---
abstract: 'Variational quantum algorithms (VQAs) offer a promising path towards using near-term quantum hardware for applications in academic and industrial research. These algorithms aim to find approximate solutions to quantum problems by optimizing a parametrized quantum circuit using a classical optimization algorithm. A successful VQA requires fast and reliable classical optimization algorithms. Understanding and optimizing how off-the-shelf optimization methods perform in this context is important for the future of the field. In this work we study the performance of four commonly used gradient-free optimization methods: SLSQP, COBYLA, CMA-ES, and SPSA, at finding ground-state energies of a range of small chemistry and material science problems. We test a telescoping sampling scheme (where the accuracy of the cost-function estimate provided to the optimizer is increased as the optimization converges) on all methods, demonstrating mixed results across our range of optimizers and problems chosen. We further hyperparameter tune two of the four optimizers (CMA-ES and SPSA) across a large range of models, and demonstrate that with appropriate hyperparameter tuning, CMA-ES is competitive with and sometimes outperforms SPSA (which is not observed in the absence of hyperparameter tuning). Finally, we investigate the ability of an optimizer to beat the 'sampling noise floor', given by the sampling noise on each cost-function estimate provided to the optimizer. Our results demonstrate the necessity for tailoring and hyperparameter-tuning known optimization techniques for inherently-noisy variational quantum algorithms, and that the variational landscape that one finds in a VQA is highly problem- and system-dependent. This provides guidance for future implementations of these algorithms in experiment.'
author:
- 'Xavier Bonet-Monroig'
- Hao Wang
- Diederick Vermetten
- Bruno Senjean
- Charles Moussa
- Thomas Bäck
- Vedran Dunjko
- 'Thomas E. O'Brien'
title: Performance comparison of optimization methods on variational quantum algorithms
---

Introduction
============

Recently we have witnessed an explosion of quantum computer prototypes accessible to researchers in academic and industrial laboratories. Existing quantum hardware has already demonstrated the ability to outperform classical computers in specific mathematically contrived tasks @Arute2019 [@zhu2021quantum]. However, it is still unclear whether noisy intermediate-scale quantum (NISQ) @preskillNISQ hardware can outperform classical computers on practically useful tasks. Here, variational quantum algorithms (VQA) @peruzzo2014variational [@mcclean2016theory; @cerezoVQA2021] were introduced as a means of preparing classically-hard quantum states by tuning parameters of a quantum circuit to optimize a cost function by utilising a classical optimizer.

The overall performance of VQAs depends on the performance of the classical optimization algorithm. Finding the limitations of these optimization methods for different VQA tasks is critical if they are to impact research and industry. For this, researchers have proposed new classical optimization algorithms that exploit periodic properties of parametrized quantum circuits @Nakanish2020SOFF [@rotosolve2021]. Other works have focused on using machine learning techniques to optimize VQAs @Wilson2021 [@Sung2020models]. These articles benchmark new optimization techniques relative to standard classical optimizers on a wide variety of systems. However, to the best of our knowledge, no extensive comparison of the most common optimization methods for these tasks has been reported yet.

In this work we study three aspects that affect the optimization performance in VQAs. We focus on four off-the-shelf optimizers (SLSQP, COBYLA, CMA-ES and SPSA) for the task of finding an approximate ground-state energy of few physical systems. We first look at two different sampling strategies and how they affect the optimization performance with default optimizer hyper-parameters. Then we focus on hyper-parameter tuning of CMA-ES and SPSA; finding a comparable performance between them given optimal hyper-parameters, with the winner depending on the details of the problem. Finally, we investigate the accuracy of the solutions in the presence of stochastic sampling noise. We define a 'sampling noise floor': a bound on the accuracy that an optimizer can reach when the optimal parameters are those of the best-ever function evaluation. Additionally, we show that CMA-ES algorithm can outperform this 'sampling noise floor' when the optimal parameters are selected from an internal estimate of the optimal candidate. Our main contribution is strong numerical evidence that the optimal parameters of a VQA should not be taken from the best-ever measured function evaluation.

Background {#sec:background}
==========

A variational quantum algorithm attempts to find approximate ground states of an $N$-qubit quantum system as the output of a circuit $U(\vec{\theta})$ with tunable parameters ${\vec{\theta}}$. This generates a variational ansatz, $$\begin{aligned}
    \ket{\Psi(\vec{\theta})} &=& U(\vec{\theta}) \ket{\Phi},
    \label{eq:parametrized_evol}\end{aligned}$$ where the parameters $\vec{\theta}\in[0,2\pi]^d$ control the rotations of single and two-qubit gates in a quantum circuit implementation of $U$ applied to an initial state $\ket{\Phi}$ (i.e., $U(\vec{\theta})=U_k(\theta_k)U_{k-1}(\theta_{k-1}) \dots U_0(\theta_0)\ket{\Phi}$). During a VQA run, these parameters are tuned to optimize a cost function ${\mathcal{C}}(\vec{\theta})$, which in our case is the expectation value of a Hermitian observable $\hat{O}$ relative to the state $|\Psi(\vec{\theta})\rangle$, $${\mathcal{C}}(\vec{\theta})=\langle\hat{O}\rangle=\langle\Psi(\vec{\theta})|\hat{O}|\Psi(\vec{\theta})\rangle.$$ To measure the expectation value of $\hat{O}$ without additional quantum circuitry, it is typical to write $\hat{O}$ as a linear combination of easy-to-measure operators, i.e., Pauli operators $\hat{P}_i\in\{{\mathbb{I}},X,Y,Z\}^{\otimes N}$ $$\hat{O}=\sum_ic_i\hat{P}_i\rightarrow{\mathcal{C}}({\vec{\theta}}) = \langle \hat{O} \rangle = \sum_i c_i \langle \hat{P}_i \rangle.
    \label{eq:pauli_decomp}$$ A VQA then passes the estimation of the cost function ${\mathcal{C}}(\vec{\theta})$ to some classical optimization routine to find the values of $\vec{\theta}$ minimizing ${\mathcal{C}}$. This optimization loop and the optimizer choices are the focus of this work.

To get an estimate of the expectation value $\langle\hat{P}_i\rangle$, one prepares and measures the state multiple times in the $\hat{P}_i$ basis and calculates the mean of the eigenvalues observed. This approximates the cost function ${\mathcal{C}}$ by an estimator ${\ensuremath{\bar{\mathcal{C}}}}$, whose distribution is dependent on the number of repetitions $M$ used to calculate $\langle \hat{P}_i \rangle$, $${\ensuremath{\bar{\mathcal{C}}}}({\vec{\theta}}, M) = \sum_i c_i \big[\langle \hat{P}_i \rangle + {\varepsilon}_i(M)\big].
    \label{eq:sampled_cost_func}$$ Here, ${\varepsilon}_i$ is a random variable drawn from a binomial distribution with variance $\sigma_i^2\sim 1/M$ that is used to simulate the experimental shot or sampling noise. Assuming that Pauli operators are measured independently, the variance of the estimator ${\ensuremath{\bar{\mathcal{C}}}}$ may be propagated directly, $$\mathrm{Var}[{\ensuremath{\bar{\mathcal{C}}}}]=\sum_ic_i^2\sigma_i^2.$$ In general, the assumption of independence is violated. One may measure mutually commuting operators in parallel @verteletskyi2019measurement [@huggins2021efficient; @bonetnearly2020; @zhao2020; @crawford2021; @cotler2020quantum]. Then the resulting measurement has non-zero covariance @romero2018strategies [@Rubin2018], which should be accounted for. However, this only introduces a constant factor to the estimation cost, and will not significantly impact the relative optimzer performance. Here, we will use $M$, defined in Eq. , as the overall cost for the quantum subroutine which takes $\vec{\theta}$ and $M$ as inputs, and outputs ${\ensuremath{\bar{\mathcal{C}}}}(\vec{\theta},M)$.

To optimize within a VQA, an access to ${\mathcal{C}}(\vec{\theta})$ is provided to a classical optimizer, which then minimizes the sampled cost function ${\ensuremath{\bar{\mathcal{C}}}}(\vec{\theta},M)$ as a function of the classical parameters $\vec{\theta}$. One can additionally provide estimates of gradients $\grad_\theta {\mathcal{C}}$ (or higher order derivatives) in order to perform gradient-based (or Newton-like) optimization. To avoid a comparison of the runtime of gradient estimation to that of estimating the raw cost function ${\ensuremath{\bar{\mathcal{C}}}}(\vec{\theta},M)$, we only compare four gradient-free optimization algorithms. Moreover, it has been shown that gradient-based optimization strategies suffer given noisy function evaluations with simple noise structures @DBLP:conf/icml/WuHXHBZ20 (e.g., stationary and isotropic noisy covariance) in the sense that 1) the convergence rate to local optima is hampered @DBLP:journals/corr/abs-2006-07904 and 2) such simple noise does not help in escaping from local optima @DBLP:conf/icml/ZhuWYWM19. We select the following optimizers (see supplemental material @bonetperformance2021sup for further details):

1.  SLSQP determines a local search direction by solving the second-order local approximation of the cost function that satisfies the constrains,

2.  COBYLA uses linear approximations of the target and constrains function to optimize a simplex within a trust region of the parameter space,

3.  CMA-ES is a population-based optimization algorithm where the points are drawn from a multivariate Gaussian distribution, whose parameters (covariance matrix and location) are adapted online,

4.  SPSA employs a stochastic perturbation vector to compute simultaneously an approximate gradient of the objective.

We compare these algorithms across multiple systems of different sizes and number of parameters considered.

Three-stage sampling adaptation
===============================

Existing state-of-the-art quantum hardware is limited by the stability of the devices, which need to be tuned within time-scales of hours up to a day. This influences a hard limit on the total number of samples we can measure before the devices changes, of the order of $\sim\!\!\!10^9$. This shot budget becomes the limiting factor in VQAs. One must carefully balance between exploring the parameter space and accurately measuring the cost function. In this spirit, Cade et al. @cade2020strategieshubbard introduced a sampling procedure for SPSA that splits the total shot budget between three stages, resulting in improved performance of VQAs.

Naively, one can fix the total number of shots per Pauli operator and run the optimization until the budget is spent (i.e., if one used 1000 shots per Pauli per function call and allocated a total shot budget of $10^7$ per Pauli, this would allow for a total of 10000 evaluations). We refer to this approach as one-stage optimization. Alternatively, one could think of an optimization strategy where the number of shots is increased as the optimization progresses towards better parameters as introduced by Cade et al. We perform a three-stage optimization procedure where the number of samples per Pauli operator increases per phase, reducing the number of total function calls. In our three-stage optimization, we fix the number of shots per Pauli for each stage (i.e., 100-1000-10000 shots for a total budget of $10^7$). The number of function evaluations is then computed from a ratio 10:3:1. For every 10 function calls at the first stage, we use 3 function calls in the second stage and 1 function call in the third stage (i.e., 7150-2145-715 evaluations for a total budget of $10^7$ shots per Pauli).

We compare the one- and three-stage protocols for the four optimization algorithms previously introduced, aimed at assessing if the three-stage protocol has any evident advantage over standard sampling strategy. For this comparison we use the relative energy error, $$\label{eq:figure}
    \Delta_{r}E = \left|\frac{{\mathcal{C}}({\vec{\theta}}_{\text{opt}}) - E_0}{E_0-c_0}\right|,$$ where ${\mathcal{C}}({\vec{\theta}}_{\text{opt}})$ is the noiseless cost function evaluated at the optimized parameters ${\vec{\theta}}_{\text{opt}}$ obtained from a noisy optimization. $E_0$ is the lowest eigenvalue of the problem. $c_0$ is the coefficient of the identity operator, which is the largest Hamiltonian term. This is the relevant figure of merit to capture the performance of the optimizer as it measures the relative error in estimating the traceless part of the Hamiltonian $H-c_0I$, requiring the quantum computer. Our numerical experiments are performed under sampling noise with a total shot budget of $10^7$, $10^8$ and $10^9$ per Pauli operator. In the one-stage method we fix the total number of function evaluations to $10^4$ and use $10^3$, $10^4$ and $10^5$ shots per Pauli operator per function call. In the three-stage procedure the function calls are also fixed at 7150-2145-715 for all budgets, and the shots per Pauli operator at every stage are $10^2$-$10^3$-$10^4$, $10^3$-$10^4$-$10^5$, and $10^4$-$10^5$-$10^6$, respectively. The optimization stops when the shot budget is reached. However, the SLSQP and COBYLA optimization algorithms have a termination criterion that in most cases results in exiting early and not utilizing their full shot budget.

We benchmark the algorithms for three different systems on 8-qubits: $H_4$ in a chain and square configuration and the 2x2 Hubbard model. The results are shown in figure \[fig:unoptimized\]. For every problem we run experiments for one-stage (blue, pink and green dots) and three-stage (red, brown, and purple dots) protocols for each optimization. Both SLSQP and COBYLA have a consistent improvement when the three-stage sampling is applied, but are inferior to the performance of CMA-ES and SPSA in all cases. Conversely, for CMA-ES the one-stage sampling has better performance than the three-stage protocol in all three test systems. Finally, SPSA behaves differently with respect to the sampling procedure depending on the system. For the square configuration of $H_4$ the three-stage sampling has a slightly better energy. In the case of the chain configuration, the one-stage method shows almost a 4-fold improvement over the three-stage method. For the 2x2 Hubbard model the three-stage methods has an order of magnitude improvement in performance. Overall, the best performance across all problems is achieved by SPSA in both one- and three-stage methods (indicated by green and orange ticks on the top of the panel).

![image](figure_1_combined_unoptimized_comparison_scatter.pdf){width="1.75\columnwidth"}

Hyperparameter tuning
=====================

Most optimization algorithms come with default (hyper-)parameters detailing the optimization. These (hyper)-parameters are either derived under idealized theoretical assumptions, or evaluated from numerical experiments on standard benchmarks. It is common to tune the hyper-parameters of the optimizers when used on a function that has not been previously studied @DoerrYHWSB20 [@nikolaus_hansen_2019_2594848]. Similarly, when performing a VQA one should consider hyper-parameter-tuning the optimizer.

Finding optimal hyper-parameters of an optimizatizer can be costly and generally problem-dependent. A sub-field of classical optimization has been devoted to automatizing such hyper-parameter tuning. Here we use the iterated racing for automatic algorithm configuration @LOPEZIBANEZ201643, IRACE (see supplemental material @bonetperformance2021sup for a description of the procedure), to tune the SPSA and CMA-ES settings for four molecular systems, $H_4$ square and chain and $H_2 O$ at equilibrium and stretched geometries (corresponding to weakly- and strongly-correlated regimes, respectively). Additionally we perform hyper-parameter tuning of CMA-ES for the Hubbard model on three different configurations; 1x6, 2x2 and 2x3. For SPSA, however, we take the results of ref. @cade2020strategieshubbard where the hyper-parameters were tuned. The hyper-parameters used for the numerical experiments can be found in Tables I and II in the supplemental material @bonetperformance2021sup.

With the tuned hyper-parameters we performed a new set of experiments including new systems: $H_2 O$ in its equilibrium and stretched geometries with 10 qubits and Hubbard models on 1x6 and 2x3 lattices requiring 12 qubits. The results of these simulations are shown in figure \[fig:optimized\]. A first observation is that CMA-ES improves on the problems tested without tuning. The overall performance between CMA-ES and SPSA is roughly similar, with SPSA doing better in weakly-correlated problems ($H_4$ chain, $H_2O$ equilibrium and 2x2 Hubbard model). For strongly-correlated systems, CMA-ES has a slightly better performance with the mean values mostly within error bars (orange ticks in fig. \[fig:optimized\]). Supporting our idea that hyper-parameter tuning is crucial to ensure a good VQA performance is the 5-fold improvement of SPSA on the $H_4$ chain problem. For the $H_4$ square only a mild improvement is observed for both SPSA and CMA-ES, both reaching an almost equal relative energy error. A reason of this can be that both optimizers find the optimal parameters accessible by the ansatz, better result requiring a larger circuit.

With regards to the Hubbard model, SPSA performs clearly better on the 2x2 configurations. However CMA-ES is capable of finding much lower points in with $10^8$ and $10^9$ optimization, suggesting that the optimization landspace is not trivial, and that better results can be found. The 1x6 and 2x3 Hubbard models yield a comparable performance between the optimizers. Again, a possible explanation for this is that they reach the actual optimum.

Finally, we observe a that CMA-ES starts to outperform SPSA as the system size and number of parameters increases. This might be an indication that SPSA is not as well-suited for large problems.

![Comparison of optimized hyper-parameters of CMA-ES (red, brown and purple dots) and SPSA (grey, orange and blue dots). Black crosses and error bars depict the mean and 95% confidence interval of 15 indepedent runs. Green and orange represent the same as in fig. \[fig:unoptimized\]. <span data-label="fig:optimized"></span>](figure_2_optimized_cma_vs_spsa_scatter_v4.pdf){width="\columnwidth"}

The sampling noise floor
========================

A successful VQA requires the optimization algorithm to return the optimal parameters of ${\ensuremath{\bar{\mathcal{C}}}}$. It is common to assign as the optimal candidate the one with the best-ever measured ${\ensuremath{\bar{\mathcal{C}}}}$. However, in VQAs, the optimization is performed using a proxy cost function ${\ensuremath{\bar{\mathcal{C}}}}$ – a sampled version of the real objective ${\mathcal{C}}$. So, it is possible that ${\ensuremath{\bar{\mathcal{C}}}}({\vec{\theta}})$ returns a value that is lower than its corresponding noiseless evaluation ${\mathcal{C}}({\vec{\theta}})$ due to statistical fluctuations. Any optimizer that assigns the optimal candidate to the point with the best-ever function evaluation of ${\ensuremath{\bar{\mathcal{C}}}}$ will, with large probability, return a candidate worse than the global minimum (assuming its existence). The region of points in parameter space that can return the best-ever measured ${\ensuremath{\bar{\mathcal{C}}}}$ we refer to as the sampling noise floor, discussed in the following.

Given a cost function ${\mathcal{C}}({\vec{\theta}})$, we attempt to optimize its sampled version ${\ensuremath{\bar{\mathcal{C}}}}$ with variance $\mathrm{Var}[{\ensuremath{\bar{\mathcal{C}}}}]$. Let us assume ${\mathcal{C}}({\vec{\theta}})$ has a global minimum ${\vec{\theta}}_{g}$ with noiseless value ${\mathcal{C}}_{g}$, and that we evaluate ${\mathcal{C}}({\vec{\theta}})$ at multiple ${\vec{\theta}}$ including one evaluation at ${\vec{\theta}}_g$. Under sampling noise, the value of the cost function evaluated at ${\vec{\theta}}_g$ is drawn with some probability $1-p$ from a confidence interval $$\Delta_{p} = [{\mathcal{C}}_g + m(p)\sqrt{\mathrm{Var}[\textbf{{\ensuremath{\bar{\mathcal{C}}}}}]}, -\infty),$$ where $m(p)\sim\log(p)$ is the size of the relevant confidence interval for our distribution of ${\ensuremath{\bar{\mathcal{C}}}}$. Assuming this distribution is symmetric, whenever $\vec{\omega}\neq{\vec{\theta}}_g$ satisfies ${\mathcal{C}}(\vec{\omega})-m(p)\sqrt{\mathrm{Var}[\textbf{{\ensuremath{\bar{\mathcal{C}}}}}]} \notin \Delta_{p}$, with probability $>(1-p)$ an evaluation of the noisy cost function will lie above this confidence region, ${\ensuremath{\bar{\mathcal{C}}}}(\vec{\omega})>\Delta_p$. Then, with confidence $>(1-p)^2$, ${\vec{\theta}}_g$ will be correctly identified (between these two candidates) as the optimal set of parameters. However, when this is not the case and ${\mathcal{C}}(\vec{\omega})-m(p)\sqrt{\mathrm{Var}[\textbf{{\ensuremath{\bar{\mathcal{C}}}}}]} \in \Delta_{p}$, we can no longer be confident that the true minimum ${\vec{\theta}}_g$ will be identified. This defines a (potentially disconnected) region in parameter space, $$\Omega(p) = \{\vec{\omega}\colon{\mathcal{C}}(\vec{\omega}) < {\mathcal{C}}({\vec{\theta}}_g) + 2m(p)\sqrt{\mathrm{Var}[\textbf{{\ensuremath{\bar{\mathcal{C}}}}}]}\},$$ from where alternative candidates can be drawn with probability $p$. This corresponds to a region of possible cost function values, $$\mathbb{C}_p=\Big[{\mathcal{C}}({\vec{\theta}}_g),{\mathcal{C}}({\vec{\theta}}_g)+2m(p)\sqrt{\mathrm{Var}[\textbf{{\ensuremath{\bar{\mathcal{C}}}}}]}\Big],$$ that an optimizer returning the best-ever measured ${\ensuremath{\bar{\mathcal{C}}}}$ may achieve. We define the width of this region, $2m(p)\sqrt{\mathrm{Var}[\textbf{{\ensuremath{\bar{\mathcal{C}}}}}]}$, as the sampling noise floor. This region is not completely defined, as we have not set a value for $p$. In practice, the value of $p$ depends upon the rate at which the optimizer can converge; an optimizer that converges slowly will encounter more parameter sets $\vec{\omega}$ near the true minimum, increasing the probability that one of these parameter sets might generate a false optimum. We do not have direct access to an estimate for the width of $\mathbb{C}_p$, but we can still demonstrate the phenomenon numerically.

SPSA and CMA-ES have been designed not to rely on the best-ever function evaluation. In particular, CMA-ES returns two different candidates; the best-ever measured and a so-called favourite. The favourite is computed by the algorithm's update function at the end of the optimization process and includes all accumulated prior information. As this information includes many more shots than a single function call, in principle it can average out the sampling noise over the optimization landscape, and beat the sampling noise floor. We investigate the effect of sampling noise on these two candidates returned by CMA-ES.

Figure \[fig:best\_fav\], shows the results of the sampling noise floor in the optimization performance. For every system we computed the following energy values: ${\ensuremath{\bar{\mathcal{C}}}}({\vec{\theta}}_{\text{best}})$, ${\mathcal{C}}({\vec{\theta}}_{\text{best}})$ and ${\mathcal{C}}({\vec{\theta}}_{\text{fav}})$. For each estimated energy we compute the relative energy error $\Delta E = \frac{{\mathcal{C}}({\vec{\theta}}_{\text{opt}}) - E_0}{\left|E_0-c_0\right|}$. Note that the best-ever function evaluation (orange points) is often below the true energy, breaking the variational principle. This is due to the effect of sampling noise, and for a fair comparison we should compare ${\mathcal{C}}({\vec{\theta}}_{\text{best}})$ (red points) and ${\mathcal{C}}({\vec{\theta}}_{\text{fav}})$ (purple points), where the true cost function is evaluated. The mean value of ${\mathcal{C}}({\vec{\theta}}_{\text{best}})$ gives an estimate for the width of the sampling noise floor. We observe that in all cases, the mean of the favourite candidate is below the mean of the best candidate, showing that the optimizer has beaten the sampling noise floor. For the Hubbard model and the $H_4$ systems, this is not significant (up to a 95$\%$ confidence interval), but for the two geometries of the water molecule, the difference is much larger. Choosing the favourite over the best yields up to a $3$-fold reduction of error. We believe the difference in performance comes from the different optimization landscapes of the different problems; studying this in detail is a target for future work.

![Best-ever versus favourite candidate from CMA-ES under noisy optimization. The optimization uses $10^7$ shots per Pauli over the course of the experimen with individual estimations of ${\ensuremath{\bar{\mathcal{C}}}}(\vec{\theta})$ are made using only $10^4$ shots per Pauli. From left to right: (orange) best-ever measured function evaluation, (red) the best-ever candidate evaluated noiseless, and (purple) favourite candidate evaluted noiseless. <span data-label="fig:best_fav"></span>](figure_3_comparison_best_vs_fav_1e7_v3.pdf){width="\columnwidth"}

Conclusion
==========

Variational quantum algorithms have recently prompted significant interest as candidates amenable to near-term hardware. However, the performance of these quantum algorithms relies on a classical optimization of a difficult cost function. This task is in general intractable to solve optimally. It is, therefore, important to benchmark the available optimizers for this purpose. We study the performance of four off-the-shelf optimization algorithms under the effect of sampling noise for the task of finding the ground-state energies. We perform a comparison using default hyper-parameters, and then extend the analysis using a three-stage sampling method from Ref. @cade2020strategieshubbard and by adding hyperparameter tuning. First find that SPSA performs best in both standard and three-stage samplings without tuning. Next, we focus on the performance of SPSA and CMA-ES in the three-stage procedure. We then hyper-parameter-optimize these two methods. With these new parameter settings, SPSA and CMA-ES have a comparable performance on the strongly-correlated systems and a small advantage for SPSA on the weakly-correlated ones. We notice that the advantages of SPSA seem to vanish as problem sizes grow.

Finally, we study the effect of sampling noise on the optimization performance using CMA-ES. We observe that the best-ever function evaluation is may not be a feasible optimal candidate, contrary to the common approach in classical optimization. Specifically, we show that the best-ever result suffers from a sampling noise floor problem that makes any of the parameters within it a potential best-ever result. In contrast, the so-called CMA-ES favourite candidate obtained from its update rule at the end of the optimization shows an overall better mean and standard deviation than its best-ever counter-part, indicating the sampling noise floor can be overcome. We expect that our analytical and numerical results of the sampling noise floor opens a new line of inquiry about optimization methods for VQAs.

The authors would like to thank Carlo Beenakker for the support on this project, and Jordi Tura, Yaroslav Herasymenko, Stefano Polla, Alicja Dutkiewicz, Saad Yalouz and Eleanor Scerri for fruitful discussions. CM, TB and VD acknowledge support from TotalEnergies. VD acknowledges the support of SURF through the QC4QC project. This work was in part supported by the Dutch Research Council (NWO/OCW), as part of the Quantum Software Consortium programme (project number 024.003.037).

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