TY - JOUR

T1 - Exact expression of the energy gap at first-order quantum phase transitions of the fully connected p-body transverse-field ising model with transverse interactions

AU - Ohkuwa, Masaki

AU - Nishimori, Hidetoshi

N1 - Funding Information:
Acknowledgments This work was supported by the JSPS KAKENHI Grant No. 26287086.
Publisher Copyright:
© 2017 The Physical Society of Japan.

PY - 2017/11/15

Y1 - 2017/11/15

N2 - We study the energy gap between the ground state and the first excited state of a mean-field-type non-stoquastic Hamiltonian by a semi-classical analysis. The fully connected mean-field model with p-body ferromagnetic interactions under a transverse field has a first-order quantum phase transition for p ≥ 3. This first-order transition is known to be reduced to second order for p ≥ 5 by an introduction of antiferromagnetic transverse interactions, which makes the Hamiltonian non-stoquastic. This reduction of the order of transition means an exponential speedup of quantum annealing by adiabatic processes because the first-order transition is shown to have an exponentially small energy gap whereas the second order case does not. We apply a semi-classical method to analytically derive the explicit expression of the rate of the exponential decay of the energy gap at first-order transitions. The result reveals how the property of first-order transition changes as a function of the system parameters. We also derive the exact closed-form expression for the critical point where the first-order transition line disappears within the ferromagnetic phase. These results help us understand how the antiferromagnetic transverse interactions affect the performance of quantum annealing by controlling the effects of non-stoquasticity in the Hamiltonian.

AB - We study the energy gap between the ground state and the first excited state of a mean-field-type non-stoquastic Hamiltonian by a semi-classical analysis. The fully connected mean-field model with p-body ferromagnetic interactions under a transverse field has a first-order quantum phase transition for p ≥ 3. This first-order transition is known to be reduced to second order for p ≥ 5 by an introduction of antiferromagnetic transverse interactions, which makes the Hamiltonian non-stoquastic. This reduction of the order of transition means an exponential speedup of quantum annealing by adiabatic processes because the first-order transition is shown to have an exponentially small energy gap whereas the second order case does not. We apply a semi-classical method to analytically derive the explicit expression of the rate of the exponential decay of the energy gap at first-order transitions. The result reveals how the property of first-order transition changes as a function of the system parameters. We also derive the exact closed-form expression for the critical point where the first-order transition line disappears within the ferromagnetic phase. These results help us understand how the antiferromagnetic transverse interactions affect the performance of quantum annealing by controlling the effects of non-stoquasticity in the Hamiltonian.

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U2 - 10.7566/JPSJ.86.114004

DO - 10.7566/JPSJ.86.114004

M3 - Article

AN - SCOPUS:85036512944

SN - 0031-9015

VL - 86

JO - Journal of the Physical Society of Japan

JF - Journal of the Physical Society of Japan

IS - 11

M1 - 114004

ER -