Efficient Quantum Imaginary Time Evolution by Drifting Real-Time Evolution: An Approach with Low Gate and Measurement Complexity

Quantum
imaginary time evolution (QITE) is one of the promising
candidates for finding the eigenvalues and eigenstates of a Hamiltonian
on a quantum computer. However, the original proposal 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 [Campbell,
E. Phys. Rev. Lett. 2019, 123, 070503]. We show that this drifting scheme
removes the depth dependency on the size of the operator pool and
converges inversely with respect 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. We
also introduce an efficient measurement reduction scheme across Trotter
steps that removes its cost dependence on the number of iterations.
We analyze the main source of error for our scheme both theoretically
and numerically. We numerically test the validity of depth reduction,
convergence performance of our algorithms, and the faithfulness of
the approximation for our measurement reduction scheme on several
benchmark molecules. In particular, the results on the LiH molecule
give circuit depths comparable to that of the advanced adaptive variational
quantum eigensolver (VQE) methods while requiring much fewer measurements.