QAA, remain advantageous under weak noise, and have a system-size-independent pulse timescale that can reduce variational complexity.

  • Summary: Pontryagin’s minimum principle predicts bang-bang protocols for VQAs with bounded linear control parameters, a result verified numerically for an SK spin glass.
  • Summary: The optimal nonadiabatic bang-bang protocols significantly reduce error versus QAA at the same running time, including under weak white noise and weak thermal-environment coupling.
  • Outlook: The characteristic time between bangs is fixed by the problem’s energy scales and is independent of system size.
  • Outlook: Bang-bang optimality and size-independent pulse durations inform hybrid classical-quantum schemes for combinatorial optimization.
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