Enhancing Hybrid Methods in Parameterized Quantum Circuit Optimization

TL;DR

This paper tackles the optimization of Parameterized Quantum Circuits (PQCs) for variational quantum algorithms on NISQ devices by introducing two cost-function–based hybrids that switch between less and more expressive single-qubit optimizers. By leveraging ideas from early stopping and moving-average criteria, the methods adaptively balance exploration and expressivity during training, improving convergence across diverse cost functions and noise profiles. Empirical results on Heisenberg and Fermi-Hubbard models show that the proposed hybrids, especially the early-stopping variant, yield faster, more robust convergence than prior gate-wise or iteration-based hybrids, with strong scalability to larger circuit sizes. The findings highlight practical pathways to enhance PQC optimization on near-term quantum hardware and motivate extending the framework to multi-optimizer cascades.

Abstract

Parameterized quantum circuits (PQCs) play an essential role in the application of variational quantum algorithms (VQAs) in noisy intermediate-scale quantum (NISQ) devices. The PQCs are a leading candidate to achieve a quantum advantage in NISQ devices and have already been applied in various domains such as quantum chemistry, quantum machine learning, combinatorial optimization, and many others. There is no single definitive way to optimize PQCs. The most commonly used methods are based on computing the gradient via the parameter-shift rule to use classical gradient descent (GD) optimizers like Adam, stochastic GD, and others. In addition, sequential single-qubit optimizers have been proposed, such as Rotosolve, Free-Axis Selection (Fraxis), Free-Quaternion Selection (FQS), and hybrid algorithms from the aforementioned optimizers. We further develop hybrid algorithms than those represented in the previous work by drawing inspiration from the early stopping method used in classical machine learning. The switch between the optimizers depends on the previous cost function values compared to the previous ones. We introduce two new hybrid algorithms that are more robust and scalable, and they outperform previous hybrid methods in terms of convergence towards the global minima across various cost functions. In addition, we find that they are feasible for NISQ devices with different noise profiles.

Figures (16)

• Figure 1: Ansatz circuit design for PQC optimization.
• Figure 2: Differences between average cost function $\expval{M}_{avg}$ values across gate optimizations compared to the new cost function value $\expval{M}$ on a logarithmic scale. Results are from a 10-qubit Heisenberg model with 10 layers, and optimizer was used. Each run is plotted in different shades of blue, the red line is the mean, and the green line represents the median. The averages $\expval{M}_{avg}$ are computed from the latest $w$ gate optimizations. On the left, $w=10$ was used, $w=100$ on the middle plot, and $w=1000$ on the right plot.
• Figure 3: Individual runs for , and cost average hybrid with window length $w=10$ and threshold $E_t=0.05$. All optimizers are initialized with the same parameters, and a 5-qubit Heisenberg model with 10 layers was used.
• Figure 4: Results for one-dimensional 10-qubit Heisenberg model with different optimizers (yellow), (orange), (dark red), and with thresholds $E_t =$ 0.1 (dark blue), 0.01 (blue), and 0.1 (cyan). The patience of the hybrid algorithm was set to $P=10$. Each line represents the mean across the 20 runs.
• Figure 5: Results for one-dimensional 10-qubit Heisenberg model with 15 layers. Each line represents a different optimizer: (yellow), (orange), (dark red), and with window lengths $w =$ 10 (dark blue), 100 (blue), and 1000 (cyan). The switching threshold for the optimizers was set to $E_t=0.1$ (left), 0.01 (middle), and 0.001 (right). Each line represents the mean across the 20 runs.