Quantum Schur Sampling Circuits can be Strongly Simulated

TL;DR

This paper demonstrates that Quantum Schur Sampling circuits, previously thought to be inherently quantum, can be efficiently simulated using classical algorithms, challenging assumptions about their non-classical nature.

Contribution

The authors develop a classical algorithm that efficiently approximates transition amplitudes of Quantum Schur Sampling circuits, extending previous results on permutational quantum computing.

Findings

Classical algorithms can efficiently approximate transition amplitudes of Quantum Schur Sampling circuits.

This challenges the belief that such circuits are inherently non-classical.

The approach extends to a broader class of quantum circuits beyond PQC.

Abstract

Permutational Quantum Computing (PQC) [\emph{Quantum~Info.~Comput.}, \textbf{10}, 470--497, (2010)] is a natural quantum computational model conjectured to capture non-classical aspects of quantum computation. An argument backing this conjecture was the observation that there was no efficient classical algorithm for estimation of matrix elements of the $S_n$ irreducible representation matrices in the Young's orthogonal form, which correspond to transition amplitudes of a broad class of PQC circuits. This problem can be solved with a PQC machine in polynomial time, but no efficient classical algorithm for the problem was previously known. Here we give a classical algorithm that efficiently approximates the transition amplitudes up to polynomial additive precision and hence solves this problem. We further extend our discussion to show that transition amplitudes of a broader class of quantum circuits -- the Quantum Schur Sampling circuits -- can be also efficiently estimated classically.