Efficient Hamiltonian-aware Quantum Natural Gradient Descent for Variational Quantum Eigensolvers

TL;DR

This work targets optimizer efficiency in Variational Quantum Eigensolvers by introducing Hamiltonian-aware Quantum Natural Gradient Descent (H-QNG), a pullback-metric optimizer defined on the subspace spanned by the Hamiltonian terms. The metric G used by H-QNG is constructed as $G_{ij}=\sum_{r=1}^v a_r^2\mathrm{tr}(\partial_i\rho_{\theta}P_r)\mathrm{tr}(\partial_j\rho_{\theta}P_r)$ with a scaling $T_{ij}=\frac{1}{2\sqrt{\sum_r a_r^2}}G_{ij}$, yielding an update $\theta^{(k+1)}=\theta^{(k)}-\eta T^{-1}\nabla f(\theta^{(k)})$. This approach reduces the quantum measurement overhead to $O(mv)$ per step—the same as vanilla gradient descent—while retaining the reparameterization-invariance of QNG and leveraging the Hamiltonian’s structure to potentially accelerate convergence. The paper provides scaling, reparameterization-invariance, and singularity analyses, and validates the method on molecular Hamiltonians (e.g., LiH and H6), where H-QNG achieves chemical accuracy faster and with fewer quantum resources than standard QNG. It also discusses robustness to shot noise and outlines avenues for extending the framework to mixed states and other variational quantum algorithms.

Abstract

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.

Figures (8)

• Figure 1: The general workflow of VQEs. In the quantum part, a parameterized quantum circuit is used to estimate the expectation value $\mathrm{tr}(\rho_{\theta^{(k)}}H)$ and its gradient, given the current parameters $\theta^{(k)}$. In the classical part, a classical optimizer iteratively updates the parameters from $\theta^{(k)}$ to $\theta^{(k+1)}$ in order to minimize the expectation value. Various optimizers can be employed in the optimization of VQEs, and the choice plays an important role in its performance.
• Figure 2: To minimize the expectation value of the Hamiltonian $H=-X-Y$, only the manifold embedded in lower dimensional space (the unit disc with boundary in (b)) attributes non-trivially to the optimization rather than the entire Bloch sphere (the sphere in (a)), since the $z$-axis coordinate does not appear in the cost function. The geodesic with respect to the spherical metric (red solid line in (a)) represents the shortest path on the Bloch sphere from the initial point to the solution. However, its projection onto the non-trivial $xy$-plain (red dotted lines in a and b) is not the shortest path compared to the true geodesic within the $xy$-plain (blue dotted lines in a and b). Therefore, in the pullback metric view, H-QNG directly employs $g_{\mathcal{M}}$ as the metric of the non-trivial manifold in \ref{['df-hqng']}, in contrast to standard QNG which considers $g_{\mathcal{M}}$ to be the metric of the full quantum state space in \ref{['pp-qng']}.
• Figure 3: The energy distance from the ground state $\Delta E$ of a 4-qubit $H_2$ Hamiltonian using vanilla gradient descent (blue line), QNG (orange line), and H-QNG (green line). All three methods start from the identical initial parameters. H-QNG reaches chemical accuracy (red dotted line) earlier than standard QNG, both in terms of optimization steps and quantum circuit evaluations (in the unit of quantity to be estimated). Both standard QNG and H-QNG significantly outperform vanilla gradient descent.
• Figure 4: Evaluating the reparameterization invariant property for vanilla gradient descent, QNG and H-QNG. The given $2$-qubit Hamiltonian is $H=X_1+Y_1+Z_1+X_2+Y_2+Z_2+X_1X_2+Y_1Y_2+Z_1Z_2$. The experiments are performed using a $2$-parameter VQE circuit composed of Pauli rotation and CNOT gates, evaluated under three optimization methods. The cost function is given by $\mathrm{tr}(\rho(\theta_1,\theta_2)H)$ , where the optimal solution can be analytically obtained as $\theta_1=-\frac{\pi}{2}$ and $\theta_2=\pi$. Optimization is carried out using three different reparameterization functions $t$ in the $\psi$-space where $t_1:(\theta_1,\theta_2)=(0.8\psi_1,1.2\psi_2)$, $t_2:(\theta_1,\theta_2)=(1.2\psi_1,0.8\psi_2)$, $t_3:(\theta_1, \theta_2)=(2\arctan(2\tan(0.5\psi_1)),2\arctan(2\tan(0.5\psi_2)))$. The initial points $\psi^{(0)}$ in $\psi$-space are chosen such that they correspond to the same point in $\theta$-space under each mapping $t$. The trajectories $\{\theta^{(k)}\}$ are obtained by applying each $t$ to the corresponding trajectories in the $\psi$-space. The trajectories of QNG and H-QNG remain invariant in the $\theta$-space (the blue curve and red curve), whereas this invariance does not hold for vanilla gradient descent (the cyan curve).
• Figure 5: The influence of $\lambda$ in regularization for H-QNG. The experiment records the energy distance from the ground state $\Delta E$ of a $4$-qubit $H_2$ Hamiltonian using different values of the regularization hyperparameter $\lambda$ in H-QNG. The learning curves are plotted until a precision of $10^{-10}$ (cyan line). A large value of $\lambda$ leads to slow convergence (purple line), while a small $\lambda$ may result in oscillations (blue line) or early convergence (orange line).

Theorems & Definitions (18)

• Definition 1
• Definition 2
• Proposition 1
• Definition 3
• Proposition 2
• proof
• Theorem 1
• proof
• Definition 4
• Proposition 3