QuantumMerlinArthurcomplete translationally invariant Hamiltonian problem and the complexity of finding groundstate energies in physical systems
Abstract
Here we present a problem related to the local Hamiltonian problem (identifying whether the groundstate energy falls within one of two ranges) which is restricted to being translationally invariant. We prove that for Hamiltonians with a fixed local dimension and O(log(N))body local terms, or local dimension N and twobody terms, there are instances where finding the groundstate energy is quantumMerlinArthurcomplete and simulating the dynamics is BQPcomplete (BQP denotes ''bounded error, quantum polynomial time''). We discuss the implications for the computational complexity of finding ground states of these systems and hence for any classical approximation techniques that one could apply including densitymatrix renormalization group, matrix product states, and multiscale entanglement renormalization ansatz. One important example is a onedimensional lattice of bosons with nearestneighbor hopping at constant filling fractioni.e., a generalization of the BoseHubbard model.
 Authors:

 Centre for Quantum Computation, DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA (United Kingdom)
 Publication Date:
 OSTI Identifier:
 21015897
 Resource Type:
 Journal Article
 Journal Name:
 Physical Review. A
 Additional Journal Information:
 Journal Volume: 76; Journal Issue: 3; Other Information: DOI: 10.1103/PhysRevA.76.030307; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 10502947
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; APPROXIMATIONS; BOSONS; DENSITY MATRIX; GROUND STATES; HAMILTONIANS; HUBBARD MODEL; LATTICE FIELD THEORY; ONEDIMENSIONAL CALCULATIONS; POLYNOMIALS; QUANTUM COMPUTERS; QUANTUM ENTANGLEMENT; QUANTUM MECHANICS; TWOBODY PROBLEM
Citation Formats
Kay, Alastair. QuantumMerlinArthurcomplete translationally invariant Hamiltonian problem and the complexity of finding groundstate energies in physical systems. United States: N. p., 2007.
Web. doi:10.1103/PHYSREVA.76.030307.
Kay, Alastair. QuantumMerlinArthurcomplete translationally invariant Hamiltonian problem and the complexity of finding groundstate energies in physical systems. United States. doi:10.1103/PHYSREVA.76.030307.
Kay, Alastair. Sat .
"QuantumMerlinArthurcomplete translationally invariant Hamiltonian problem and the complexity of finding groundstate energies in physical systems". United States. doi:10.1103/PHYSREVA.76.030307.
@article{osti_21015897,
title = {QuantumMerlinArthurcomplete translationally invariant Hamiltonian problem and the complexity of finding groundstate energies in physical systems},
author = {Kay, Alastair},
abstractNote = {Here we present a problem related to the local Hamiltonian problem (identifying whether the groundstate energy falls within one of two ranges) which is restricted to being translationally invariant. We prove that for Hamiltonians with a fixed local dimension and O(log(N))body local terms, or local dimension N and twobody terms, there are instances where finding the groundstate energy is quantumMerlinArthurcomplete and simulating the dynamics is BQPcomplete (BQP denotes ''bounded error, quantum polynomial time''). We discuss the implications for the computational complexity of finding ground states of these systems and hence for any classical approximation techniques that one could apply including densitymatrix renormalization group, matrix product states, and multiscale entanglement renormalization ansatz. One important example is a onedimensional lattice of bosons with nearestneighbor hopping at constant filling fractioni.e., a generalization of the BoseHubbard model.},
doi = {10.1103/PHYSREVA.76.030307},
journal = {Physical Review. A},
issn = {10502947},
number = 3,
volume = 76,
place = {United States},
year = {2007},
month = {9}
}