TY - JOUR
T1 - Exponential enhancement of the efficiency of quantum annealing by non-stoquastic Hamiltonians
AU - Nishimori, Hidetoshi
AU - Takada, Kabuki
N1 - Funding Information:
Most of the technical results described in this article appeared in Seki and Nishimori (2012, 2015) and Seoane and Nishimori (2012) albeit from a little different viewpoint than presented here. One of the authors (HN) sincerely thanks Yuya Seki and Beatriz Seoane for stimulating collaboration. Also acknowledged are useful comments by Tameem Albash, Jacob Biamonte, Eddie Farhi, Itay Hen, Layla Hormozi, Helmut Katzgraber, and Daniel Lidar. This work was funded by the ImPACT Program of Council for Science, Technology and Innovation, Cabinet Office, Government of Japan and by the JSPS KAKENHI Grant No. 26287086
Publisher Copyright:
© 2017 Nishimori and Takada.
PY - 2017
Y1 - 2017
N2 - Non-stoquastic Hamiltonians have both positive and negative signs in off-diagonal elements in their matrix representation in the standard computational basis and thus cannot be simulated efficiently by the standard quantum Monte Carlo method due to the sign problem. We describe our analytical studies of this type of Hamiltonians with infinite-range non-random as well as random interactions from the perspective of possible enhancement of the efficiency of quantum annealing or adiabatic quantum computing. It is shown that multi-body transverse interactions like XX and XXXXX with positive coefficients appended to a stoquastic transverse-field Ising model render the Hamiltonian non-stoquastic and reduce a first-order quantum phase transition in the simple transverse-field case to a second-order transition. This implies that the efficiency of quantum annealing is exponentially enhanced, because a first-order transition has an exponentially small energy gap (and therefore exponentially long computation time) whereas a second-order transition has a polynomially decaying gap (polynomial computation time). The examples presented here represent rare instances where strong quantum effects, in the sense that they cannot be efficiently simulated in the standard quantum Monte Carlo, have analytically been shown to exponentially enhance the efficiency of quantum annealing for combinatorial optimization problems.
AB - Non-stoquastic Hamiltonians have both positive and negative signs in off-diagonal elements in their matrix representation in the standard computational basis and thus cannot be simulated efficiently by the standard quantum Monte Carlo method due to the sign problem. We describe our analytical studies of this type of Hamiltonians with infinite-range non-random as well as random interactions from the perspective of possible enhancement of the efficiency of quantum annealing or adiabatic quantum computing. It is shown that multi-body transverse interactions like XX and XXXXX with positive coefficients appended to a stoquastic transverse-field Ising model render the Hamiltonian non-stoquastic and reduce a first-order quantum phase transition in the simple transverse-field case to a second-order transition. This implies that the efficiency of quantum annealing is exponentially enhanced, because a first-order transition has an exponentially small energy gap (and therefore exponentially long computation time) whereas a second-order transition has a polynomially decaying gap (polynomial computation time). The examples presented here represent rare instances where strong quantum effects, in the sense that they cannot be efficiently simulated in the standard quantum Monte Carlo, have analytically been shown to exponentially enhance the efficiency of quantum annealing for combinatorial optimization problems.
KW - Exponential speedup
KW - Non-stoquastic Hamiltonian
KW - Quantum adiabatic algorithm
KW - Quantum annealing
KW - Stoquastic Hamiltonian
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U2 - 10.3389/fict.2017.00002
DO - 10.3389/fict.2017.00002
M3 - Article
AN - SCOPUS:85047392888
VL - 4
JO - Frontiers in ICT
JF - Frontiers in ICT
SN - 2297-198X
IS - FEB
M1 - 2
ER -