An improved Quantum Max Cut approximation via matching
Eunou Lee, Ojas Parekh
TL;DR
The paper addresses approximating the maximum energy state of a quantum Hamiltonian, focusing on Quantum Max Cut (QMC) on graphs with $H=\sum_{(i,j)\in E} w_{ij}H_{ij}$ and $H_{ij}=(I-X_i X_j - Y_i Y_j - Z_i Z_j)/4$. It introduces a classical 0.595-approximation that solves the level-2 Quantum Lasserre SDP and rounds via two routes: (i) a product-state rounding in the style of GP19, and (ii) a matching-based construction that places the $2$-qubit singlet $|\psi^-\rangle=(|01\rangle-|10\rangle)/\sqrt{2}$ on a maximum weight matching and uses maximally mixed states on unmatched qubits. The result improves upon prior guarantees for general graphs (0.562) and triangle-free graphs (0.582) while yielding a simpler, largely product-state output; the analysis hinges on monogamy constraints and matching relaxations derived from SDP hierarchies. Overall, the work demonstrates a provably strong classical approximation for QMC and sheds light on how SDP-guided rounding and matching-based strategies can outperform entangled-state approaches in this setting.
Abstract
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.