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Classical Optimizers for Noisy Intermediate-Scale Quantum Devices

Wim Lavrijsen, Ana Tudor, Juliane Müller, Costin Iancu, Wibe de Jong

arXiv:2004.03004v2quant-ph

TL;DR

Noise can invalidate the assumptions behind commonly used classical optimizers in hybrid quantum algorithms, threatening their convergence and solution quality. The paper evaluates derivative-free optimizers in VQE simulations and packages selected methods in scikit-qant. Its findings indicate that noise-aware optimization and matching or composing optimizers are important for obtaining high-quality results in the studied settings.

  • Problem

    Quantum noise can invalidate assumptions behind classical optimizers’ convergence and optimality guarantees, causing premature convergence, noise-induced minima, or nonconvergence.

    Source: §1 · p. 1

  • Method

    The authors evaluate noise-aware derivative-free optimizers in VQE simulations and package selected methods in scikit-qant.

    Source: §1 · p. 1, §3.3 · p. 4

  • Results

    The study finds that specialized noise-robust optimizers are essential on NISQ hardware, and that optimizer composition works best for final solution quality when domain information is unavailable.

    Source: §7 · p. 10

  • Takeaways & Limitations

    VQE success on NISQ devices depends on noise-aware classical optimizers that work at the required scales, an open research area for hybrid algorithms.

    Source: §7 · p. 10

    If noise makes the output state no longer represent the intended physical system, such as through electron loss or spin flips from T1 noise, the estimated energy can fall below that system’s ground energy.

    Source: §6 · p. 6

Abstract

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We present a collection of optimizers tuned for usage on Noisy Intermediate-Scale Quantum (NISQ) devices. Optimizers have a range of applications in quantum computing, including the Variational Quantum Eigensolver (VQE) and Quantum Approximate Optimization (QAOA) algorithms. They are also used for calibration tasks, hyperparameter tuning, in machine learning, etc. We analyze the efficiency and effectiveness of different optimizers in a VQE case study. VQE is a hybrid algorithm, with a classical minimizer step driving the next evaluation on the quantum processor. While most results to date concentrated on tuning the quantum VQE circuit, we show that, in the presence of quantum noise, the classical minimizer step needs to be carefully chosen to obtain correct results. We explore state-of-the-art gradient-free optimizers capable of handling noisy, black-box, cost functions and stress-test them using a quantum circuit simulation environment with noise injection capabilities on individual gates. Our results indicate that specifically tuned optimizers are crucial to obtaining valid science results on NISQ hardware, and will likely remain necessary even for future fault tolerant circuits.

1 INTRODUCTION

The introduction argues that noise can undermine classical optimization in hybrid quantum algorithms, making optimizer choice important alongside quantum-circuit design. The study evaluates noise-aware methods and reports that suitable optimizer choices depend on the problem and available domain information.

1 INTRODUCTION

  • Noisy objectives invalidate premises behind common optimizers’ convergence and optimality guarantees, risking premature convergence, noise-induced local minima, or failure to converge.Earlier hardware studies avoided this optimizer problem through full phase-space exploration and zero-noise backfitting, an approach not tractable for many parameters or expected large qubit concurrency.Source: §1 · p. 1
  • The study contributes a Python-usable optimizer suite and tuning guide, and tests optimizer sensitivity to noise across simulations and science problems.The evaluation considers initial-solution quality, parameter bounds, scaling, noise levels, and effects on the full VQE algorithm.Source: §1 · p. 1
  • Standard SciPy optimizers are not noise-aware and can fail under quantum noise, although some methods gain robustness from their underlying techniques.The study notes smoothing from local modeling in trust-region methods, though not usually by explicit noise-aware design.Source: §1 · p. 1
  • The authors package ported optimizers with consistent interfaces and plugins for Qiskit and Cirq in scikit-qant.The package also includes tutorial notebooks and an evaluation harness for assessing applicability to new problems.Source: §1 · p. 2
  • Across three VQE problems, results favor matching optimizers to problem-specific strengths or composing local and global optimizers when domain information is unavailable.Good initial parameters, local-minimum knowledge, or inaccessible-region information can change which optimizer performs best.Source: §1 · p. 2

2 NUMERICAL OPTIMIZATION

This section explains how noise changes optimization surfaces and lays out criteria for choosing derivative-free optimizers. It compares common baselines and finds Cobyla outperformed the other evaluated methods.

2 NUMERICAL OPTIMIZATION

  • Noise can turn the ethylene VQE surface from convex and smooth into non-convex and non-smooth, while more complex circuits and parameter spaces add further challenges.The 8-qubit, 14-parameter Hubbard example at σ=0.01 rad illustrates the effects of noise on a more complex surface.Source: §2 · p. 2, Figure 1 · p. 3

2.1 Optimizer Selection Criteria

  • Optimizer selection weighs solution quality, scalability, objective-query count and precision, and implementation performance and parallelization.The criteria link quantum-side queries to chip time and implementation factors to classical-side scaling and runtime.Source: §2.1 · p. 2
  • Quantum simulations average many probabilistic circuit shots over local noise, so noisy optimization can target the expected response rather than the worst case.The section identifies expected-value and worst-case optimization as two common strategies.Source: §2.1 · p. 2

2.2 Baseline Optimizers

  • SciPy BFGS reconstructs derivatives from objective evaluations, so noisy results can quickly misdirect its estimates despite its common use as a baseline.The paper retains BFGS for comparison because it is well known and commonly used in prior studies.Source: §2.2 · p. 2
  • Cobyla outperformed the other evaluated readily available Python methods and was chosen as the second baseline.The alternatives included Nelder-Mead, RBFOpt, DYCORS, and CMA-ES.Source: §2.2 · pp. 2–3

3 SCIKIT-QUANT OPTIMIZERS

The section introduces scikit-qant’s derivative-free optimizer collection and describes the methods’ distinct search strategies and tuning needs. It highlights that bounds, initial parameters, and problem-specific tuning shape performance.

3 SCIKIT-QUANT OPTIMIZERS

  • The toolkit packages NOMAD, ImFil, SnobFit, and BOBYQA, selected using prior evaluations, open-source availability, and ease of Python porting.Prior work reports differing performance by surface shape, while the toolkit’s derivative-free methods use sampling and can parallelize quantum-step evaluations.Source: §3 · p. 3

3.1 NOMAD

  • NOMAD alternates mesh-based search and poll steps; its evaluation count grows at least geometrically with parameter count, making bounded search spaces important.The poll step underpins convergence analysis; tuning can change poll directions, size, or number of polling points.Source: §3.1 · p. 3

3.2 ImFil

  • ImFil evaluates bounded clusters of points, uses their best value and first-order interpolation to guide later clusters, and is relatively insensitive to its starting point.For studied noisy objectives, requested step sizes between clusters drive total evaluations more than explicit convergence detection.Source: §3.2 · p. 4
  • For high-quality initial parameters, ImFil needs tight bounds to avoid evaluations in regions unlikely to contain the global minimum.Without such bounds, its boundary-determined initial clusters can lead it to explore unnecessarily broad regions.Source: §3.2 · p. 4

3.3 SnobFit

  • SnobFit balances local and global search for noisy, expensive objectives, recursively selecting new evaluation points from supplied points and their function values.The authors did not use its uncertainty inputs because estimating whole-circuit systematic errors on real hardware remains an open problem.Source: §3.3 · p. 4

3.4 BOBYQA

  • BOBYQA performs well on noisy VQE problems when initial parameters are near-optimal and trust-region sizes are set appropriately, but fares worse on Hubbard problems with many shallow local minima.A good initial guess can be obtained relatively cheaply from classical simulation for the specific VQE case discussed.Source: §3.4 · p. 4

3.5 Validation and Tuning

  • The study validated implementations with unit tests and tuned defaults away from original settings, trading wall-clock performance against science performance.Changes included more samples per iteration for PyBobyqa and NOMAD, tighter SnobFit convergence criteria, and smaller ImFil step scales to reach chemical accuracy.Source: §3.5 · p. 4

4 HYBRID QUANTUM-CLASSICAL ALGORITHMS

These hybrid algorithms alternate classical parameter updates with quantum evaluations of an objective. The section describes how VQE’s ansatz constrains the search and how its quantum processor produces measurements used to calculate energy.

4 HYBRID QUANTUM-CLASSICAL ALGORITHMS

  • Classical optimizers vary parameters, quantum processors prepare and measure states, and the resulting probability distribution is mapped to an objective value for the next update.In VQE, that objective is the expected energy associated with the Hamiltonian.Source: §4 · p. 5, Figure 3 · p. 5
  • The section’s findings are presented as applicable to other hybrid algorithms, including the more complex QAOA algorithms.The authors base this expectation on sweeping noise levels through optimizer breaking points.Source: §4 · p. 5

4.1 Role of the Ansatz in VQE

  • The ansatz restricts VQE’s valid input states and shapes its optimization surface; the study uses the practical unitary coupled cluster ansatz for all experiments.UCC represents individual-particle and interacting-particle movements and can use symmetries to reduce parameters.Source: §4.1 · p. 5
  • UCC has a natural but somewhat inefficient qubit mapping, and the study examines 4- and 8-qubit problems with 2 to 14 parameters.More compact representations can require more native gates and make problems harder to solve.Source: §4.1 · p. 5

4.2 VQE Quantum Processor Step

  • For VQE, the processor splits a Hamiltonian represented as Pauli operators into components evaluated independently, then combines them classically.Measurements produce bit-string probability distributions; post-processing such as Bayesian unfolding can make the final energy mapping nonlinear under on-chip noise.Source: §4.2 · p. 5

5 IMPACT OF NOISE

The section examines noise in hybrid VQE, using simulations to vary noise magnitude and study its effects on optimization. It contrasts partial compensation of coherent errors with stochastic noise that disrupts access to the true minimum.

5 IMPACT OF NOISE

  • Although VQE’s iterative nature is considered somewhat robust to noise, studying the full hybrid algorithm’s dynamics was identified as a prerequisite for successful NISQ-hardware use.The section focuses on overall noise magnitude and changes to optimization-surface shape because accurate predictive models are not available.Source: §5 · p. 5, §5.1 · p. 5

5.1 Accounting for Noise Sources

  • The simulations sweep noise magnitude across problems with different optimization surfaces, injecting Gaussian-distributed gate over- and under-rotations with an orthogonal component.The model excludes coherent or correlated noise and measurement errors; two-qubit gates contribute more noise than one-qubit gates.Source: §5.1 · pp. 5–6

5.2 Interplay with Minimizer

  • Coherent errors can be compensated by the optimizer, but stochastic noise moves executions away from the intended state and makes the true global minimum less reachable as noise increases.For VQE, the ansatz restriction makes coherent errors more impactful than in hybrid algorithms generally; stochastic evaluations near the minimum can return energies above the ground-state energy.Source: Figure 4 · p. 6, §5.2 · p. 6
  • For ethylene bond breaking, increasing stochastic error broadens evaluation distributions and lifts their average until the true global minimum is no longer reachable.The figure marks the chemical-accuracy cutoff for useful scientific results.Source: Figure 5 · p. 6

6 RESULTS

Across the VQE tests, noise-aware optimizers outperform across noise levels, with ImFil giving the best overall results; the study also shows how problem-specific initialization, bounds, and stopping criteria affect solution quality and evaluation use.

6 RESULTS

  • The tests cover ethylene C–C rotation and bond stretching/breaking, plus 4-site Hubbard simulations with 2 or 4 electrons.The Hubbard simulations use hopping term 1.0, Coulomb term 2.0, and, for 4 electrons, chemical potential 0.25.Source: §6 · pp. 6–7

6.1 Experimental Setup

  • The noise-injection setup adds independently sampled rotation noise at circuit gates and sweeps its strength; numerical noise magnitudes are Gaussian standard deviations.Runs use objective-call budgets, repeat the full algorithm, and compare average and minimum results with classical ab-initio calculations.Source: §6.1 · p. 7

6.2 Optimization Solution Quality

  • When the objective-function floor exceeds chemical accuracy, the quantum processor—not the optimizer—limits the result; the deeper 8-qubit circuit suffers more from noise.Chemical accuracy is 0.00159 hartrees; the figure compares ethylene rotation and bond breaking.Source: §6.2 · p. 7, Figure 8 · p. 8
  • Noise-aware optimizers outperform across noise levels, with ImFil best overall; at low noise, methods perform similarly and Cobyla is likely preferred.Stratification appears only at the highest noise levels; iteration counts then help determine which method is best.Source: §6.2 · pp. 7–8
  • On the shallow Hubbard optimization surface, NOMAD and especially SnobFit underperform or fail with overly large bounds, while tighter bounds greatly improve both.Tighter bounds do not improve BFGS or Cobyla in this comparison.Source: §6.2 · pp. 8–9

6.3 Leveraging Domain Science Knowledge

  • In ethylene rotation, ImFil uses bounds for a mostly global first-stencil search and moves away from a bad start to the global optimum, while PyBobyqa can get stuck at a pronounced local minimum.The initial points are (0.1, 0.1) and (0.3, −0.3), and the global optimum is (0.00012, 0.04).Source: §6.3 · p. 9
  • Using ImFil to derive tighter bounds lets SnobFit improve beyond ImFil’s failed stencil, and the combined run remains within budget when noise is high.ImFil fails earlier at high noise, leaving budget for SnobFit.Source: §6.3 · p. 9

6.4 Performance Considerations

  • In the Hubbard tests, fixed-threshold PyBobyqa can exhaust its budget as noise keeps successive differences above threshold; noise-aware stopping can conserve evaluations for optimizer switching.Figure 10 compares objective-function call counts across noise levels for ethylene rotation and the 4-electron Hubbard simulation.Source: Figure 10 · p. 8, §6.4 · pp. 9–10

7 DISCUSSION

The discussion recommends matching noise-robust optimizers to problem conditions and available scientific information. It argues that suitable minimizers are important for reliable VQE on NISQ devices, while noting that challenges remain open.

7 DISCUSSION

  • SciPy BFGS/Cobyla are fastest noise-free but unusable under small noise; bounds favor ImFil/NOMAD or SnobFit when tight, while good starts favor PyBobyqa.ImFil offers direct control over scales and searches; the study says optimizer combinations work best for final solution quality.Source: §7 · p. 10
  • Minimizers must balance distance to the global minimum against evaluation limits, since calibration can drift; noise can also create many local minima.The VQE ansatz affects the optimization surface and noise, so the study can inform practical ansatz design.Source: §7 · p. 10
  • There is no single best minimizer: a suite matched to each problem can use available domain-science information.The study identifies good initial parameters, knowledge of local minima, and inaccessible search regions as potentially useful information.Source: §7 · p. 10
  • Noise affects the objective nonlinearly; robust minima may preserve correct parameters, but complex surfaces can trap optimizers and call for more or better constraints.The authors expect most methods to scale to hundreds of parameters and hybrid performance on NISQ hardware to be limited by the quantum part.Source: §7 · p. 10
  • The authors describe noise-aware optimization at needed scales as an open research area and offer a toolkit for practitioners and optimization researchers.They present the toolkit as a starting point for developing methods tailored to VQE and other hybrid algorithms.Source: §7 · p. 10

8 RELATED WORK

Prior work explored noisy optimization in small VQE problems and used zero-noise extrapolation on small circuits. The authors identify unresolved questions about optimizer robustness in QAOA and larger, more complex optimization problems.

8 RELATED WORK

  • McClean et al. [24] found noisy-output challenges when comparing optimizers on a four-qubit H2 VQE problem with one parameter.The passage notes that current QAOA studies still use BFGS and Nelder-Mead and mostly concentrate on the quantum algorithm.Source: §8 · p. 10
  • The authors expect QAOA to need noise-robust optimizers, since most approaches lack the physical or chemical bounds available for VQE.They state that QAOA studies have not addressed the effects of noise on classical optimizers.Source: §8 · p. 10
  • Zero-noise extrapolation runs VQE, calibrates system-noise impact, then extrapolates results offline; published examples involve small circuits with two optimization parameters.Whether these techniques relax robust-optimizer requirements for higher-dimensional problems with more complex surfaces remains unresolved.Source: §8 · p. 10

9 CONCLUSION

The conclusion presents noise-aware classical optimization as necessary for hybrid algorithms on current NISQ hardware. It introduces a toolkit whose consistent interfaces support combining optimizers with complementary strengths.

9 CONCLUSION

  • Successful hybrid algorithms on current NISQ hardware require noise-aware optimizers; the collected scikit-qant suite outperforms standard SciPy optimizers.The conclusion reports that the suite's optimizers work particularly well, without giving a quantitative comparison here.Source: §9 · p. 10
  • The toolkit offers practitioners a directly usable collection of classical optimizers for hybrid quantum-classical algorithms.The conclusion describes the collected suite as particularly effective relative to standard SciPy optimizers.Source: §9 · p. 10
  • Consistent programming interfaces make it possible to combine optimizers and exploit their respective strengths.The study used VQE and expects its findings to apply more broadly; with the suite, the classical step is no longer the weakest link on NISQ-era hardware.Source: §9 · p. 11
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