QuantumMerlinArthurcomplete problems for stoquastic Hamiltonians and Markov matrices
Abstract
We show that finding the lowest eigenvalue of a 3local symmetric stochastic matrix is QuantumMerlinArthurcomplete (QMAcomplete). We also show that finding the highest energy of a stoquastic Hamiltonian is QMAcomplete 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 frustrationfree Hamiltonian is universal. Our results give a QMAcomplete problem arising in the classical setting of Markov chains and adiabatically universal Hamiltonians that arise in many physical systems.
 Authors:

 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 10502947
 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, 6304, 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. QuantumMerlinArthurcomplete 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, 6304, 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. QuantumMerlinArthurcomplete 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, 6304, 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 .
"QuantumMerlinArthurcomplete problems for stoquastic Hamiltonians and Markov matrices". United States. doi:10.1103/PHYSREVA.81.032331.
@article{osti_21408402,
title = {QuantumMerlinArthurcomplete 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, 6304, 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 3local symmetric stochastic matrix is QuantumMerlinArthurcomplete (QMAcomplete). We also show that finding the highest energy of a stoquastic Hamiltonian is QMAcomplete 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 frustrationfree Hamiltonian is universal. Our results give a QMAcomplete 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 = {10502947},
number = 3,
volume = 81,
place = {United States},
year = {2010},
month = {3}
}
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