  Quantum imaginary time evolution (QITE) is one of the promising candidates
for finding eigenvalues and eigenstates of a Hamiltonian. However, the original
QITE proposal [Nat. Phys. 16, 205-210 (2020)], which approximates the imaginary
time evolution by real time evolution, suffers from large circuit depth and
measurements due to the size of the Pauli operator pool and Trotterization. To
alleviate the requirement for deep circuits, we propose a time-dependent
drifting scheme inspired by the qDRIFT algorithm [Phys. Rev. Lett 123, 070503
(2019)], which randomly draws a Pauli term out of the approximated unitary
operation generators of QITE according to the strength and rescales that term
by the total strength of the Pauli terms. We show that this drifting scheme
removes the depth dependency on size of the operator pool and converges inverse
linearly to the number of steps. We further propose a deterministic algorithm
that selects the dominant Pauli term to reduce the fluctuation for the ground
state preparation. Meanwhile, we introduce an efficient measurement reduction
scheme across Trotter steps, which removes its cost dependence on the number of
iterations, and a measurement distribution protocol for different observables
within each time step. We also analyze the main source of error for our scheme
both theoretically and numerically. We numerically test the validity of depth
reduction, convergence performance, and faithfulness of measurement reduction
approximation of our algorithms on LiH, BeH$_2$ and N$_2$ molecules. In
particular, the results on LiH molecule give circuit depths comparable to that
of the advanced adaptive variational quantum eigensolver~(VQE) methods while
requiring much fewer measurements.
