Title: Message-passing algorithm of quantum annealing with nonstoquastic Hamiltonian,
Abstract: Quantum annealing (QA) is a generic method for solving optimization problems
using fictitious quantum fluctuation. The current device performing QA involves
controlling the transverse field; it is classically simulatable by using the
standard technique for mapping the quantum spin systems to the classical ones.
In this sense, the current system for QA is not powerful despite utilizing
quantum fluctuation. Hence, we developed a system with a time-dependent
Hamiltonian consisting of a combination of the formulated Ising model and the
"driver" Hamiltonian with only quantum fluctuation. In the previous study, for
a fully connected spin model, quantum fluctuation can be addressed in a
relatively simple way. We proved that the fully connected antiferromagnetic
interaction can be transformed into a fluctuating transverse field and is thus
classically simulatable at sufficiently low temperatures. Using the fluctuating
transverse field, we established several ways to simulate part of the
nonstoquastic Hamiltonian on classical computers. We formulated a
message-passing algorithm in the present study. This algorithm is capable of
assessing the performance of QA with part of the nonstoquastic Hamiltonian
having a large number of spins. In other words, we developed a different
approach for simulating the nonstoquastic Hamiltonian without using the quantum
Monte Carlo technique. Our results were validated by comparison to the results
obtained by the replica method.