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Title: Quantum-Merlin-Arthur-complete problems for stoquastic Hamiltonians and Markov matrices

Abstract

We show that finding the lowest eigenvalue of a 3-local symmetric stochastic matrix is Quantum-Merlin-Arthur-complete (QMA-complete). We also show that finding the highest energy of a stoquastic Hamiltonian is QMA-complete and that adiabatic quantum computation using certain excited states of a stoquastic Hamiltonian is universal. We also show that adiabatic evolution in the ground state of a stochastic frustration-free Hamiltonian is universal. Our results give a QMA-complete problem arising in the classical setting of Markov chains and adiabatically universal Hamiltonians that arise in many physical systems.

Authors:
; ;  [1]
  1. Institute for Quantum Information, California Institute of Technology, Pasadena, California 91125 (United States)
Publication Date:
OSTI Identifier:
21408402
Resource Type:
Journal Article
Journal Name:
Physical Review. A
Additional Journal Information:
Journal Volume: 81; Journal Issue: 3; Other Information: DOI: 10.1103/PhysRevA.81.032331; (c) 2010 The American Physical Society; Journal ID: ISSN 1050-2947
Country of Publication:
United States
Language:
English
Subject:
74 ATOMIC AND MOLECULAR PHYSICS; EIGENVALUES; EVOLUTION; EXCITED STATES; GROUND STATES; HAMILTONIANS; MARKOV PROCESS; MATRICES; QUANTUM COMPUTERS; SYMMETRY; COMPUTERS; ENERGY LEVELS; MATHEMATICAL OPERATORS; QUANTUM OPERATORS; STOCHASTIC PROCESSES

Citation Formats

Jordan, Stephen P, Gosset, David, Love, Peter J, Center for Theoretical Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, 6-304, Cambridge, Massachusetts 02139, and Department of Physics, Haverford College, 370 Lancaster Avenue, Haverford, Pennsylvania 19041, USA, and Institute for Quantum Information, California Institute of Technology, Pasadena, California 91125. Quantum-Merlin-Arthur-complete problems for stoquastic Hamiltonians and Markov matrices. United States: N. p., 2010. Web. doi:10.1103/PHYSREVA.81.032331.
Jordan, Stephen P, Gosset, David, Love, Peter J, Center for Theoretical Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, 6-304, Cambridge, Massachusetts 02139, & Department of Physics, Haverford College, 370 Lancaster Avenue, Haverford, Pennsylvania 19041, USA, and Institute for Quantum Information, California Institute of Technology, Pasadena, California 91125. Quantum-Merlin-Arthur-complete problems for stoquastic Hamiltonians and Markov matrices. United States. doi:10.1103/PHYSREVA.81.032331.
Jordan, Stephen P, Gosset, David, Love, Peter J, Center for Theoretical Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, 6-304, Cambridge, Massachusetts 02139, and Department of Physics, Haverford College, 370 Lancaster Avenue, Haverford, Pennsylvania 19041, USA, and Institute for Quantum Information, California Institute of Technology, Pasadena, California 91125. Mon . "Quantum-Merlin-Arthur-complete problems for stoquastic Hamiltonians and Markov matrices". United States. doi:10.1103/PHYSREVA.81.032331.
@article{osti_21408402,
title = {Quantum-Merlin-Arthur-complete problems for stoquastic Hamiltonians and Markov matrices},
author = {Jordan, Stephen P and Gosset, David and Love, Peter J and Center for Theoretical Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, 6-304, Cambridge, Massachusetts 02139 and Department of Physics, Haverford College, 370 Lancaster Avenue, Haverford, Pennsylvania 19041, USA, and Institute for Quantum Information, California Institute of Technology, Pasadena, California 91125},
abstractNote = {We show that finding the lowest eigenvalue of a 3-local symmetric stochastic matrix is Quantum-Merlin-Arthur-complete (QMA-complete). We also show that finding the highest energy of a stoquastic Hamiltonian is QMA-complete and that adiabatic quantum computation using certain excited states of a stoquastic Hamiltonian is universal. We also show that adiabatic evolution in the ground state of a stochastic frustration-free Hamiltonian is universal. Our results give a QMA-complete problem arising in the classical setting of Markov chains and adiabatically universal Hamiltonians that arise in many physical systems.},
doi = {10.1103/PHYSREVA.81.032331},
journal = {Physical Review. A},
issn = {1050-2947},
number = 3,
volume = 81,
place = {United States},
year = {2010},
month = {3}
}