Importance sampling for stochastic quantum simulations

Simulating many-body quantum systems is a promising task for computers. However, the depth of most algorithms, such as product formulas, scales with number terms in Hamiltonian, and can therefore be challenging to implement on near-term, well early fault-tolerant devices. An efficient solution given by stochastic compilation protocol known qDrift, which builds random formulas sampling from Hamiltonian according coefficients. In this work, we unify qDrift importance sampling, allowing us sample arbitrary probability distributions, while controlling both bias, statistical fluctuations. We show that simulation cost reduced achieving same accuracy, considering individual during stage. Moreover, incorporate recent work composite channel compute rigorous bounds bias variance, showing how choose samples, experiments, time steps target accuracy. These results lead more implementation protocol, without use channels. Theoretical are confirmed numerical simulations performed lattice nuclear effective field theory.