Fast quantum imaginary time evolution

A fast implementation of the quantum imaginary time evolution (QITE) algorithm called Fast QITE is proposed. The algorithmic cost of QITE typically scales exponentially with the number of particles it nontrivially acts on in each Trotter step. In contrast, a Fast QITE implementation reduces this to only a linear scaling. It is shown that this speed up leads to a quantum advantage when sampling diagonal elements of a matrix exponential, which cannot be achieved using the standard implementation of the QITE algorithm. Finally the cost of implementing Fast QITE for finite temperature simulations is also discussed.