Beyond Single Trajectories: Optimal Control and Jordan-Lie Algebra in Hybrid Quantum Walks for Combinatorial Optimization

TL;DR

This paper introduces a hybrid quantum walk framework that superposes multiple trajectories for combinatorial optimization, outperforming traditional QAOA in speed and accuracy by leveraging optimal control and algebraic structures.

Contribution

It develops a novel HQW ansatz with a dynamical coin operator, extending QAOA, and provides algebraic and control-theoretic insights into its enhanced performance.

Findings

HQW outperforms QAOA in convergence speed and solution accuracy.

Optimal coin operators differ from static gates, enhancing expressivity.

Numerical results on Max-Cut and Independent Set problems demonstrate systematic improvements.

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) follows a single, fixed evolution path, overlooking the potential computational advantage of coherently superposing multiple trajectories. Here we overcome this limitation with a hybrid quantum walk (HQW) ansatz that super poses multiple Hamiltonian-driven paths coherently within each circuit layer via a dynamical coin operator. QAOA emerges as a special case of this framework with a static Pauli-X coin. Using Pontryagin's minimum principle, we derive the optimal form of the coin operator, demonstrating that it generally differs from a constant gate. A dynamical Lie algebra analysis reveals that HQW generates a strictly larger Jordan-Lie algebra, providing an algebraic foundation for its enhanced expressivity. Especially, we reveal the connection between the unique Jordan product negativity in HQW's DLA and its performance advantages. Numerical experiments on Max-Cut and Maximum Independent Set problems show that HQW systematically outperforms QAOA in convergence speed, solution accuracy, and robustness. Our work establishes a path-superposition paradigm for quantum optimization, combining optimal control theory with algebraic structure to guide the design of advanced quantum algorithms.