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

Journal Article · Sat Sep 15 04:00:00 EDT 2007 · Physical Review. A

DOI:https://doi.org/10.1103/PHYSREVA.76.030307· OSTI ID:21015897

Kay, Alastair ^[1]

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1. Centre for Quantum Computation, DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA (United Kingdom)

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.

• OSTI ID: 21015897
• Journal Information: Physical Review. A, Journal Name: Physical Review. A Journal Issue: 3 Vol. 76; ISSN 1050-2947; ISSN PLRAAN
• Country of Publication: United States
• Language: English

Related Subjects

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