Direct interpolative construction of the discrete Fourier transform as a matrix product operator

TL;DR

This paper introduces a straightforward, error-guaranteed method for constructing the quantum Fourier transform as a matrix product operator using interpolative decomposition, enhancing efficiency in quantum simulations.

Contribution

It provides a closed-form, near-optimal MPO construction for the QFT with guaranteed error bounds, improving upon previous methods.

Findings

Speeds up QFT and DFT applications in quantum simulations.

Connects interpolative MPO construction to the approximate quantum Fourier transform.

Offers a practical, error-controlled approach for MPO construction.

Abstract

The quantum Fourier transform (QFT), which can be viewed as a reindexing of the discrete Fourier transform (DFT), has been shown to be compressible as a low-rank matrix product operator (MPO) or quantized tensor train (QTT) operator. However, the original proof of this fact does not furnish a construction of the MPO with a guaranteed error bound. Meanwhile, the existing practical construction of this MPO, based on the compression of a quantum circuit, is not as efficient as possible. We present a simple closed-form construction of the QFT MPO using the interpolative decomposition, with guaranteed near-optimal compression error for a given rank. This construction can speed up the application of the QFT and the DFT, respectively, in quantum circuit simulations and QTT applications. We also connect our interpolative construction to the approximate quantum Fourier transform (AQFT) by demonstrating that the AQFT can be viewed as an MPO constructed using a different interpolation scheme.