Improving Variational Quantum Circuit Optimization via Hybrid Algorithms and Random Axis Initialization (2503.20728v1)

Abstract: Variational quantum circuits (VQCs) are an essential tool in applying noisy intermediate-scale quantum computers to practical problems. VQCs are used as a central component in many algorithms, for example, in quantum machine learning, optimization, and quantum chemistry. Several methods have been developed to optimize VQCs. In this work, we enhance the performance of the well-known Rotosolve method, a gradient-free optimization algorithm specifically designed for VQCs. We develop two hybrid algorithms that combine an improved version of Rotosolve with the free quaternion selection (FQS) algorithm, which is the main focus of this study. Through numerical simulations, we observe that these hybrid algorithms achieve higher accuracy and better average performance across different ansatz circuit sizes and cost functions. For shallow variational circuits, we identify a trade-off between the expressivity of the variational ansatz and the speed of convergence to the optimum: a more expressive ansatz ultimately reaches a closer approximation to the true minimum, but at the cost of requiring more circuit evaluations for convergence. By combining the less expressive but fast-converging Rotosolve with the more expressive FQS, we construct hybrid algorithms that benefit from the rapid initial convergence of Rotosolve while leveraging the superior expressivity of FQS. As a result, these hybrid approaches outperform either method used independently.

Analysis of Optimization Techniques for Variational Quantum Circuits: Hybrid Algorithms and Random Initialization

The paper, "Improving Variational Quantum Circuit Optimization via Hybrid Algorithms and Random Axis Initialization," provides an in-depth exploration of optimization strategies tailored for variational quantum circuits (VQCs). This paper is particularly relevant for quantum computational paradigms involving noisy intermediate-scale quantum (NISQ) devices, where the optimization of circuit parameters plays a crucial role. The authors propose enhancements in the form of hybrid algorithmic strategies and a novel initialization technique known as random axis initialization.

VQCs are essential in many NISQ applications, such as quantum chemistry, optimization, and machine learning. Optimizing these circuits presents unique challenges, particularly due to the so-called "barren plateau" problem, where parameter gradients vanish with increasing circuit depth, hindering optimization.

The authors enhance the well-known Rotosolve algorithm—a gradient-free optimization strategy—by integrating it with the Free Quaternion Selection (FQS) method. They specifically develop two hybrid algorithms that synthesize the quick convergence typical of Rotosolve with the robust expressivity characteristic of FQS. This combination is intended to create optimization strategies that leverage the rapid initial progress of Rotosolve while ultimately reaching closer approximations to problem solutions, benefiting from the broader search coverage afforded by FQS.

Methodology and Hybrid Algorithm Proposals

The paper introduces a method where the generator of single-qubit gates is randomly sampled from the Haar distribution to form a modified Rotosolve variant named Rotosolve-Haar. This random initialization aims to enhance the initial gate settings across quantum circuit evaluations, potentially mitigating the barren plateau problem.

Two hybrid algorithms are synthesized with this enhancement:

1. Cycle-Specific Hybrid Algorithm: This method alternates between Rotosolve-Haar and FQS based on a specified cycle period, balancing speed and expressivity over repeated cycles.
2. Gate-Specific Hybrid Algorithm: Within this framework, gate optimizations probabilistically alternate between Rotosolve-Haar and FQS, effectively embedding the flexibility of switching based on individual gate initialization effectiveness.

Empirical Results

The performance of these hybrid algorithms was evaluated through simulations using several test cases, including the Heisenberg models in one-dimensional and two-dimensional configurations and molecular Hamiltonians of HeH$^+$ and H$_2$. Simulations demonstrate that the hybrid approaches frequently outperform individual methods in both speed and accuracy across various VQC architectures. Specific findings include:

• One-Dimensional Heisenberg Model: Hybrid algorithms demonstrated superior accuracy and convergence speed compared to standalone Rotosolve, FQS, and Fraxis (another gradient-free optimizer).
• Two-Dimensional Heisenberg Model: Algorithms engaging expressivity-kept superior performance, with hybrids often topping in convergence.
• Molecular Hamiltonians: While known techniques like Fraxis showed fast performance for certain molecular systems, hybrids surpassed others in terms of reaching lower energy states in deeper circuit configurations.

Speculation on Future Developments

The results suggest that hybrid methodologies combining distinct optimization philosophies may be particularly effective for complex quantum circuits, which necessitate wide expressivity and rapid convergence. Future research could expand these hybrid approaches to more complex, multi-qubit gate architectures and explore other forms of random initialization to further mitigate expressivity-related anomalies such as barren plateaus.

By enhancing the fidelity and efficacy of VQC optimizations in practice, these innovations contribute to advancing real-world applicability and the fundamental capability of quantum algorithms. Such progress is crucial, eventually enabling NISQ devices to expand beyond current computational capabilities.