A Lyapunov Framework for Quantum Algorithm Design in Combinatorial Optimization with Approximation Ratio Guarantees

• URL: http://arxiv.org/abs/2512.21716v1
• Date: Thu, 25 Dec 2025 15:38:24 GMT
• Title: A Lyapunov Framework for Quantum Algorithm Design in Combinatorial Optimization with Approximation Ratio Guarantees
• Authors: Shengminjie Chen, Ziyang Li, Hongyi Zhou, Jialin Zhang, Wenguo Yang, Xiaoming Sun,
• Abstract summary: We develop a framework aiming at designing quantum algorithms for optimization problems.<n>We provide theoretical guarantees on their approximation ratios.<n>We apply our framework to Max-Cut problem, implementing it as an adaptive variational quantum algorithm.
• Score: 15.259020859762556
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this work, we develop a framework aiming at designing quantum algorithms for combinatorial optimization problems while providing theoretical guarantees on their approximation ratios. The principal innovative aspect of our work is the construction of a time-dependent Lyapunov function that naturally induces a controlled Schrödinger evolution with a time dependent Hamiltonian for maximizing approximation ratios of algorithms. Because the approximation ratio depends on the optimal solution, which is typically elusive and difficult to ascertain a priori, the second novel component is to construct the upper bound of the optimal solution through the current quantum state. By enforcing the non-decreasing property of this Lyapunov function, we not only derive a class of quantum dynamics that can be simulated by quantum devices but also obtain rigorous bounds on the achievable approximation ratio. As a concrete demonstration, we apply our framework to Max-Cut problem, implementing it as an adaptive variational quantum algorithm based on a Hamiltonian ansatz. This algorithm avoids ansatz and graph structural assumptions and bypasses parameter training through a tunable parameter function integrated with measurement feedback.

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