  The imaginary-time evolution method is widely known to be efficient for
obtaining the ground state in quantum many-body problems on a classical
computer. A recently proposed quantum imaginary-time evolution method (QITE)
faces problems of deep circuit depth and difficulty in the implementation on
noisy intermediate-scale quantum (NISQ) devices. In this study, a nonlocal
approximation is developed to tackle this difficulty. We found that by removing
the locality condition or local approximation (LA), which was imposed when the
imaginary-time evolution operator is converted to a unitary operator, the
quantum circuit depth is significantly reduced. We propose two-step
approximation methods based on a nonlocality condition: extended LA (eLA) and
nonlocal approximation (NLA). To confirm the validity of eLA and NLA, we apply
them to the max-cut problem of an unweighted 3-regular graph and a weighted
fully connected graph; we comparatively evaluate the performances of LA, eLA,
and NLA. The eLA and NLA methods require far fewer circuit depths than LA to
maintain the same level of computational accuracy. Further, we developed a
``compression'' method of the quantum circuit for the imaginary-time steps as a
method to further reduce the circuit depth in the QITE method. The eLA, NLA,
and the compression method introduced in this study allow us to reduce the
circuit depth and the accumulation of error caused by the gate operation
significantly and pave the way for implementing the QITE method on NISQ
devices.
