PaperlayerView exact revision on arXiv

Source-linked AI summary

Accelerating Noisy VQE Optimization with Gaussian Processes

Juliane Mueller, Wim Lavrijsen, Costin Iancu, Wibe de Jong

arXiv:2204.07331v3quant-phmath.OC

TL;DR

Noisy VQE combines classical optimization with expensive quantum-chip evaluations, requiring tens of thousands or more evaluations for interesting applications. GP+ImFil uses an adaptive Gaussian-process surrogate for global search and seeds multiple local Implicit Filtering optimizations near promising regions. For noisy problems, GP+ImFil finds improvements faster and overall better solutions than ImFil across Hubbard-model instances spanning 2–20 optimization parameters.

  • Problem

    Noisy VQE combines classical optimization with expensive quantum-chip evaluations, requiring tens of thousands or more evaluations for interesting applications.

    Source: §I · p. 1

  • Method

    GP+ImFil uses an adaptive Gaussian-process surrogate for global search and seeds multiple local Implicit Filtering optimizations near promising regions.

    Source: §I · p. 2, §IV · p. 3, C. From Global to Local Search · p. 5

  • Results

    For noisy problems, GP+ImFil finds improvements faster and overall better solutions than ImFil across Hubbard-model instances spanning 2–20 optimization parameters.

    Source: §VI · p. 10

  • Takeaways & Limitations

    Gaussian-process-guided local searches are beneficial for noisy VQE optimization, particularly on larger or rugged problems where careful starting-point selection helps.

    Source: §I · p. 2, C. Hubbard Model with Measurement Noise · p. 8, §VI · p. 10

    On smooth unimodal deterministic problems with small bounding boxes, GP iterations are not advantageous, and multistart improvements are not consistent enough to replace exhaustive methods generally.

    Source: §VI · p. 10

Abstract

from arXiv · show

Hybrid variational quantum algorithms, which combine a classical optimizer with evaluations on a quantum chip, are the most promising candidates to show quantum advantage on current noisy, intermediate-scale quantum (NISQ) devices. The classical optimizer is required to perform well in the presence of noise in the objective function evaluations, or else it becomes the weakest link in the algorithm. We introduce the use of Gaussian Processes (GP) as surrogate models to reduce the impact of noise and to provide high quality seeds to escape local minima, whether real or noise-induced. We build this as a framework on top of local optimizations, for which we choose Implicit Filtering (ImFil) in this study. ImFil is a state-of-the-art, gradient-free method, which in comparative studies has been shown to outperform on noisy VQE problems. The result is a new method: "GP+ImFil". We show that when noise is present, the GP+ImFil approach finds results closer to the true global minimum in fewer evaluations than standalone ImFil, and that it works particularly well for larger dimensional problems. Using GP to seed local searches in a multi-modal landscape shows mixed results: although it is capable of improving on ImFil standalone, it does not do so consistently and would only be preferred over other, more exhaustive, multistart methods if resources are constrained.

I. INTRODUCTION AND MOTIVATION

VQE and related hybrid algorithms face noisy, expensive black-box optimization landscapes where local methods depend strongly on initialization. The paper proposes Gaussian-process-guided global search with targeted local optimization and positions ImFil as the noisy baseline.

I. INTRODUCTION AND MOTIVATION

  • Noisy VQE requires many quantum-chip evaluations while classical computation is easier to scale, motivating methods that reduce objective queries.Source: §I · p. 1
  • Multi-modal, black-box energy landscapes lack reliable gradients, and local optimizers can become trapped without good starting points.Source: §I · p. 1
  • Noise can cause incorrect local-search decisions and premature convergence to noise-induced minima, while multistart may repeat solutions and has no clear stopping budget.Source: §I · p. 1
  • The proposed method uses an adaptive Gaussian process to identify local attraction regions and seed local optimizations on parameter-space subdomains.Source: §I · pp. 1–2

II. RELATED WORK

  • The study uses ImFil as its local optimizer and noisy baseline because prior comparisons and the authors’ testing found it strong under noise.Source: §II · p. 2
  • Prior GP and Bayesian-optimization approaches for VQE did not adjust kernels for noisy function values, whereas this work uses GP surrogates to guide promising local-search starts.Source: §II · p. 2

III. SIMULATION OF VQE

The study simulates VQE as Hamiltonian-energy minimization using scalable Hubbard-model instances, with noise-free simulations serving as a reference. It evaluates mapped circuits and directly computes expectation values in the noise-free setup.

III. SIMULATION OF VQE

  • VQE minimizes the Hamiltonian expectation value over parameterized quantum states, with θ in a bounded d-dimensional domain and an Ansatz determined by hardware or symmetry constraints.Source: §III · p. 2
  • The Hubbard Model provides a scalable, multi-modal benchmark for studying when surrogate models improve VQE optimization across problem scale and circuit depth.Source: §III · p. 2

A. Noise-free simulation

  • The authors simulate both noise-free and noisy settings because simulations can scan larger scales and depths than currently available hardware.Source: §III · p. 2, A. Noise-free simulation · p. 2
  • In the noise-free setup, sampling error is ignored as reducible through more samples, while mapped circuit approximation error remains relevant and is simulated in Qiskit.Source: A. Noise-free simulation · p. 3
  • The noise-free energy is obtained directly from the final-state expectation value rather than simulating partial tomography.Source: A. Noise-free simulation · p. 3

B. Noisy simulation

Noise affects VQE through state-preparation, measurement, and sampling mechanisms, with measurement errors especially likely to create noise-induced local minima. The benchmark uses scalable Fermi-Hubbard systems with UCCSD circuits and neighboring-site hopping.

B. Noisy simulation

  • State-preparation errors raise energy estimates and blur the optimization surface by increasing the parameter change needed for statistically significant differences.Source: B. Noisy simulation · p. 3
  • Measurement misclassification biases average energy estimates, while individual results can over- or undershoot the actual minimum depending on the Hamiltonian.Source: B. Noisy simulation · p. 3
  • Measurement errors are the most important modeled noise source because they increase the chance that optimizers become trapped in noise-induced local minima.Source: B. Noisy simulation · p. 3

C. The Fermi-Hubbard Model

  • The Fermi-Hubbard benchmark models interacting particles on a periodic lattice and is relevant because classical solution algorithms scale exponentially without further approximations.Source: C. The Fermi-Hubbard Model · p. 3
  • Its interaction term U applies to opposite-spin electrons sharing a site, while hopping term t represents kinetic energy between neighboring sites.Source: Figure 2 · p. 3, C. The Fermi-Hubbard Model · p. 3
  • The simulations use an unoptimized UCCSD Ansatz to match model symmetries and preserve consistent interpretation when noise is added.Source: C. The Fermi-Hubbard Model · p. 3

IV. SURROGATE MODEL BASED OPTIMIZATION

The paper uses surrogate models, especially Gaussian processes, to approximate expensive black-box energy objectives and guide evaluations. GP modeling incorporates uncertainty and white-noise handling to support optimization of noisy VQE landscapes.

A. General Surrogate Model Optimization Algorithms

  • Surrogate models cheaply approximate expensive black-box energy objectives and guide iterative sampling decisions toward promising parameter values.The workflow constructs a model from evaluated input-output pairs, selects new points through an auxiliary optimization problem, and updates the model iteratively.Source: §IV · pp. 3–4, A. General Surrogate Model… · p. 4
  • Gaussian processes are used because they model noisy data while providing uncertainty estimates alongside objective-value predictions.Their computational overhead can become demanding as training sets grow, but limited quantum evaluations make that cost acceptable in the studied setting.Source: §IV · p. 3, A. General Surrogate Model… · p. 4

B. Gaussian Process Models

  • The GP represents the objective as a stochastic process whose predictions depend on the selected kernel and fitted mean, variance, and length-scale parameters.The squared exponential kernel determines how quickly points become uncorrelated across each dimension, with parameters estimated by maximum likelihood.Source: B. Gaussian Process Models · p. 4
  • Expected improvement selects new evaluations by balancing predicted improvement against uncertainty, but its multimodality means the selected point is not guaranteed to be globally optimal.The criterion is maximized over the parameter domain and is zero at previously evaluated points.Source: B. Gaussian Process Models · p. 4
  • Adding a white-noise kernel prevents interpolation of noisy values, allowing the GP to capture underlying global trends and making optimization easier.For noise-free simulations, the added white kernel does not significantly deteriorate the approximation; Fig. 3 contrasts fits without and with this kernel at σnoise = 0.05.Source: B. Gaussian Process Models · pp. 4–5, Figure 3 · p. 5

C. From Global to Local Search

The method uses an adaptive GP search to identify several promising and separated starting points, then performs bounded ImFil local searches around them. Evaluation budgets govern the division between global modeling and local refinement.

C. From Global to Local Search

  • The GP-based optimizer identifies multiple suitable starting points for local searches, while ImFil supplies the noisy, bound-constrained local optimization.ImFil uses coordinate search and approximate gradients to inform iterative sampling decisions.Source: C. From Global to Local Search · p. 5
  • The algorithm begins with a space-filling design, fits a GP, and selects new evaluations by maximizing expected improvement until the GP budget is reached.The initialization also sets local-search and multistart budgets, the weighting pattern, and the GP kernel.Source: C. From Global to Local Search · p. 5
  • ImFil searches are run in reduced bound regions around selected starts, updating the shared sample set until the local-search budget or maximum number of searches is exhausted.The lowest function value found across the local searches is returned as the solution.Source: C. From Global to Local Search · p. 5
  • Starting points are chosen to combine low predicted function values with spatial separation, reducing the risk that local searches converge to the same minimum.The best evaluated point starts the set, and additional points are selected using weighted value and distance scores.Source: C. From Global to Local Search · p. 5
  • The method exposes a trade-off between initial-design size and GP iterations, and the GP can stop in a local optimum, especially in high-dimensional spaces.The affordable number of local searches also depends on the local-search budget and evaluations required from each starting point.Source: C. From Global to Local Search · p. 6

V. NUMERICAL STUDY

The numerical study compares GP+ImFil with standalone ImFil and GP-only optimization under a common 1000-evaluation limit. Three stochastic runs per problem estimate variability, with matched initial designs for fair comparison.

V. NUMERICAL STUDY

  • Each algorithm is limited to 1000 function evaluations and is run three times per test problem to characterize variability from stochastic initialization and expected-improvement optimization.The comparison includes GP+ImFil, standalone ImFil, and GP-only Bayesian optimization.Source: §V · p. 6
  • GP-based methods use an initial design of 2(d+1) points, while GP+ImFil allocates an additional 8(d+1) evaluations to GP iterations before local search.Standalone ImFil receives the same initial design points to seed its multiple restarts.Source: §V · p. 6

A. Test Problem Setup

The study evaluates six Hubbard-model problems spanning dimensionality and optimization difficulty, with deterministic and noisy versions. Grid size and filling vary the number of parameters and landscape complexity.

A. Test Problem Setup

  • For a 2x1 Hubbard grid, noise changes the energy landscape and can introduce local optima compared with the noise-free case.The examples use different fillings, and the displayed contours are based on a GP approximation combining squared exponential and white-noise kernels.Source: A. Test Problem Setup · p. 6, Figure 4 · p. 6
  • Six Hubbard-model examples vary in dimensionality and optimization difficulty, allowing method scaling to be studied without changing the underlying scientific problem.Larger grids require more qubits and optimization parameters, while filling controls parameter count through the model’s symmetry.Source: A. Test Problem Setup · p. 6
  • The test set contains deterministic instances and noisy instances with measurement noise σs = 0.003, with each evaluation restricted to 8192 samples.The noisy setting is intended to reflect measurement conditions after unfolding data and the default sampling level on IBM devices.Source: A. Test Problem Setup · p. 6
  • Table I records the specifications of the Hubbard-model test problems, distinguishing deterministic cases marked “d” from noisy cases marked “n”.The table organizes the instances used for the numerical study across their differing problem settings.Source: Table I · p. 6

B. Deterministic Hubbard Model

On deterministic Hubbard models, GP+ImFil combines space-filling GP exploration with localized ImFil searches, but its benefit over ImFil is limited mainly to the largest problem.

B. Deterministic Hubbard Model

  • ImFil-only sampling covers the full space and concentrates more densely near the optimum than GP-only Bayesian optimization.Bayesian optimization spreads samples through exploration-exploitation trade-offs and may not sample densely enough for ImFil-level accuracy.Source: Figure 5 · p. 7, B. Deterministic Hubbard Model · p. 7
  • GP+ImFil improved on ImFil only for H6-d, while all methods found the optimum on simple H1-d and H2-d instances.The small problems used search spaces containing only one minimum.Source: B. Deterministic Hubbard Model · p. 7, Table II · p. 7
  • GP samples broadly before launching multiple ImFil searches from selected points within smaller subdomains around promising regions.This produces distributed GP samples and clustered, localized ImFil samples.Source: Figure 6 · p. 7, B. Deterministic Hubbard Model · p. 7
  • GP and GP+ImFil reduce progress-plot variability and find early improvements faster than ImFil, although ImFil eventually reaches better final solutions on all 2x2-grid problems.The restricted local-search box may exclude the optimum, motivating adaptive trust regions.Source: B. Deterministic Hubbard Model · p. 7, Figure 7 · p. 8, Figure 8 · p. 8

C. Hubbard Model with Measurement Noise

With measurement noise, GP+ImFil combines broad GP exploration with repeated local searches and performs best on higher-dimensional Hubbard problems, while GP-only optimization can stall.

C. Hubbard Model with Measurement Noise

D. Effectiveness of Seeding

GP seeding improves the reliability of local searches across a multimodal landscape, but its success is nondeterministic and remains less effective than exhaustive multistart.

D. Effectiveness of Seeding

  • In two of three configurations, standalone ImFil without a good initial point became trapped in a local minimum, whereas GP+ImFil found the true global minimum in all cases.GP+ImFil success was not guaranteed because of GP’s nondeterministic behavior, and two configurations required multiple runs.Source: D. Effectiveness of Seeding · p. 9
  • Boundary-adjacent seeds can create artificial local minima, so seed selection should reduce the weight of extrapolated points near search boundaries.The authors identify boundary slopes as the most common reason for GP+ImFil failure.Source: D. Effectiveness of Seeding · p. 9
  • Because full-space searches contain multiple global minima from symmetry and periodicity, reported results are averages rather than evaluations at identical parameter vectors.The authors recommend rerunning a good seed far from the bounds when seeking the best final result.Source: D. Effectiveness of Seeding · p. 9
  • Exhaustive multistart offers a stronger success guarantee than GP seeding, but at substantially higher resource cost.Exhaustive multistart searches each region of the full parameter space, whereas ImFil local searches are confined to small subdomains.Source: D. Effectiveness of Seeding · p. 9
  • Even at the lowest observed 20% success rate, repeated GP-seeded runs cost far less than exhaustive multistart because each local search covers only a small subdomain.The local search box was chosen as ±0.2 in each parameter.Source: D. Effectiveness of Seeding · p. 9
  • Standalone ImFil remains preferable when it converges without becoming trapped in a local minimum.The paper notes that this preference follows from its lower resource use when convergence is successful.Source: D. Effectiveness of Seeding · p. 9

E. A Note on Computational Time

GP-based sampling adds substantial computational overhead for cheap objectives but becomes more attractive when quantum-function evaluations are expensive and noisy.

E. A Note on Computational Time

  • For small problems with fast function evaluations, GP+ImFil is not recommended because GP training and acquisition optimization add overhead without significantly improving performance.ImFil approximates gradients through coordinate searches, whereas GP sampling requires maximum-likelihood training and a further optimization problem.Source: E. A Note on Computational Time · p. 10
  • When function evaluations are expensive, GP overhead becomes negligible relative to evaluation cost, motivating its use because performance improvements are especially evident in noisy cases.Source: E. A Note on Computational Time · p. 10

F. Other Kernels

The kernel study finds promising deterministic behavior for Matérn GP+ImFil, while the broader results favor GP guidance mainly for noisy, rugged problems and identify several paths for refinement.

F. Other Kernels

  • The Matérn kernel outperforms the RBF+White Kernel in exploratory noise-free 2x2 tests and converges faster than ImFil for one illustrated problem.The Matérn kernel did not improve the noisy version, so further investigation is needed before drawing definite conclusions.Source: F. Other Kernels · p. 10, Table V · p. 10, Figure 14 · p. 10
  • Kernel selection remains consequential, with the Matérn kernel requiring further study because its benefit appeared in deterministic but not noisy tests.Source: F. Other Kernels · p. 10, §VI · p. 10

VI. DISCUSSION, CONCLUSIONS, AND FUTURE DIRECTIONS

  • GP+ImFil couples GP-based global search with multiple local ImFil searches for Hubbard-model VQE across noise-free and noisy problems with 2–20 optimization parameters.The study evaluates whether appropriately chosen kernels can reduce objective evaluations relative to ImFil.Source: §VI · p. 10
  • For noisy functions, GP+ImFil finds improvements faster and achieves better overall solutions than ImFil, especially by avoiding noise-induced minima far from the global optimum.For smooth unimodal deterministic problems with small bounding boxes, GP iterations are not advantageous.Source: §VI · p. 10
  • On deterministic problems, ImFil usually finds better solutions than GP+ImFil even though its progress toward the optimum is significantly slower.Source: §VI · p. 10
  • Future improvements include dynamically adjusting GP and local-search parameters, detecting artificial minima, testing noisy-problem local optimizers such as SnobFit, and exploiting parallel GP training.The authors also identify kernel choice and configuration as important directions for future work.Source: §VI · p. 10

Related papers

Loading 2204.07331v3…