Modified Conjugate Quantum Natural Gradient

TL;DR

The paper tackles the challenge of efficiently optimizing variational quantum algorithms by introducing Modified Conjugate Quantum Natural Gradient (CQNG), an optimizer that blends the Quantum Natural Gradient with nonlinear conjugate-gradient ideas. CQNG adaptively selects both the step size $\alpha_t$ and the conjugate coefficient $\beta_t$ at each iteration, improving convergence over standard QNG while maintaining the same $O(m^2)$ metric-cost with a negligible $O(1)$ overhead. Through three experimental settings—two-qubit H2, the Heisenberg model, and molecular Hamiltonians (H4, H2O, C2)—CQNG demonstrates faster convergence and reduced quantum-resource usage, including earlier escape from cost plateaus. The results suggest CQNG as a practical, geometry-aware optimizer for VQAs with potential extensions to noisy circuits, stochastic information approaches, and time-dependent quantum optimization.

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 [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.

Figures (15)

• Figure 1: Energy convergence and fidelity to the ground state as a function of training iterations for GD, QNG, and CQNG optimizers. The initial parameters are set to $[-0.2,-0.2,0]$, and a learning rate $\eta = 0.05$ is used for GD and QNG, and CQNG. The initial values $\beta_0 = 0.1$ and $\alpha_0 = 0.05$ are used for the COBYLA optimizer, which dynamically optimizes $\alpha_t$ and $\beta_t$ at each step according to Eq. (\ref{['eq:update_beta']}).
• Figure 2: Same experimental conditions as in Fig. \ref{['fig:ex1_energy']}, but with the initial point set to $[\pi/2,\pi/2,0]$. This setup is more challenging because the starting state lies adjacent to a nearly flat region of the cost landscape, causing optimizers to become trapped in local minima. As shown in the results, CQNG escapes these regions more quickly than the other methods.
• Figure 3: The average of 100 runs from different initial starting points. The hyperparameters, including the learning rate for GD and QNG, as well as the initial step size $\alpha_0$ used in COBYLA for CQNG, are tuned using a grid search.
• Figure 4: EfficientSU2 ansatz with two layers.
• Figure 5: Average cost function value as a function of training iterations for GD, QNG, and CQNG, evaluated on 12, 14, 16, and 20 qubits with a fixed circuit depth of 5 layers—corresponding to 120, 140, 160, and 200 trainable parameters, respectively. Each data point represents an average over 30 runs with different initial parameters. The number of measurement shots is set to 10024, and a learning rate of 0.01 is used. Initial values $\beta_0 = 0.1$ and $\alpha_0 = 0.01$ are used for the COBYLA optimizer, which dynamically updates $\alpha_t$ and $\beta_t$ at each step according to Eq. (\ref{['eq:update_beta']}).