  In this study, we propose a quantum-classical hybrid scheme for performing
orbital-free density functional theory (OFDFT) using probabilistic
imaginary-time evolution (PITE), designed for the era of fault-tolerant quantum
computers (FTQC), as a material calculation method for large-scale systems.
PITE is applied to the part of OFDFT that searches the ground state of the
Hamiltonian in each self-consistent field (SCF) iteration, while the other
parts such as electron density and Hamiltonian updates are performed by
existing algorithms on classical computers. When the simulation cell is
discretized into $N_\mathrm{g}$ grid points, combined with quantum phase
estimation (QPE), it is shown that obtaining the ground state energy of
Hamiltonian requires a circuit depth of $O(\log N_\mathrm{g})$. The ground
state calculation part in OFDFT is expected to be accelerated, for example, by
creating an appropriate preconditioner from the estimated ground state energy
for the locally optimal block preconditioned conjugate gradient (LOBPCG)
method.
