  A probabilistic imaginary-time evolution (PITE) method was proposed as a
nonvariational method to obtain a ground state on a quantum computer. In this
formalism, the success probability of obtaining all imaginary-time evolution
operators acting on the initial state decreases as the imaginary time proceeds.
To alleviate the undesirable nature, we propose quantum circuits for PITE
combined with the quantum amplitude amplification (QAA) method. We reduce the
circuit depth in the combined circuit with QAA by introducing a
pre-amplification operator. We successfully demonstrated that the combination
of PITE and QAA works efficiently and reported a case in which the quantum
acceleration is achieved. Additionally, we have found that by optimizing a
parameter of PITE, we can reduce the number of QAA operations and that
deterministic imaginary-time evolution (deterministic ITE) can be achieved
which avoids the probabilistic nature of PITE. We applied the deterministic ITE
procedure to multiple imaginary-time steps and discussed the computational cost
for the circuits. Finally, as an example, we demonstrate the numerical results
of the PITE circuit combined with QAA in the first- and second-quantized
Hamiltonians.
