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A Novel Noise-Aware Classical Optimizer for Variational Quantum Algorithms

Jeffrey Larson, Matt Menickelly, Jiahao Shi

TL;DR

The paper tackles robust optimization for variational quantum algorithms by developing a noise-aware derivative-free trust-region method. Grounded in a generalized zeroth-order noise model and Cao's noise-aware framework, it introduces ANATRA, which uses minimum Frobenius-norm quadratic interpolation and careful interpolation-set management with decoupled sampling and trust-region radii. Theoretical results establish high-probability convergence to an $\epsilon$-neighborhood with a worst-case rate of $\mathcal{O}(\epsilon^{-2})$ under both bounded and subexponential noise, while numerical experiments show ANATRA outperforming several baselines, especially in high-noise regimes typical of low-shot VQA evaluations. The findings suggest that explicitly accounting for noise in classical optimizers can materially improve the reliability and efficiency of VQA workflows on noisy quantum hardware.

Abstract

A key component of variational quantum algorithms (VQAs) is the choice of classical optimizer employed to update the parameterization of an ansatz. It is well recognized that quantum algorithms will, for the foreseeable future, necessarily be run on noisy devices with limited fidelities. Thus, the evaluation of an objective function (e.g., the guiding function in the quantum approximate optimization algorithm (QAOA) or the expectation of the electronic Hamiltonian in variational quantum eigensolver (VQE)) required by a classical optimizer is subject not only to stochastic error from estimating an expected value but also to error resulting from intermittent hardware noise. Model-based derivative-free optimization methods have emerged as popular choices of a classical optimizer in the noisy VQA setting, based on empirical studies. However, these optimization methods were not explicitly designed with the consideration of noise. In this work we adapt recent developments from the ``noise-aware numerical optimization'' literature to these commonly used derivative-free model-based methods. We introduce the key defining characteristics of these novel noise-aware derivative-free model-based methods that separate them from standard model-based methods. We study an implementation of such noise-aware derivative-free model-based methods and compare its performance on demonstrative VQA simulations to classical solvers packaged in \texttt{scikit-quant}.

A Novel Noise-Aware Classical Optimizer for Variational Quantum Algorithms

TL;DR

The paper tackles robust optimization for variational quantum algorithms by developing a noise-aware derivative-free trust-region method. Grounded in a generalized zeroth-order noise model and Cao's noise-aware framework, it introduces ANATRA, which uses minimum Frobenius-norm quadratic interpolation and careful interpolation-set management with decoupled sampling and trust-region radii. Theoretical results establish high-probability convergence to an -neighborhood with a worst-case rate of under both bounded and subexponential noise, while numerical experiments show ANATRA outperforming several baselines, especially in high-noise regimes typical of low-shot VQA evaluations. The findings suggest that explicitly accounting for noise in classical optimizers can materially improve the reliability and efficiency of VQA workflows on noisy quantum hardware.

Abstract

A key component of variational quantum algorithms (VQAs) is the choice of classical optimizer employed to update the parameterization of an ansatz. It is well recognized that quantum algorithms will, for the foreseeable future, necessarily be run on noisy devices with limited fidelities. Thus, the evaluation of an objective function (e.g., the guiding function in the quantum approximate optimization algorithm (QAOA) or the expectation of the electronic Hamiltonian in variational quantum eigensolver (VQE)) required by a classical optimizer is subject not only to stochastic error from estimating an expected value but also to error resulting from intermittent hardware noise. Model-based derivative-free optimization methods have emerged as popular choices of a classical optimizer in the noisy VQA setting, based on empirical studies. However, these optimization methods were not explicitly designed with the consideration of noise. In this work we adapt recent developments from the ``noise-aware numerical optimization'' literature to these commonly used derivative-free model-based methods. We introduce the key defining characteristics of these novel noise-aware derivative-free model-based methods that separate them from standard model-based methods. We study an implementation of such noise-aware derivative-free model-based methods and compare its performance on demonstrative VQA simulations to classical solvers packaged in \texttt{scikit-quant}.
Paper Structure (25 sections, 9 theorems, 64 equations, 5 figures)

This paper contains 25 sections, 9 theorems, 64 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Existing Methods
  3. Our Contribution
  4. Paper Organization
  5. Setting
  6. A Noise Model
  7. A Noise-Aware Trust-Region Algorithmic Framework
  8. Preliminary Assumptions and Analysis
  9. A Practical Noise-Aware Derivative-Free Algorithm Based on \ref{['alg.TR.noise']}
  10. Geometry of Interpolation Sets
  11. Decoupling the Sampling Radius from the Trust Region Radius
  12. Careful Maintenance of Poisedness
  13. Statement of DFO Algorithm
  14. Theoretical Results
  15. Numerical Results
  16. ...and 10 more sections

Key Result

Theorem 1

Suppose ass.L.smoothass.H.upper.bounded are satisfied. Suppose we have access to a zeroth-order oracle of type1 with parameter $\epsilon_f$ and a first-order oracle with parameters $\kappa_{eg}, \epsilon_g,$ and $p_1$. Let $\{\Theta_k\}$ denote the sequence of random variables with realizations $\{\ it holds that for any $T\in \mathcal{O}\left(\frac{1}{\epsilon^2}\right)$.

Figures (5)

  • Figure 1: Results for $d=2$-dimensional quadratic problems \ref{['eq:synthetic_quadratic']}. Top row corresponds to uniform noise; bottom row corresponds to Gaussian noise. Throughout all of these plots, the solid lines correspond to the median ground truth objective value over 30 runs of the best point evaluated by the solver.
  • Figure 2: Same as \ref{['fig:2dquadratic']} but with $d=10$-dimensional quadratic functions.
  • Figure 3: Comparing solvers on \ref{['eq:synthetic_rosenbrock']}. Top plots correspond to uniform noise; bottom plots correspond to Gaussian noise.
  • Figure 4: Comparing solvers on QAOA MaxCut problems. Top plots correspond to the toy graph, and bottom plots correspond to the Chvátal graph.
  • Figure 5: Comparing the performance of ANATRA and PyBOBYQA on the QAOA MaxCut problems with the toy graph, but on IBM Algiers, as opposed to an idealized QASM simulator.

Theorems & Definitions (21)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Remark 1
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 3
  • proof
  • ...and 11 more