Efficient Quantum Fourier Transforms For Semisimple Algebras

TL;DR

This paper extends the quantum Fourier transform to finite-dimensional semisimple algebras, providing efficient quantum algorithms for specific algebras with applications in physics and quantum computing.

Contribution

It introduces a generalized QFT for semisimple algebras, including efficient approximation algorithms with poly(n, log d, log(1/ε)) complexity.

Findings

Efficient quantum algorithms for Fourier transforms over partition, Brauer, and walled Brauer algebras.

Approximate Fourier transforms are well-approximated by unitaries when d is large.

Properties of the Fourier basis in semisimple algebras are established, potentially of independent interest.

Abstract

The quantum Fourier transform (QFT) is a fundamental primitive in quantum computation and quantum information. In this work, we generalize the QFT for finite groups to a QFT for finite-dimensional semisimple algebras, and give efficient quantum Fourier transforms for the partition algebra $P_n(d)$, Brauer algebra $B_n(d)$, and walled Brauer algebra $B_{r,s}(d)$. These algebras play important roles in generalized Schur-Weyl duality, statistical physics and many-body systems, and have recently found several applications in quantum algorithms. Unlike the group case, the Fourier transform over a semisimple algebra can be non-unitary. Nevertheless, we show that when the parameter $d$ is sufficiently large, the Fourier transform is well approximated by a unitary operator. Furthermore, we show that for each of the algebras $A$ from above, such an approximate Fourier transform can be implemented efficiently: we give a quantum algorithm with gate complexity $\mathrm{poly}(n,\log d,\log(1/\varepsilon))$ for approximating the Fourier transform to error $(d^{-1/2} + \varepsilon) \cdot \mathrm{poly}(|A|)$. Along the way, we establish several properties of the Fourier basis of semisimple algebras that may be of independent interest.