Importance sampling for stochastic quantum simulations

Oriel Kiss1,2, Michele Grossi1, and Alessandro Roggero3,4

1European Organization for Nuclear Research (CERN), Geneva 1211, Switzerland
2Department of Nuclear and Particle Physics, University of Geneva, Geneva 1211, Switzerland
3Physics Department, University of Trento, Via Sommarive 14, I-38123 Trento, Italy
4INFN-TIFPA Trento Institute of Fundamental Physics and Applications, Trento, Italy

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Simulating many-body quantum systems is a promising task for quantum computers. However, the depth of most algorithms, such as product formulas, scales with the number of terms in the Hamiltonian, and can therefore be challenging to implement on near-term, as well as early fault-tolerant quantum 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 coefficients. In this work, we unify the qDrift protocol with importance sampling, allowing us to sample from arbitrary probability 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.

In this work, we unify important sampling with random product formula and show that a guaranteed cost reduction in terms of two-qubit gates can be achieved without sacrificing accuracy. Moreover, we give rigorous concentration bounds for general qDRIFT channels and show that qDRIFT channels can be efficiently parallelized. Finally, we look into nuclear models, more particularly effective field theory on a lattice, and numerically show that we can save an order of magnitude in CNOT gates per circuit at the same accuracy. We hope this paper will be a starting point for elaborating qDRFIT formulas tailored for specific applications and hardware.

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