Variational Quantum Eigensolver: A Comparative Analysis of Classical and Quantum Optimization Methods

Duc-Truyen Le, Vu-Linh Nguyen, Ha C. Nguyen, Hung Q. Nguyen, Van-Duy Nguyen

Published May 6, 2025• 2 citations

Abstract

Abstract In this study, we study the Variational Quantum Eigensolver (VQE) application for the Ising model as a test bed model, in which we pivotally delved into several optimization methods, both classical and quantum, and analyzed the quantum advantage that each of these methods offered, and then we proposed a new combinatorial optimization scheme, deemed as QN-SPSA+PSR which combines calculating approximately Fubini-study metric (QN-SPSA) and the exact evaluation of gradient by Parameter-Shift Rule (PSR). The QN-SPSA+PSR method integrates the QN-SPSA computational efficiency with the precise gradient computation of the PSR, improving both stability and convergence speed while maintaining low computational consumption. Our results provide a new potential quantum supremacy in the VQAs's 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.

DOI: 10.21203/rs.3.rs-6497487/v1

Related papers

Variational Quantum Eigensolver: A Comparative Analysis of Classical and Quantum Optimizer Methods

Duc-Truyen Le, Vu-Linh Nguyen, Ha C. Nguyen +2 more

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.

arXiv:2412.19176IEEE Transactions on Quantum EngineeringJan 1, 20260 citations

An Empirical Comparison of Optimizers for Quantum Machine Learning with SPSA-Based Gradients

Marco Wiedmann, Marc Hölle, Maniraman Periyasamy +5 more

Variational quantum algorithms (VQAs) have attracted a lot of attention from the quantum computing community for the last few years. Their hybrid quantum-classical nature with relatively shallow quantum circuits makes them a promising platform for demonstrating the capabilities of noisy intermediate scale quantum (NISQ) devices. Although the classical machine learning community focuses on gradient-based parameter optimization, finding near-exact gradients for variational quantum circuits (VQCs) with the parameter-shift rule introduces a large sampling overhead. Therefore, gradient-free optimizers have gained popularity in quantum machine learning circles. Among the most promising candidates is the simultaneous perturbation stochastic approximation (SPSA) algorithm, due to its low computational cost and inherent noise resilience. We introduce a novel approach that uses the approximated gradient from SPSA in combination with state-of-the-art gradient-based classical optimizers. We demonstrate numerically that this out-performs both standard SPSA and the parameter-shift rule in terms of convergence rate and absolute error in simple regression tasks. The improvement of our novel approach over SPSA with stochastic gradient descent (SGD) is even amplified when shot- and hardware-noise are taken into account. We also demonstrate that error mitigation does not significantly affect our results.

arXiv:2305.00224IEEE International Conference on Quantum Computing and Engineering (QCE)Sep 17, 202325 citations

Enhancing Hybrid Methods in Parameterized Quantum Circuit Optimization

Joona V. Pankkonen, Matti Raasakka, Andrea Marchesin +1 more

Parameterized quantum circuits (PQCs) play an essential role in the application of variational quantum algorithms (VQAs) in noisy intermediate-scale quantum (NISQ) devices. The PQCs are a leading candidate to achieve a quantum advantage in NISQ devices and have already been applied in various domains such as quantum chemistry, quantum machine learning, combinatorial optimization, and many others. There is no single definitive way to optimize PQCs. The most commonly used methods are based on computing the gradient via the parameter-shift rule to use classical gradient descent (GD) optimizers like Adam, stochastic GD, and others. In addition, sequential single-qubit optimizers have been proposed, such as Rotosolve, Free-Axis Selection (Fraxis), Free-Quaternion Selection (FQS), and hybrid algorithms from the aforementioned optimizers. We further develop hybrid algorithms than those represented in the previous work by drawing inspiration from the early stopping method used in classical machine learning. The switch between the optimizers depends on the previous cost function values compared to the previous ones. We introduce two new hybrid algorithms that are more robust and scalable, and they outperform previous hybrid methods in terms of convergence towards the global minima across various cost functions. In addition, we find that they are feasible for NISQ devices with different noise profiles.

arXiv:2510.08142Oct 9, 20250 citations

Variational Quantum Eigensolver Applications in Quantum Machine Learning

M Amshavalli, Saurabha Srivastava

The Variational Quantum Eigensolver (VQE) has emerged as a transformative approach in quantum computing, providing a practical means to tackle complex quantum systems. This chapter delves into the essential components and strategies for enhancing VQE performance, focusing on key areas such as hybrid optimization techniques, quantum circuit design, and measurement efficiency. By integrating classical and quantum resources, hybrid optimization methods significantly improve convergence rates while reducing the computational burden on quantum devices. The chapter explores the design of quantum circuits, emphasizing the importance of selecting appropriate variational ansätze to achieve optimal results. Challenges related to Hamiltonian simulation and measurement are critically examined, alongside innovative solutions to mitigate noise and enhance measurement fidelity. Through these comprehensive insights, this chapter aims to illuminate the potential of VQE in advancing quantum machine learning applications across diverse scientific domains.

Dec 7, 20240 citations

Variational Parameter Optimization of Quantum-classical Hybrid Heuristics on Near-term Quantum Computer

Tsukasa Miki, Daisuke Tsukayama, Ryo Okita +2 more

Currently available quantum processors are noisy intermediate-scale quantum (NISQ) devices. The variational quantum eigensolver (VQE) is an algorithm that is closer to near-term applicability due to lower quantum hardware requirements. In VQE, trial states with variational parameters are prepared by quantum computers, and the optimal parameters are determined by a classical optimizer. This optimization is known to be an NP-hard problem. In this work, we make a comparison between various gradient-free optimizers in terms of approximation ratio and function evaluations. As a result, we find that COBYLA method is the best to find the approximated solution with a lower number of executions of quantum computers.

IEEE International Conference on Manipulation, Manufacturing and Measurement on the Nanoscale (3M-NANO)Aug 8, 20223 citations

Practical Utility of Hybrid Quantum-Classical Machine Learning: A Rigorous Benchmarking of VQE and QAOA on NISQ-Era Simulators

Shahriar Ahsan Taisiq, Tahera Shafina Toma

Abstract The emergence of noisy intermediate-scale quantum (NISQ) devices has fueled the development of hybrid quantum-classical algorithms for machine learning (ML), promising potential advantages over classical methods. However, the practical efficacy of these algorithms remains uncertain due to challenges like noise susceptibility, trainability issues, and a lack of rigorous benchmarking. This study presents a comprehensive empirical evaluation of two flagship hybrid algorithms the Variational Quantum Eigensolver (VQE) for classification and the Quantum Approximate Optimization Algorithm (QAOA) for combinatorial optimization against state-of-the-art classical baselines under equivalent resource constraints. Through extensive simulations incorporating realistic noise models, we demonstrate that while these quantum approaches achieve competitive performance (e.g., 91.2% accuracy for VQE on a binary classification task), they consistently underperformed optimized classical models like Support Vector Machines (94.8% accuracy) and require orders of magnitude more computational time. Furthermore, we provide empirical validation of critical barriers to scalability, including the barren plateau phenomenon and significant performance degradation under noise, even with advanced error mitigation. Our results indicate that for the tasks studied, current NISQ era algorithms do not yet provide a tangible quantum advantage. This work contributes a rigorous benchmarking framework and a critical, realistic assessment of the field, suggesting that future progress hinges on fundamental algorithmic innovations and hardware improvements rather than incremental refinements of existing paradigms.

Apr 9, 20260 citations