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Title: Quantum-Merlin-Arthur-complete translationally invariant Hamiltonian problem and the complexity of finding ground-state energies in physical systems

Abstract

Here we present a problem related to the local Hamiltonian problem (identifying whether the ground-state 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 two-body terms, there are instances where finding the ground-state energy is quantum-Merlin-Arthur-complete and simulating the dynamics is BQP-complete (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 density-matrix renormalization group, matrix product states, and multiscale entanglement renormalization ansatz. One important example is a one-dimensional lattice of bosons with nearest-neighbor hopping at constant filling fraction--i.e., a generalization of the Bose-Hubbard model.

Authors:
 [1]
  1. 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 1050-2947
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; ONE-DIMENSIONAL CALCULATIONS; POLYNOMIALS; QUANTUM COMPUTERS; QUANTUM ENTANGLEMENT; QUANTUM MECHANICS; TWO-BODY PROBLEM

Citation Formats

Kay, Alastair. Quantum-Merlin-Arthur-complete translationally invariant Hamiltonian problem and the complexity of finding ground-state energies in physical systems. United States: N. p., 2007. Web. doi:10.1103/PHYSREVA.76.030307.
Kay, Alastair. Quantum-Merlin-Arthur-complete translationally invariant Hamiltonian problem and the complexity of finding ground-state energies in physical systems. United States. doi:10.1103/PHYSREVA.76.030307.
Kay, Alastair. Sat . "Quantum-Merlin-Arthur-complete translationally invariant Hamiltonian problem and the complexity of finding ground-state energies in physical systems". United States. doi:10.1103/PHYSREVA.76.030307.
@article{osti_21015897,
title = {Quantum-Merlin-Arthur-complete translationally invariant Hamiltonian problem and the complexity of finding ground-state energies in physical systems},
author = {Kay, Alastair},
abstractNote = {Here we present a problem related to the local Hamiltonian problem (identifying whether the ground-state 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 two-body terms, there are instances where finding the ground-state energy is quantum-Merlin-Arthur-complete and simulating the dynamics is BQP-complete (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 density-matrix renormalization group, matrix product states, and multiscale entanglement renormalization ansatz. One important example is a one-dimensional lattice of bosons with nearest-neighbor hopping at constant filling fraction--i.e., a generalization of the Bose-Hubbard model.},
doi = {10.1103/PHYSREVA.76.030307},
journal = {Physical Review. A},
issn = {1050-2947},
number = 3,
volume = 76,
place = {United States},
year = {2007},
month = {9}
}