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

TL;DR

The paper introduces a quantum Lyapunov control framework to design quantum approximation algorithms for combinatorial optimization with provable guarantees. By constructing a time-dependent Lyapunov function $E(t)$ that incorporates the problem Hamiltonian $\mathbf{H}_f$ and an upper bound on the optimal value, it derives a time-dependent Hamiltonian that ensures monotone improvement and yields lower bounds on the achievable approximation ratio. The framework has one- and two-parameter variants, decomposes the evolution into commuting and non-commuting parts, and uses measurement-feedback to adaptively set parameters, demonstrated on Max-Cut with adaptive variational circuits. Numerical experiments on diverse graph classes show the true ratio converges toward 1 with more iterations, while the Lyapunov-based bounds remain reliable and can be tighter with a two-parameter design. This approach broadens theoretical guarantees for quantum optimization beyond standard QAOA analyses and provides a practical pathway for adaptive, feedback-driven quantum approximation algorithms with provable performance bounds.

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.

Figures (5)

• Figure 1: Discrete Implementation of the Algorithm Derived from the Lyapunov Framework
• Figure 2: The algorithm defined in Eq. (\ref{['eq::using_FALQON_Hamilton']}) was evaluated on nine experimental settings, corresponding to 3-regular graphs, Erd≈ës-R√©nyi random graphs, and bipartite graphs with node sizes $n={12,16,20}$ (from left to right). Each experiment contains four bar plots, where the color of each bar encodes the value of the corresponding parameter across different iteration rounds. Within each experiment, the four bars from left to right represent, respectively, $\langle \psi(t_j)| \mathbf{H}_f | \psi(t_j) \rangle / \langle \psi^*| \mathbf{H}_f | \psi^* \rangle,\quad \lambda(t_j) - \lambda(0),\quad (y(t_j) - y(0)) / x(t_j),\quad \langle \psi(t_j)| \mathbf{H}_f | \psi(t_j) \rangle / m$.
• Figure 3: The right panel illustrates one possible edge-coloring scheme of the Petersen graph based on Vizing's theorem, while the left panel depicts a single-layer iterative circuit of the algorithm defined in Eq. (\ref{['eq::using_FALQON_Hamilton']}). Edges sharing the same color can be processed in parallel, resulting in a circuit of constant depth.
• Figure 4: The scatter points represent the number of iterations required for the algorithm to achieve an approximation ratio on instances. The blue line denotes the linear fit of these data points. The purple line corresponds to the linear fit of the largest iteration counts observed for each graph size $n$. The light-blue shaded region shows the interquartile spread of the iteration counts (25th‚Äì75th percentiles) for each graph size $n$. For comparison, the yellow and green lines represent the functions $R(n) = n^3$ and $R(n) = n^2$, respectively.
• Figure 5: The results of running the algorithm defined in Eq. (\ref{['light-cone_Hamilton']}) on 3-regular graphs with $n = \{12, 16, 20\}$ are presented. For different $n$, the four bars from left to right represent, respectively, $\langle \psi(t_j)| \mathbf{H}_f | \psi(t_j) \rangle / \langle \psi^*| \mathbf{H}_f | \psi^* \rangle,\quad \lambda(t_j) - \lambda(0),\quad (y(t_j) - y(0)) / x(t_j),\quad \langle \psi(t_j)| \mathbf{H}_f | \psi(t_j) \rangle / m$.

Theorems & Definitions (18)

• Theorem 1
• Theorem 2
• Lemma 1
• Definition 1: Potential Function
• Lemma 2
• proof
• Lemma 3: A Rephrased Version of Lemma 1 in Berry2020, Appendix B in Tran2019
• proof
• Lemma 4
• proof