Quantum Max-Cut is NP hard to approximate
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911199894962176 |
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| author | Piddock, Stephen |
| author_facet | Piddock, Stephen |
| contents | 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 $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_07995 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantum Max-Cut is NP hard to approximate Piddock, Stephen Quantum Physics Computational Complexity 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 $k$. |
| title | Quantum Max-Cut is NP hard to approximate |
| topic | Quantum Physics Computational Complexity |
| url | https://arxiv.org/abs/2510.07995 |