research-article

Complexity Classification of Local Hamiltonian Problems

Authors:

Ashley MontanaroAuthors Info & Claims

SIAM Journal on Computing, Volume 45, Issue 2

Pages 268 - 316

https://doi.org/10.1137/140998287

Published: 01 January 2016 Publication History

Abstract

The calculation of ground-state energies of physical systems can be formalized as the $k$-local Hamiltonian problem, which is a natural quantum analogue of classical constraint satisfaction problems. One way of making the problem more physically meaningful is to restrict the Hamiltonian in question by picking its terms from a fixed set $\mathcal{S}$ and scaling them by arbitrary weights. Examples of such special cases are the Heisenberg and Ising models from condensed-matter physics. In this work we characterize the complexity of this problem for all 2-local qubit Hamiltonians. Depending on the subset $\mathcal{S}$, the problem falls into one of the following categories: in $\mathsf{ P}$; $\mathsf{NP}$-complete; polynomial-time equivalent to the Ising model with transverse magnetic fields; or $\mathsf{QMA}$-complete. The third of these classes has been shown to be $\mathsf{StoqMA}$-complete by Bravyi and Hastings. The characterization holds even if $\mathcal{S}$ does not contain any 1-local terms; for example, we prove for the first time $\mathsf{QMA}$-completeness of the Heisenberg and XY interactions in this setting. If $\mathcal{S}$ is assumed to contain all 1-local terms, which is the setting considered by previous work, we have a characterization that goes beyond 2-local interactions: for any constant $k$, all $k$-local qubit Hamiltonians whose terms are picked from a fixed set $\mathcal{S}$ correspond to problems either in $\mathsf{P}$; polynomial-time equivalent to the Ising model with transverse magnetic fields; or $\mathsf{QMA}$-complete. These results are a quantum analogue of the maximization variant of Schaefer's dichotomy theorem for Boolean constraint satisfaction problems.

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Cited By

Gharibian S(2024)Guest Column: The 7 faces of quantum NPACM SIGACT News10.1145/3639528.363953554:4(54-91)Online publication date: 3-Jan-2024
https://dl.acm.org/doi/10.1145/3639528.3639535
Aharonov DIrani SLeonardi SGupta A(2022)Hamiltonian complexity in the thermodynamic limitProceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3519935.3520067(750-763)Online publication date: 9-Jun-2022
https://dl.acm.org/doi/10.1145/3519935.3520067

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cover image SIAM Journal on Computing

SIAM Journal on Computing Volume 45, Issue 2

DOI:10.1137/smjcat.45.2

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© 2016, Society for Industrial and Applied Mathematics.

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Society for Industrial and Applied Mathematics

United States

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Published: 01 January 2016

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Cited By

Gharibian S(2024)Guest Column: The 7 faces of quantum NPACM SIGACT News10.1145/3639528.363953554:4(54-91)Online publication date: 3-Jan-2024
https://dl.acm.org/doi/10.1145/3639528.3639535
Aharonov DIrani SLeonardi SGupta A(2022)Hamiltonian complexity in the thermodynamic limitProceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3519935.3520067(750-763)Online publication date: 9-Jun-2022
https://dl.acm.org/doi/10.1145/3519935.3520067

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