Application of Langevin Dynamics to Advance the Quantum Natural Gradient Optimization Algorithm

TL;DR

The paper addresses the challenge of optimizing variational quantum circuits where the Quantum Natural Gradient (QNG) can stall in nonconvex landscapes. By formulating a discrete-time Langevin dynamics with a QNG stochastic force, the authors derive a Momentum-QNG update that injects momentum and noise into the optimization, analogous to SGD with momentum. Through benchmarks on Variational Quantum Eigensolvers, Quantum Approximate Optimization Algorithm, and the Sherrington-Kirkpatrick model, Momentum-QNG demonstrates improved exploration and often superior performance over basic QNG, with Adam providing robust convergence in many cases; notably, Momentum-QNG excels in small-field SK regimes. The work highlights a practical Langevin-inspired enhancement to QNG for variational quantum circuits and provides open-source code to enable reproducibility and broader adoption.

Abstract

A Quantum Natural Gradient (QNG) algorithm for optimization of variational quantum circuits has been proposed recently. In this study, we employ the Langevin equation with a QNG stochastic force to demonstrate that its discrete-time solution gives a generalized form of the above-specified algorithm, which we call Momentum-QNG. Similar to other optimization algorithms with the momentum term, such as the Stochastic Gradient Descent with momentum, RMSProp with momentum and Adam, Momentum-QNG is more effective to escape local minima and plateaus in the variational parameter space and, therefore, demonstrates an improved performance compared to the basic QNG. In this paper we benchmark Momentum-QNG together with the basic QNG, Adam and Momentum optimizers and explore its convergence behaviour. Among the benchmarking problems studied, the best result is obtained for the quantum Sherrington-Kirkpatrick model in the strong spin glass regime. Our open-source code is available at https://github.com/borbysh/Momentum-QNG

Figures (3)

• Figure 1: Benchmarking Momentum-QNG together with QNG, Momentum and Adam on the portfolio optimization problem. The vertical axis shows the mean (symbols) and the standard deviation (shaded regions) of the difference between the optimized and the ground state energy (a) ($N=6$), (c) ($N=11$), (e) ($N=12$) and of the number of steps to convergence (b) ($N=6$), (d) ($N=11$), (f) ($N=12$) in a series of 200 trials, while the horizontal axis shows the learning rate $\eta$ of four different optimizers under consideration.
• Figure 2: Benchmarking Momentum-QNG together with QNG, Momentum and Adam on the Sherrington-Kirkpatrick model at three different values of transverse field $g$ (indicated at figure captions). The vertical axis shows the mean (symbols) and the standard deviation (shaded regions) of the difference (in percents) between the optimized and the true ground state energy (a) ($g=0.1$), (c) ($g=10^{-3}$), (e) ($g=10^{-5}$) and of the number of steps to convergence (b) ($g=0.1$), (d) ($g=10^{-3}$), (f) ($g=10^{-5}$) in a series of 200 trials, while the horizontal axis shows the learning rate $\eta$ of four different optimizers under consideration.
• Figure 3: Benchmarking Momentum-QNG together with QNG, Momentum and Adam on the Minimum Vertex Cover problem. The vertical axis shows the mean (symbols) and the standard deviation (shaded regions) of the quality ratio (a) ($N=4$), (c) ($N=8$) and of the number of steps to convergence (b) ($N=4$), (d) ($N=8$) in a series of 200 trials, while the horizontal axis shows the learning rate $\eta$ of four different optimizers under consideration.