Importance sampling for stochastic quantum simulations

Simulating complex quantum systems is a promising task for digital quantum computers. However, the depth of popular product formulas scales with the number of summands in the Hamiltonian, which can therefore be challenging to implement on near-term as well as fault-tolerant devices. An efficient solution is given by the stochastic compilation protocol known as qDrift, which builds random product formulas by sampling from the Hamiltonian according to the magnitude of their coefficients. In this work, we unify the qDrift protocol with importance sampling, allowing us to sample from arbitrary distributions while controlling both the bias as well as the statistical fluctuations. We show that the simulation cost can be reduced while achieving the same accuracy by considering the individual simulation cost during the sampling stage.

Moreover, we incorporate recent work on composite channel and compute rigorous bounds on the bias and variance showing how to choose the number of samples, experiments, and time steps for a given target accuracy. These results lead to a more efficient implementation of the qDrift protocol, both with and without the use of composite channels. Theoretical results are confirmed by numerical simulations performed on a lattice nuclear effective field theory.