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arXiv:2504.04932 · Quantum Science and Technology

Weighted approximate quantum natural gradient for variational quantum eigensolver

Chenyu Shi, Vedran Dunjko, Hao Wang

Published Jan 30, 2026• 1 citation
arXivHTMLPDF

Abstract

Abstract The variational quantum eigensolver (VQE) is one of the most prominent algorithms using near-term quantum devices, designed to find the ground state of a Hamiltonian. In VQE, a classical optimizer iteratively updates the parameters in the quantum circuit. Among various optimization methods, the quantum natural gradient descent (QNG) stands out as a promising optimization approach for VQE. However, standard QNG only leverages the quantum Fisher information of the entire system and treats each subsystem equally in the optimization process, without accounting for the different weights and contributions of each subsystem corresponding to each local term in the Hamiltonian. To address this limitation, we propose a Weighted Approximate QNG (WA-QNG) method tailored for k -local Hamiltonians. In this paper, we theoretically analyze the potential advantages of WA-QNG compared to QNG from three distinct perspectives and reveal its connection with the Gauss–Newton method. We also show it outperforms the standard QNG descent in the numerical simulations for seeking the ground state of the Hamiltonian.

DOI
10.1088/2058-9565/ae3fc8

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•Jan 1, 2025•0 citations
Abstract The variational quantum eigensolver (VQE) is one of the most prominent algorithms using near-term quantum devices, designed to find the ground state of a Hamiltonian. In VQE, a classical optimizer iteratively updates the parameters in the quantum circuit. Among various optimization methods, the quantum natural gradient descent (QNG) stands out as a promising optimization approach for VQE. However, standard QNG only leverages the quantum Fisher information of the entire system and treats each subsystem equally in the optimization process, without accounting for the different weights and contributions of each subsystem corresponding to each local term in the Hamiltonian. To address this limitation, we propose a Weighted Approximate QNG (WA-QNG) method tailored for k -local Hamiltonians. In this paper, we theoretically analyze the potential advantages of WA-QNG compared to QNG from three distinct perspectives and reveal its connection with the Gauss–Newton method. We also show it outperforms the standard QNG descent in the numerical simulations for seeking the ground state of the Hamiltonian.
The Variational Quantum Eigensolver (VQE) is one of the most promising algorithms for current quantum devices. It employs a classical optimizer to iteratively update the parameters of a variational quantum circuit in order to search for the ground state of a given Hamiltonian. The efficacy of VQEs largely depends on the optimizer employed. Recent studies suggest that Quantum Natural Gradient Descent (QNG) can achieve faster convergence than vanilla gradient descent (VG), but at the cost of additional quantum resources to estimate Fubini-Study metric tensor in each optimization step. The Fubini-Study metric tensor used in QNG is related to the entire quantum state space and does not incorporate information about the target Hamiltonian. To take advantage of the structure of the Hamiltonian and address the limitation of additional computational cost in QNG, we propose Hamiltonian-aware Quantum Natural Gradient Descent (H-QNG). In H-QNG, we propose to use the Riemannian pullback metric induced from the lower-dimensional subspace spanned by the Hamiltonian terms onto the parameter space. We show that H-QNG inherits the desirable features of both approaches: the low quantum computational cost of VG and the reparameterization-invariance of QNG. We also validate its performance through numerical experiments on molecular Hamiltonians, showing that H-QNG achieves faster convergence to chemical accuracy while requiring fewer quantum computational resources.
We investigate the performance of the quantum natural gradient (QNG) optimizer in the presence of noise. Specifically, we evaluate the efficacy of QNG within the quantum approximate optimization algorithm for finding the ground state of the transverse field Ising model. Its performance is benchmarked against the vanilla gradient descent optimizer across two prominent quantum computing platforms: Rydberg atoms and superconducting circuits. Our analysis includes simulations under both idealized noise-free conditions and realistic noisy environments based on calibration data from actual devices. Results demonstrate that QNG consistently outperforms vanilla gradient descent, exhibiting faster convergence on average and greater robustness against random initializations of parameters. This robustness is attributed to the distance regularization in parameter space inherent to QNG. Additionally, QNG achieves a higher convergence rate to the solution, effectively avoiding certain local minima. These findings highlight QNG as a promising tool for optimizing variational quantum algorithms in noisy intermediate-scale quantum devices.
Abstract The efficient optimization of variational quantum algorithms (VQAs) is critical for their successful application in quantum computing. The Quantum Natural Gradient (QNG) method, which leverages the geometry of quantum state space, has demonstrated improved convergence compared to standard gradient descent (Stokes et al. in Quantum 4:269, 2020). In this work, we introduce the Modified Conjugate Quantum Natural Gradient (CQNG), an optimization algorithm that integrates QNG with principles from the nonlinear conjugate-gradient method. Unlike QNG, which employs a fixed learning rate, CQNG dynamically adjusts its hyperparameters at each step, enhancing both efficiency and flexibility. Numerical simulations show that CQNG achieves faster convergence and reduces quantum-resource requirements compared to QNG across various optimization scenarios, even when strict conjugacy conditions are not fully satisfied—hence the term “Modified Conjugate.” These results highlight CQNG as a promising optimization technique for improving the performance of VQAs.
Optimization problems are prevalent in various fields, and the gradient-based gradient descent algorithm is a widely adopted optimization method. However, in classical computing, computing the numerical gradient for a function with d variables necessitates at least d + 1 function evaluations, resulting in a computational complexity of O ( d ). As the number of variables increases, the classical gradient estimation methods require substantial resources, ultimately surpassing the capabilities of classical computers. Fortunately, leveraging the principles of superposition and entanglement in quantum mechanics, quantum computers can achieve genuine parallel computing, leading to exponential acceleration over classical algorithms in some cases. In this paper, we propose a novel quantum-based gradient calculation method that requires only a single oracle calculation to obtain the numerical gradient result for a multivariate function. The complexity of this algorithm is just O (1). Building upon this approach, we successfully implemented the quantum gradient descent algorithm and applied it to the variational quantum eigensolver (VQE), creating a pure quantum variational optimization algorithm. Compared with classical gradient-based optimization algorithm, this quantum optimization algorithm has remarkable complexity advantages, providing an efficient solution to optimization problems. The proposed quantum-based method shows promise in enhancing the performance of optimization algorithms, highlighting the potential of quantum computing in this field.
The variational quantum eigensolver (VQE) is an algorithm for finding the ground states of a given Hamiltonian. Its application to binary-formulated combinatorial optimization (CO) has been widely studied in recent years. However, typical VQE approaches for CO problems often suffer from local minima or barren plateaus, limiting their ability to achieve optimal solutions. The quantum imaginary time evolution (QITE) offers an alternative approach for effective ground-state preparation but requires large circuits to approximate non-unitary operations. Although compressed QITE (cQITE) reduces circuit depth, accumulated errors eventually cause energy increases. To address these challenges, we propose an Adaptive Weighted QITE-VQE (AWQV) algorithm that integrates the VQE gradients with the cQITE updates through an adaptive weighting scheme during optimization. In numerical simulations for MaxCut on unweighted regular graphs, AWQV achieves near-optimal approximation ratios, while for weighted Erdős-Rényi instances, it outperforms the classical Goemans-Williamson algorithm.
The efficient optimization of variational quantum algorithms (VQAs) is critical for their successful application in quantum computing. The Quantum Natural Gradient (QNG) method, which leverages the geometry of quantum state space, has demonstrated improved convergence compared to standard gradient descent [Quantum 4, 269 (2020)]. In this work, we introduce the Modified Conjugate Quantum Natural Gradient (CQNG), an optimization algorithm that integrates QNG with principles from the nonlinear conjugate gradient method. Unlike QNG, which employs a fixed learning rate, CQNG dynamically adjusts hyperparameters at each step, enhancing both efficiency and flexibility. Numerical simulations show that CQNG achieves faster convergence than QNG across various optimization scenarios, even when strict conjugacy conditions are not always satisfied -- hence the term ``Modified Conjugate.'' These results highlight CQNG as a promising optimization technique for improving the performance of VQAs.