  Ground-state preparation is an important task in quantum computation. The
probabilistic imaginary-time evolution (PITE) method is a promising candidate
for preparing the ground state of the Hamiltonian, which comprises a single
ancilla qubit and forward- and backward-controlled real-time evolution
operators. The ground state preparation is a challenging task even in the
quantum computation, classified as complexity-class quantum Merlin-Arthur.
However, optimal parameters for PITE could potentially enhance the
computational efficiency to a certain degree. In this study, we analyze the
computational costs of the PITE method for both linear and exponential
scheduling of the imaginary-time step size for reducing the computational cost.
First, we analytically discuss an error defined as the closeness between the
states acted on by exact and approximate imaginary-time evolution operators.
The optimal imaginary-time step size and rate of change of imaginary time are
also discussed. Subsequently, the analytical discussion is validated using
numerical simulations for a one-dimensional Heisenberg chain. From the results,
we find that linear scheduling works well in the case of unknown eigenvalues of
the Hamiltonian. For a wide range of eigenstates, the linear scheduling returns
smaller errors on average. However, the linearity of the scheduling causes
problems for some specific energy regions of eigenstates. To avoid these
problems, incorporating a certain level of nonlinearity into the scheduling,
such as by inclusion of an exponential character, is preferable for reducing
the computational costs of the PITE method. The findings of this study can make
a significant contribute to the field of ground-state preparation of many-body
Hamiltonians on quantum computers.
