A Fourier analysis framework for approximate classical simulations of quantum circuits

What makes a class of quantum circuits efficiently classically simulable on average? I present a framework that applies harmonic analysis of groups to circuits with a structure encoded by group parameters. Expanding the circuits in a suitable truncated multi-path operator basis gives algorithms to evaluate the Fourier coefficients of output distributions or expectation values that are viewed as functions on the group. Under certain conditions, a truncated Fourier series can be efficiently estimated with guaranteed mean-square convergence. For classes of noisy circuits, it leads to algorithms for sampling and mean value estimation under error models with a spectral gap, where the complexity increases exponentially with the gap's inverse and polynomially with the circuit's size. This approach unifies and extends existing algorithms for noisy parametrised or random circuits using Pauli basis paths. For classes of noiseless circuits, mean values satisfying Lipschitz continuity can be on average approximated using efficient sparse Fourier decompositions. I also discuss generalisations to homogeneous spaces, qudit systems and a way to analyse random circuits via matrix coefficients of irreducible representations.