Abstract We compare the use of adaptive moment estimation (ADAM), simultaneous perturbation stochastic approximation (SPSA), Nakanishi–Fujii–Todo method (NFT), and CoolMomentum in a variational quantum eigensolver. Using a random weighted max-cut problem, we numerically analyze these methods and confirm that CoolMomentum performs better than the other methods. ADAM and SPSA tend to get trapped in local minima or exhibit infeasible optimization durations. Although NFT exhibits fast convergence, it tends to suffer from local minima similar to ADAM and SPSA. Contrarily, CoolMomentum shows a higher accuracy under noiseless and realistic hardware noise conditions.
Abstract We compare the use of adaptive moment estimation (ADAM), simultaneous perturbation stochastic approximation (SPSA), Nakanishi–Fujii–Todo method (NFT), and CoolMomentum in a variational quantum eigensolver. Using a random weighted max-cut problem, we numerically analyze these methods and confirm that CoolMomentum performs better than the other methods. ADAM and SPSA tend to get trapped in local minima or exhibit infeasible optimization durations. Although NFT exhibits fast convergence, it tends to suffer from local minima similar to ADAM and SPSA. Contrarily, CoolMomentum shows a higher accuracy under noiseless and realistic hardware noise conditions.
The current state of quantum processing units (QPU) is characterized by Noisy Intermediate-Scale Quantum (NISQ) devices. Among the methods to utilize NISQ devices, the Variational Quantum Eigensolver (VQE) is prominent for finding ground state energy of quantum systems. However, VQE parameter optimization is NP-Hard, posing significant challenges. To address this, we introduce the CoolMomentum method, a global optimization approach based on Langevin dynamics with Simulated Annealing. We compared CoolMomentum with ADAM, SPSA, and NFT in VQE using a random weighted Max-Cut problem for up to 20 qubits. Our results confirm that CoolMomentum consistently outperforms the other methods. ADAM and SPSA tend to get trapped in local minima or exhibit infeasible optimization durations, whereas NFT, despite its fast convergence, also suffers from local minima. In contrast, CoolMomentum demonstrates higher accuracy and robustness across various numbers of qubits, highlighting its potential as a superior optimization strategy for VQE.
In this study, we investigated the Variational Quantum Eigensolver (VQE) application for the Ising model as a testbed model, in which we thoroughly delved into several optimizers, both classical and quantum, and analyzed the extent to which each of these methods would offer a benefit. We then investigated a new combinatorial optimization scheme, termed QN-SPSA+PSR, in which the Fubini-Study metric is approximated within the Quantum Natural Gradient (QN) framework, with its inner gradient estimated by the Simultaneous Perturbation Stochastic Approximation (SPSA), while the outer gradient of the cost function is evaluated exactly by the Parameter-Shift Rule (PSR). The QN-SPSA+PSR method integrates the QN-SPSA computational efficiency with the precise gradient computation of the PSR, improving the stability of QN-SPSA-based and convergence speed per parameter update while maintaining low computational consumption. Our results provide a potential performance improvement in the VQAs' optimization subroutine, even in Quantum Machine Learning's optimization section, and enhance viable paths toward efficient quantum simulations on Noisy Intermediate-Scale Quantum Computing (NISQ) devices. Additionally, we also conducted a detailed study of quantum circuit ansatz structures in order to find the one that would work best with the Ising model and NISQ, in which we utilized the properties of the investigated model.
Sequential minimal optimization methods, such as the Rotosolve and the Nakanishi-Fujii-Todo algorithm (NFT), are widely used for Variational Quantum Eigensolvers (VQEs). These methods optimize one parameter direction at a time, requiring measurements at only a few locations along that direction. In the presence of measurement shot noise, however, their performance depends critically on the choice of measurement locations, and recent studies suggest that equidistant measurements are optimal. However, we often observe that equidistant measurements are not always optimal in practice. We argue that this discrepancy between theory and practice arises from the fact that two assumptions underlying previous analyses do not generally hold: (1) the absence of prior knowledge about the energy minimizer, and (2) the use of the uncertainty of the estimated energy as a proxy for optimization performance. In this paper, we develop a new theory for determining optimal measurement locations. First, we show that incorporating prior information about the minimizer is beneficial. Early in optimization, when little is known about the pivot, i.e., the current minimizer, equidistant measurements are indeed near-optimal, but as the prior belief sharpens the optimal locations move away from equidistant. Second, rather than analyzing the uncertainty of the estimated minimum energy, we study the uncertainty of the estimator of the minimizer itself, which leads to substantially different strategies. Based on this analysis, we propose Prior-informed Adaptive Shifts (PAS), a method that automatically adjusts measurement locations during optimization. Numerical experiments across different shot counts and problems validate our theoretical findings and demonstrate that PAS adaptively recovers whichever fixed shift is best in each regime without it being specified in advance.
Abstract We numerically benchmark 30 optimisers on 372 instances of the variational quantum eigensolver for solving the Fermi–Hubbard system with the Hamiltonian variational ansatz. We rank the optimisers with respect to metrics such as final energy achieved and function calls needed to get within a certain tolerance level, and find that the best-performing optimisers are variants of gradient descent such as Momentum and ADAM (using finite difference), SPSA, CMAES, and BayesMGD. We perform gradient analysis, and observe that the step size for finite difference has a very significant impact. We also consider using simultaneous perturbation (inspired by SPSA) as a gradient subroutine: here finite difference can lead to a more precise estimate of the ground state but uses more calls, whereas simultaneous perturbation can converge quicker but may be less precise in the later stages. Finally, we study the quantum natural gradient algorithm: we implement this method for one-dimensional Fermi–Hubbard systems, and find that whilst it can reach a lower energy with fewer iterations, this improvement is typically lost when taking total function calls into account. Our method involves performing careful hyperparameter sweeping on 4 instances. We present a variety of analysis and figures, detailed optimiser notes, and discuss future directions.
Variational quantum algorithms (VQAs) offer a promising path toward 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 [sequential least-squares quadratic programming, constrained optimization by linear approximations, the covariance matrix adaptation evolutionary strategy (CMA-ES), and the simultaneous perturbation stochastic approximation (SPSA)] to find 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) for 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 of 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 experiments.
The Nakanishi Fujii Todo (NFT) algorithm, also known as Rotosolve, implements Sequential Minimal Optimization for Variational Quantum Eigensolvers (SMO-VQE) by exploiting the trigonometric dependence of the energy on individual circuit parameters. This enables analytical one-dimensional minimization using only a few , typically two, energy evaluations, but introduces bias in the estimated energy. Although performing additional measurements every few tens of iterations can mitigate bias accumulation, we find that such corrections often degrade optimization performance. In this paper, we analyze the origin and accumulation of bias during the SMO-VQE process. Specifically, we show that the bias can be accurately estimated without additional measurements. Furthermore, we find that bias correction destabilizes optimization along directions with small curvature, whereas the original biased estimator implicitly acts as a regularizer. Based on these insights, we propose a simple regularization method that implements error accumulation while maintaining unbiased energy estimation. The resulting algorithm consistently improves performance across different system sizes, circuit depths, target Hamiltonians, and measurement shots, with minimal hyperparameter tuning.