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Simultaneous Perturbation Stochastic Approximation of the Quantum Fisher Information

Julien Gacon, Christa Zoufal, Giuseppe Carleo, Stefan Woerner

arXiv:2103.09232v2quant-ph

TL;DR

Full QFIM evaluation scales quadratically with the number of parameters, a cost relevant to QNG and VarQITE. The paper uses SPSA to approximate the QFIM at constant circuit-evaluation cost, then applies QN-SPSA to ground-state preparation and Variational Quantum Boltzmann Machines, reporting favorable optimization results in the tested settings.

  • Problem

    Evaluating the QFIM at every iteration of VarQITE or QNG has quadratic cost in the number of circuit parameters, becoming expensive for complex objectives with many parameters.

    Source: §1 · p. 1

  • Method

    The paper applies SPSA to approximate the QFIM, yielding a VarQITE approach with a constant number of circuit evaluations per iteration.

    Source: §1 · p. 2

  • Results

    QN-SPSA combined QNG's fast convergence and robustness to initial parameters with SPSA's computational-cost benefits, and the tested ground-state runs made it the most effective tested optimization method; VarQBM training reproduced a Bell-state target distribution.

    Source: §5 · p. 10

  • Takeaways & Limitations

    The reduced circuit-evaluation requirement enables simulation and investigation of larger systems than previously possible, while complex Hamiltonians measured in many bases can make QN-SPSA's overhead over SPSA negligible.

    Source: §5 · pp. 10–11

    The inverse of the smoothed Hessian estimator is biased, and bias-removal techniques that might improve convergence are outside the paper's scope.

    Source: §2 · p. 3

Abstract

from arXiv · show

The Quantum Fisher Information matrix (QFIM) is a central metric in promising algorithms, such as Quantum Natural Gradient Descent and Variational Quantum Imaginary Time Evolution. Computing the full QFIM for a model with $d$ parameters, however, is computationally expensive and generally requires $\mathcal{O}(d^2)$ function evaluations. To remedy these increasing costs in high-dimensional parameter spaces, we propose using simultaneous perturbation stochastic approximation techniques to approximate the QFIM at a constant cost. We present the resulting algorithm and successfully apply it to prepare Hamiltonian ground states and train Variational Quantum Boltzmann Machines.

1 Introduction

Variational quantum algorithms use a classical–quantum feedback loop, and VarQITE and QNG are promising applications tied to quantum state evolution and optimization. Their repeated QFIM evaluations scale quadratically with parameter count; the paper proposes an SPSA-based approximation with constant circuit evaluations per iteration.

1 Introduction

  • Variational quantum algorithms use a classical–quantum feedback loop: the quantum circuit evaluates an objective and a classical counterpart updates its parameters.They are described as particularly suitable for near-term noisy quantum devices.Source: §1 · p. 1
  • VarQITE maps quantum imaginary-time state evolution to parameter evolution in a parameterized-circuit ansatz, with applications including Hamiltonian ground states and Gibbs-state preparation.The cited applications include quantum chemistry, combinatorial optimization, many-body simulation, and Quantum Boltzmann Machines.Source: §1 · p. 1
  • QNG follows steepest descent in quantum information geometry rather than ℓ2 geometry, offering re-parameterization invariance and updates adjusted to parameter-wise loss sensitivity.VarQITE coincides with QNG when the loss is the system energy.Source: §1 · p. 1
  • QFIM evaluation at every VarQITE or QNG iteration scales quadratically with circuit parameters, making it costly for complex objectives with many variational parameters.Diagonal or block-diagonal approximations may fail to capture parameter correlations, particularly when those correlations are strong.Source: §1 · pp. 1–2
  • The proposed SPSA-based QFIM approximation gives VarQITE a constant number of circuit evaluations per iteration, while additional evaluations can approximate its exact path arbitrarily closely.The method is particularly efficient when precise state evolution is unnecessary, such as in ground-state approximation.Source: §1 · p. 2

2 SPSA

SPSA reduces dimension-dependent evaluation costs by perturbing all parameters together in a random direction; its second-order extension estimates the Hessian stochastically. The section describes the estimator, stabilization measures, and conditions associated with convergence.

2 SPSA

  • SPSA estimates the gradient by simultaneously perturbing all parameters in a random direction, using two objective evaluations rather than O(d) evaluations.With suitable random-direction sampling, the gradient estimate is unbiased; SPSA is also suited to noisy objectives.Source: §2 · p. 2
  • 2-SPSA extends SPSA to approximate the Hessian, using samples from two random directions to form a stochastic Hessian estimate.The Hessian estimate is smoothed across previous samples; the second-order update uses that estimate.Source: §2 · pp. 2–3
  • The sampled Hessian point estimate is unbiased, but the inverse of its smoothed estimator is not; removing that bias might improve convergence and is beyond the paper's scope.Averaging additional resampled directions can systematically improve the Hessian approximation to arbitrary accuracy.Source: §2 · p. 3
  • Near a minimum, the Hessian approximation can be made positive semidefinite, and a positive diagonal regularizer is added to ensure invertibility.A blocking condition can further mitigate instabilities from a nearly singular estimate.Source: §2 · p. 3
  • A blocking rule accepts a candidate update only when its loss is below the current loss plus a tolerance; otherwise, the Hessian and gradient are resampled.For sample-based loss estimates, the cited guidance sets the tolerance to twice the loss standard deviation.Source: §2 · p. 3
  • SPSA and 2-SPSA convergence is guaranteed when conditions on evaluation noise, loss differentiability, and the metaparameters η, ϵ, and ∆ are satisfied.The passage refers readers to the cited convergence proof for the conditions and details.Source: §2 · p. 3

3 SPSA of the QFIM

QN-SPSA extends 2-SPSA to estimate the QFIM, then applies that approximation to QNG ground-state preparation and Gibbs-state preparation for VarQBMs. The approach replaces a quadratic-cost metric calculation with a constant-cost estimate, while VarQITE may trade some accuracy for reduced costs.

3 SPSA of the QFIM

  • QN-SPSA extends 2-SPSA to estimate the QFIM instead of the loss Hessian, for algorithms that access the QFIM at every iteration.The section applies it to QNG ground-state preparation and Gibbs-state preparation used to evaluate and train VarQBMs.Source: §3 · p. 3

3.1 Quantum Natural Gradient

  • Four expectation-value evaluations give a 2-SPSA stochastic approximation to the Fubini-Study metric, replacing a calculation that generally requires O(d^2) expectation values.The Fubini-Study metric and QFIM are equivalent up to a constant factor; the QFIM equals 4g.Source: §3.1 · p. 4
  • For the metric Hessian estimate, 2-SPSA perturbs only the metric’s second argument while keeping its first argument fixed.The metric can be represented using the squared state overlap, with smoothing and positive-semidefinite estimation retained from the preceding method.Source: §3.1 · p. 4

3.2 Quantum Boltzmann Machines

  • For Gibbs-state preparation, replacing g with the QN-SPSA estimate reduces costs while sacrificing some accuracy; VarQITE averages multiple point estimates per time step to track exact evolution more closely.This approximation is used for VarQBM evaluation and training.Source: §3 · p. 3, §3.2 · p. 5

4 Numerical Results

The numerical studies compare analytic and SPSA-based optimizers on ground-state and Gibbs-state tasks. QN-SPSA approaches QNG’s performance in several settings while using far fewer circuit evaluations in the 22-qubit example.

4.1.1 Speed of convergence

  • For 11 qubits, QN-SPSA outperforms SPSA and vanilla gradient descent and approaches analytic QNG’s loss, but with greater variance.Regularization keeps its mean loss from reaching QNG’s; the authors describe β as interpolating between natural and vanilla gradients.Source: §4.1.1 · p. 6
  • With 132 parameters, QNG requires approximately 2.6 million circuit executions, while QN-SPSA needs only 2100.The analytic gradients are omitted in this 22-qubit experiment because of their computational cost.Source: §4.1.1 · pp. 6–7

4.1.2 Region of convergence

  • QN-SPSA converges for the most sampled initial points, while stochastic methods are less affected by vanishing gradient components than their analytic counterparts.The SPSA variants are counted as converged when at least one of ten runs reaches the global optimum.Source: §4.1.2 · p. 7

4.2 Molecular Hamiltonian

  • QN-SPSA converges faster than SPSA and reaches a lower final energy for LiH on hardware, though its energy remains approximately 200mH above the exact energy.The comparison averages five experiments of 300 iterations each; hardware noise is offered as a likely explanation for the residual energy difference.Source: §4.2 · pp. 7–8
  • For the diagonal Hamiltonian, QN-SPSA takes fewer iterations but performs similarly to SPSA; for non-diagonal LiH, its added cost is negligible and convergence is significantly faster.The circuit-evaluation comparison covers the 11-qubit Pauli two-design and LiH experiments.Source: §4.2 · pp. 8–9, Figure 4 · p. 10

4.3 Quantum Boltzmann Machines

  • QN-SPSA VarQBMs reliably converge to the same loss as optimization with exact Gibbs-state preparation, and their output sampling distribution approximates the Bell distribution well.The target probabilities for |00⟩ and |11⟩ fall within the trained-state standard deviation; |01⟩ and |10⟩ have minuscule amplitudes, also present after exact preparation.Source: §4.3 · p. 10

5 Conclusion

QN-SPSA approximates the QFIM at constant circuit-execution cost and reproduces reported results on ground-state preparation and VarQBMs. The conclusion highlights its ground-state optimization performance, a Bell-distribution learning result, and limitations that merit further study.

5 Conclusion

  • SPSA reduces QFIM circuit executions from quadratic scaling in parameter count to constant cost, and QN-SPSA reproduced literature results using the analytic QFIM.The conclusion reports tests on ground-state preparation and VarQBMs.Source: §5 · p. 10
  • QN-SPSA retained QNG's fast convergence and robustness to initial parameters while achieving the tested algorithms' most effective ground-state optimization.The authors also report that QN-SPSA's overhead over SPSA becomes negligible when a complex Hamiltonian requires measurements in many bases.Source: §5 · p. 10
  • VarQBMs successfully trained a Gibbs state to reproduce a Bell-state target distribution; in this small example, the speed-up was less significant than for ground-state calculations due to required re-sampling.Source: §5 · pp. 10–11
  • QN-SPSA can use first-order SPSA initially for a stable QFIM estimate, calibrate finite-difference perturbations, and adapt sampled dimensions to step acceptance.Source: §5 · p. 11
  • Estimating QFIM point-values requires overlaps between differently parameterized ansatze, and available overlap algorithms may duplicate circuit depth or width or fail to scale.Reducing this overhead remains an open research question, particularly for noisy-device use.Source: §5 · p. 11
  • The authors present QN-SPSA as a promising efficient optimizer, citing fewer evaluations than original QNG as a step toward practically relevant problem sizes.Source: §5 · p. 11

A Variational Quantum Imaginary Time Evolution

VarQITE approximates imaginary-time evolution with a time-discretized variational ansatz. McLachlan's principle yields a linear system whose numerical solution defines the parameter evolution.

A Variational Quantum Imaginary Time Evolution

  • A time-discretized variational ansatz represents the evolving quantum state, with circuit parameters associated with time.Source: A Variational Quantum Imaginary Time… · p. 15
  • McLachlan's variational principle minimizes the distance between the Wick-rotated Schrödinger equation's two sides over the variational state space.Source: A Variational Quantum Imaginary Time… · p. 15
  • The variational principle produces a linear system that, with an initial parameter value, defines an initial-value problem solvable by an ODE solver.Source: A Variational Quantum Imaginary Time… · p. 15
  • With sufficiently small time steps and an expressive circuit, VarQITE steps do not increase system energy; ground-state search requires initial overlap with the ground state.In the infinite-time limit, higher-energy contributions are damped, but practical simulation requires choosing a sufficiently large finite time.Source: A Variational Quantum Imaginary Time… · p. 15

B Minimization-formulation of Vanilla and Natural Gradient Descent

This appendix shows that the vanilla and natural gradient descent updates follow from their respective minimization formulations. In each case, differentiating the objective with respect to the update and setting it to zero recovers the update rule.

B Minimization-formulation of Vanilla and Natural Gradient Descent

C Comparison of the QFIM Formulas

The appendix establishes that the two QFI formulas are equivalent and justifies the coefficient 1/2 in Eq. (12). It uses normalization of the quantum state to rewrite a second-derivative term.

C Comparison of the QFIM Formulas

D Influence of QFIM Regularization

QN-SPSA can produce singular QFIM estimates, which complicates the required inversion. The paper regularizes the estimate to stabilize optimization while balancing numerical stability against fidelity to the QFIM.

D Influence of QFIM Regularization

  • QN-SPSA's up-to-rank-2 QFI estimate can be singular, complicating inversion or solving a linear system.The issue can arise even when optimization starts from the full-rank identity matrix.Source: D Influence of QFIM Regularization · p. 17
  • Taking absolute eigenvalues and adding β > 0 to the diagonal makes each estimate symmetric positive definite.The normalized form (|A| + βI)/(1 + β) avoids the step size approaching zero as β grows.Source: D Influence of QFIM Regularization · p. 17
  • β trades numerical stability against faithful QFIM approximation, interpolating between natural and vanilla gradient descent.A small β yields an update closer to QNG but is more prone to numerical instability; a large β moves toward vanilla gradient descent.Source: D Influence of QFIM Regularization · p. 17

E Convergence efficiency of natural gradients

In a weighted MAXCUT example, QN-SPSA outperforms SPSA in convergence even after accounting for additional evaluation costs. The comparison also illustrates how natural-gradient updates account for objective sensitivity, whereas vanilla-gradient learning rates require careful tuning.

E Convergence efficiency of natural gradients

  • Natural gradients converge faster than vanilla gradients in iterations, but their extra circuit-evaluation costs can offset that advantage.QN-SPSA's small overhead may make it more efficient than first-order SPSA even when the diagonal Hamiltonian requires only one measurement per loss evaluation.Source: E Convergence efficiency of natural… · p. 17
  • QN-SPSA outperforms SPSA on weighted MAXCUT, including manually calibrated SPSA, when additional evaluation costs are counted.Figure 7(c) compares convergence by required function evaluations; the loss summaries are means and standard deviations across 25 runs.Source: Figure 7 · p. 18, E Convergence efficiency of natural… · p. 18
  • Vanilla-gradient methods require learning-rate tuning to objective sensitivity, while natural-gradient learning rates control model change and can be set largely independently of the model.The MAXCUT example uses a loss landscape with sharp regions near global minima, where gradients can be much larger than in flatter surroundings.Source: E Convergence efficiency of natural… · pp. 18–19
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