The classical limit of Quantum Max-Cut
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917963847696384 |
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| author | Bulchandani, Vir B. Piddock, Stephen |
| author_facet | Bulchandani, Vir B. Piddock, Stephen |
| contents | It is well-known in physics that the limit of large quantum spin $S$ should be understood as a semiclassical limit. This raises the question of whether such emergent classicality facilitates the approximation of computationally hard quantum optimization problems, such as the local Hamiltonian problem. We demonstrate this explicitly for spin-$S$ generalizations of Quantum Max-Cut ($\mathrm{QMaxCut}_S$), equivalent to the problem of finding the ground state energy of an arbitrary spin-$S$ quantum Heisenberg antiferromagnet ($\mathrm{QHA}_S$). We prove that approximating the value of $\mathrm{QHA}_S$ to inverse polynomial accuracy is QMA-complete for all $S$, extending previous results for $S=1/2$. We also present two distinct families of classical approximation algorithms for $\mathrm{QMaxCut}_S$ based on rounding the output of a semidefinite program to a product of Bloch coherent states. The approximation ratios for both our proposed algorithms strictly increase with $S$ and converge to the Briët-Oliveira-Vallentin approximation ratio $α_{\mathrm{BOV}} \approx 0.956$ from below as $S \to \infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_12968 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The classical limit of Quantum Max-Cut Bulchandani, Vir B. Piddock, Stephen Quantum Physics Disordered Systems and Neural Networks Statistical Mechanics Strongly Correlated Electrons It is well-known in physics that the limit of large quantum spin $S$ should be understood as a semiclassical limit. This raises the question of whether such emergent classicality facilitates the approximation of computationally hard quantum optimization problems, such as the local Hamiltonian problem. We demonstrate this explicitly for spin-$S$ generalizations of Quantum Max-Cut ($\mathrm{QMaxCut}_S$), equivalent to the problem of finding the ground state energy of an arbitrary spin-$S$ quantum Heisenberg antiferromagnet ($\mathrm{QHA}_S$). We prove that approximating the value of $\mathrm{QHA}_S$ to inverse polynomial accuracy is QMA-complete for all $S$, extending previous results for $S=1/2$. We also present two distinct families of classical approximation algorithms for $\mathrm{QMaxCut}_S$ based on rounding the output of a semidefinite program to a product of Bloch coherent states. The approximation ratios for both our proposed algorithms strictly increase with $S$ and converge to the Briët-Oliveira-Vallentin approximation ratio $α_{\mathrm{BOV}} \approx 0.956$ from below as $S \to \infty$. |
| title | The classical limit of Quantum Max-Cut |
| topic | Quantum Physics Disordered Systems and Neural Networks Statistical Mechanics Strongly Correlated Electrons |
| url | https://arxiv.org/abs/2401.12968 |