NDLI: Complexity Classification of Local Hamiltonian Problems

Content Provider	IEEE Xplore Digital Library
Author	Cubitt, T. Montanaro, A.
Copyright Year	2014
Description	Author affiliation: Dept. of Comput. Sci., Univ. of Bristol, Bristol, UK (Montanaro, A.) \|\| Dept. of Appl. Math. & Theor. Phys., Univ. of Cambridge, Cambridge, UK (Cubitt, T.)
Abstract	The calculation of ground-state energies of physical systems can be formalised as the k-local Hamiltonian problem, which is the 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 S. Examples of such special cases are the Heisenberg and Ising models from condensed-matter physics. In this work we characterise the complexity of this problem for all 2-local qubit Hamiltonians. Depending on the subset S, the problem falls into one of the following categories: in P, NP-complete, polynomial-time equivalent to the Ising model with transverse magnetic fields, or QMA-complete. The third of these classes contains NP and is contained within StoqMA. The characterisation holds even if S does not contain any 1-local terms, for example, we prove for the first time QMA-completeness of the Heisenberg and XY interactions in this setting. If S is assumed to contain all 1-local terms, which is the setting considered by previous work, we have a characterisation that goes beyond 2-local interactions: for any constant k, all k-local qubit Hamiltonians whose terms are picked from a fixed set S correspond to problems either in P, polynomial-time equivalent to the Ising model with transverse magnetic fields, or QMA-complete. These results are a quantum analogue of Schaefer's dichotomy theorem for boolean constraint satisfaction problems.
Sponsorship	IEEE Comput. Soc. Tech. Comm. Math. Found. Comput.
Starting Page	120
Ending Page	129
File Size	338965
Page Count	10
File Format	PDF
ISBN	9781479965175
ISSN	02725428
DOI	10.1109/FOCS.2014.21
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2014-10-18
Publisher Place	USA
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Complexity theory Physics Eigenvalues and eigenfunctions Lattices Matrix decomposition Computer science Quantum computing QMA-completeness Hamiltonian complexity
Content Type	Text
Resource Type	Article

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