Weighted Approximate Quantum Natural Gradient for Variational Quantum Eigensolver

TL;DR

This work targets the optimization bottleneck in variational quantum eigensolvers by recognizing that local Hamiltonian terms contribute unequally to the objective. It introduces Weighted Approximate Quantum Natural Gradient (WA-QNG), which replaces the full-system quantum Fisher information with a weighted sum of subsystem Hilbert-Schmidt metric tensors, capturing the differential impact of each local term in $H=\sum_m h_m H_m$; the HS tensors approximate the Fisher information when subsystems are near-pure, and the method aligns with QNG in the global-term limit. The authors connect WA-QNG to Gauss-Newton under near-pure conditions, provide optimization and geometric interpretations, and demonstrate faster convergence than standard QNG in Ising and Heisenberg benchmarks, with robustness to weight imbalance and increasing locality. They also discuss complexity and measurement considerations, highlighting that classical shadows can efficiently estimate the required metrics for $k$-local subsystems, potentially reducing shot cost in practical settings. Overall, WA-QNG offers a principled, scalable improvement for VQE optimization by leveraging locality structure and subsystem weights, with potential applicability to other variational quantum algorithms.

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 Quantum Natural Gradient (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 quantum natural gradient descent in the numerical simulations for seeking the ground state of the Hamiltonian.

Figures (11)

• Figure 1: Left: An illustration of the variational quantum circuit in VQE. A parameterized quantum circuit $U(\theta)$ is used to prepare a variational quantum state $\rho_{\theta}$. By adjusting the parameter $\theta$, the quantum circuit aims to prepare the state $\rho_{\theta}$ that approximates the ground state $\rho_{GS}$ by minimizing the objective function $f(\theta)$. Right: The general working principle of VQE. The quantum computer evaluates the expectation value $f(\theta)=\mathrm{tr}(\rho_{\theta}H)$, while the classical computer employs an optimizer to iteratively update the parameters $\theta$ to minimize the objective function $f(\theta)$.
• Figure 2: An illustration of distance distortion in the parametrization. The original parameter space is an Euclidean space. The parametrization maps the point in the parameter space to a point on the sphere with radius $r=1$ by the coordinate transformation $x=\sin(\theta)\cos(\phi)$, $y=\sin(\theta)\sin(\phi)$, $z=\cos(\theta)$. The distances from $A$ to $B$ (red line) and from $C$ to $D$ (purple line) are the same in the original parameter space. However, after parametrization to the sphere, the distances are distorted and become different.
• Figure 3: An illustration when the Hamiltonian is $H=Z_1Z_2+2Z_1Z_2+X_1+0.5X_2+1.5X_3$. The small rectangles indicate that the corresponding Hamiltonian term acts on that subsystem. And the width of rectangles reflects the magnitude of each coefficient $h_m$. Each subsystem contributes differently to the output due to the coefficients, suggesting that they should be assigned different weights in the optimization process.
• Figure 4: An example of a $4$-qubit EfficientSU2 circuit. It consists of single-qubit rotation gates $R_x$ and $R_y$, followed by a series of CNOT gates to enhance entanglement.
• Figure 5: The learning curves of the three methods for Ising model of $10$, $12$, $14$ qubits on 1-layer EfficientSU2 circuit.