Year: 2026
Type: preprint
Abstract: The quantum Fourier transform (QFT) is a fundamental primitive in quantum computation and quantum information. In this work, we extend the framework for the QFT from finite groups to finite-dimensional semisimple algebras, and give efficient QFTs 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.
Source: arXiv (Cornell University)
Authors: Foxman, Ben, Nehoran, Barak, Ding, Yongshan
Institutions: Yale University, Columbia University
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: CAREER: Applications and Architectures with Heterogeneous Superconducting Qubits, G1523888516, NQVL:QSTD:Pilot: Erasure Qubits and Dynamic Circuits for Quantum Advantage (ERASE), G241279073, QLCI-CI: NSF Quantum Leap Challenge Institute for Present and Future Quantum Computing, G357145406, Collaborative Research: FET: Medium: Design and Implementation of Quantum Databases, G5765906131, NSF Engines Development Award: Advancing quantum technologies (CT), G7463374569, G872144517
OpenAlex ID: W7160596565