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

TL;DR

This paper introduces a Lyapunov-based framework for designing quantum algorithms with guaranteed approximation ratios for combinatorial optimization, demonstrated on the Max-Cut problem using an adaptive variational approach.

Contribution

It develops a novel Lyapunov function-based method for quantum algorithm design that provides theoretical approximation guarantees and practical implementation strategies.

Findings

Framework achieves controlled quantum evolution with approximation guarantees.

Applied to Max-Cut, the algorithm avoids ansatz and graph assumptions.

Provides rigorous bounds on approximation ratios for quantum algorithms.

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 Schr\"odinger 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.