Quantum Fourier Transform Has Small Entanglement

Abstract

The quantum Fourier transform (QFT) is a key component of many important quantum algorithms, most famously being the essential ingredient in Shor’s algorithm for factoring products of primes. Given its remarkable capability, one would think it can introduce large entanglement to qubit systems and would be difficult to simulate classically. While early results showed the QFT indeed has maximal operator entanglement, we show that this is entirely due to the bit reversal in the QFT. The core part of the QFT has Schmidt coefficients decaying exponentially quickly, and thus it can generate only a constant amount of entanglement regardless of the number of qubits. In addition, we show the entangling power of the QFT is the same as the time evolution of a Hamiltonian with exponentially decaying interactions, and thus a variant of the area law for dynamics can be used to understand the low entanglement intuitively. Using the low entanglement property of the QFT, we show that classical simulations of the QFT on a matrix product state with low bond dimension take time linear in the number of qubits, providing a potential speedup over the classical fast Fourier transform on many classes of functions. We demonstrate this speedup in test calculations on some simple functions. For data vectors of length 106–108, the speedup can be a few orders of magnitude.2 MoreReceived 2 January 2023Accepted 25 September 2023DOI:https://doi.org/10.1103/PRXQuantum.4.040318Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasEntanglement entropyEntanglement measuresEntanglement productionQuantum algorithms & computationQuantum circuitsQuantum computationQuantum correlations in quantum informationQuantum correlations, foundations & formalismQuantum entanglementQuantum information theoryTechniquesMatrix product statesTensor network methodsQuantum Information, Science & TechnologyEnergy Science & Technology