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.
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.
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.
Variational quantum algorithms (VQAs) combining the advantages of parameterized quantum circuits and classical optimizers, promise practical quantum applications in the noisy intermediate-scale quantum era. The performance of VQAs heavily depends on the optimization method. Compared with gradient-free and ordinary gradient descent methods, the quantum natural gradient (QNG), which mirrors the geometric structure of the parameter space, can achieve faster convergence and avoid local minima more easily, thereby reducing the cost of circuit executions. We utilized a fully programmable photonic chip to experimentally estimate the QNG in photonics for the first time, to the best of our knowledge. We obtained the dissociation curve of the He-H + cation and achieved chemical accuracy, verifying the outperformance of QNG optimization on a photonic device. Our work opens up a vista of utilizing QNG in photonics to implement practical near-term quantum applications.
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.
ABSTRACT The lack of both a consistently effective and broadly reliable classical optimizer for variational quantum algorithms and a clear physical understanding of how the Fubini‐Study (FS) metric evolves during optimization remains a central challenge for practical variational quantum eigensolver (VQE) applications. In this work, we systematically investigate the quantum natural gradient (QNG) as a geometry‐aware optimization strategy and benchmark its performance against commonly used classical optimizers, including gradient‐based, derivative‐free, and adaptive methods. Using representative molecular systems, we analyze energy convergence, parameter‐space trajectories, and the dynamical evolution of the FS metric tensor. QNG consistently achieves faster and more stable convergence in the studied molecules by aligning parameter updates with the intrinsic curvature of the quantum state manifold. The evolution of the metric reveals a rapid suppression of irrelevant parameter directions and a reduced state distinguishability near the ground state. Obtained under ideal and simplified noise conditions, these results clarify the geometric origin of QNG advantage and highlight the potential suitability of geometry‐aware optimization for robust VQE implementation on near‐term quantum devices.
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 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.
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.