Scale invariance and efficient classical simulation of the quantum Fourier transform

TL;DR

This paper demonstrates that the quantum Fourier transform can be efficiently approximated and simulated using matrix product operators, revealing scale invariance and enabling faster classical simulations of quantum states.

Contribution

It introduces a method to efficiently represent the quantum Fourier transform as a matrix product operator and analyzes its convergence and simulation complexity.

Findings

Quantum Fourier transform can be represented with slowly growing matrix product operator size.

Tensors in the operator converge to a common tensor as qubits increase.

Simulation of the transform on matrix product states is feasible with polynomial time complexity.

Abstract

We provide numerical evidence that the quantum Fourier transform can be efficiently represented in a matrix product operator with a size growing relatively slowly with the number of qubits. Additionally, we numerically show that the tensors in the operator converge to a common tensor as the number of qubits in the transform increases. Together these results imply that the application of the quantum Fourier transform to a matrix product state with $n$ qubits of maximum Schmidt rank $\chi$ can be simulated in $O(n (log(n))^2 \chi^2)$ time. We perform such simulations and quantify the error involved in representing the transform as a matrix product operator and simulating the quantum Fourier transform of periodic states.