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arXiv · 2510.07995

Quantum Max-Cut is NP hard to approximate

Stephen Piddock
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Abstract

We unconditionally prove that it is NP-hard to compute a constant multiplicative approximation to the QUANTUM MAX-CUT problem on an unweighted graph of constant bounded degree. The proof works in two stages: first we demonstrate a generic reduction to computing the optimal value of a quantum problem, from the optimal value over product states. Then we prove an approximation preserving reduction from MAX-CUT to PRODUCT-QMC the product state version of QUANTUM MAX-CUT. More precisely, in the second part, we construct a PTAS reduction from MAX-CUT$_k$ (the rank-k constrained version of MAX-CUT) to MAX-CUT$_{k+1}$, where MAX-CUT and PRODUCT-QMC coincide with MAX-CUT$_1$ and MAX-CUT$_3$ respectively. We thus prove that Max-Cut$_k$ is APX-complete for all constant $k$.

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Stephen Piddock. 2025-10-09. Quantum Max-Cut is NP hard to approximate. https://arxiv.org/abs/2510.07995

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