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arXiv:2504.11120

Improved approximation ratios for the Quantum Max-Cut problem on general, triangle-free and bipartite graphs

Sander Gribling, Lennart Sinjorgo, Renata Sotirov

Published Apr 15, 2025• 0 citations
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Abstract

We study polynomial-time approximation algorithms for the Quantum Max-Cut (QMC) problem. Given an edge-weighted graph $G$ on n vertices, the QMC problem is to determine the largest eigenvalue of a particular $2^n \times 2^n$ matrix that corresponds to $G$. We provide a sharpened analysis of the currently best-known QMC approximation algorithm for general graphs. This algorithm achieves an approximation ratio of $0.599$, which our analysis improves to $0.603$. Additionally, we propose two new approximation algorithms for the QMC problem on triangle-free and bipartite graphs, that achieve approximation ratios of $0.61383$ and $0.8162$, respectively. These are the best-known approximation ratios for their respective graph classes.

DOI
10.48550/arxiv.2504.11120

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We study polynomial-time approximation algorithms for the Quantum Max-Cut (QMC) problem. Given an edge-weighted graph GGG on n vertices, the QMC problem is to determine the largest eigenvalue of a particular 2n×2n2^n \times 2^n2n×2n matrix that corresponds to GGG. We provide a sharpened analysis of the currently best-known QMC approximation algorithm for general graphs. This algorithm achieves an approximation ratio of 0.5990.5990.599, which our analysis improves to 0.6030.6030.603. Additionally, we propose two new approximation algorithms for the QMC problem on triangle-free and bipartite graphs, that achieve approximation ratios of 0.613830.613830.61383 and 0.81620.81620.8162, respectively. These are the best-known approximation ratios for their respective graph classes.
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The Quantum Max-ddd-Cut (ddd-QMC) problem is a special instance of a 222-local Hamiltonian problem, representing the quantum analog of the classical Max-ddd-Cut problem. The ddd-QMC problem seeks the largest eigenvalue of a Hamiltonian defined on a graph with nnn vertices, where edges correspond to swap operators acting on (Cd)⊗n(\mathbb{C}^d)^{\otimes n}(Cd)⊗n. In recent years, progress has been made by investigating the algebraic structure of the ddd-QMC Hamiltonian. Building on this approach, this article solves the ddd-QMC problem for complete tripartite graphs for small local dimensions, d≤3d \le 3d≤3.
Finding a high (or low) energy state of a given quantum Hamiltonian is a potential area to gain a provable and practical quantum advantage. A line of recent studies focuses on Quantum Max Cut, where one is asked to find a high energy state of a given antiferromagnetic Heisenberg Hamiltonian. In this work, we present a classical approximation algorithm for Quantum Max Cut that achieves an approximation ratio of 0.595, outperforming the previous best algorithms of Lee [Eunou Lee, 2022] (0.562, generic input graph) and King [King, 2023] (0.582, triangle-free input graph). The algorithm is based on finding the maximum weighted matching of an input graph and outputs a product of at most 2-qubit states, which is simpler than the fully entangled output states of the previous best algorithms.
Finding a high (or low) energy state of a given quantum Hamiltonian is a potential area to gain a provable and practical quantum advantage. A line of recent studies focuses on Quantum Max Cut, where one is asked to find a high energy state of a given antiferromagnetic Heisenberg Hamiltonian. In this work, we present a classical approximation algorithm for Quantum Max Cut that achieves an approximation ratio of 0.595, outperforming the previous best algorithms of Lee [Lee, 2022] (0.562, generic input graph) and King [King, 2023] (0.582, triangle-free input graph). The algorithm is based on finding the maximum weighted matching of an input graph and outputs a product of at most 2-qubit states, which is simpler than the fully entangled output states of the previous best algorithms.
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Triangles capture higher-order structures in graphs and are fundamental to applications such as clustering and network analysis. To enable efficient use of such structures at scale, we study the problem of triangle cut sparsification, which aims to reduce the graph size while approximately preserving triangle counts across every cut. We investigate quantum algorithms for this problem, using triangle listing as our main technical ingredient. In particular, we present a quantum algorithm for triangle listing that, for a graph with nnn vertices, mmm edges, and ttt triangles, runs in time Tq-list=T_{\mathrm{q\text{-}list}} =Tq-list​= O~(min⁡(n5/4t7/12+n7/6t7/9,m+m3/4t1/2,\widetilde{O}\bigl(\min(n^{5/4}t^{7/12} + n^{7/6}t^{7/9}, m + m^{3/4}t^{1/2},O(min(n5/4t7/12+n7/6t7/9,m+m3/4t1/2, n3/2t1/2))n^{3/2}t^{1/2})\bigr)n3/2t1/2)), improving upon the best known classical bounds over a broad range of parameters. Our algorithm is based on a heavy-light vertex partition and an extension of triangle detection via quantum walks and Grover search. Leveraging this result, we design a quantum algorithm for constructing ε\varepsilonε-triangle cut sparsifiers of size O~(n/ε2)\widetilde{O}(n/\varepsilon^2)O(n/ε2) in time O~(Tq-list+mn/ε)\widetilde{O}(T_{\mathrm{q\text{-}list}} + \sqrt{mn}/\varepsilon)O(Tq-list​+mn​/ε). Finally, we demonstrate applications to clustering algorithms based on triangle-related measures and prove a lower bound of Ω(n/ε2)Ω(n/\varepsilon^2)Ω(n/ε2) on the size of any ε\varepsilonε-triangle cut sparsifiers.