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.

翻译：量子傅里叶变换（QFT）可视为离散傅里叶变换（DFT）的重索引，已被证明可压缩为低秩矩阵乘积算子（MPO）或量化张量列（QTT）算子。然而，该结论的原始证明并未提供具有明确误差界的MPO构造方法。与此同时，现有基于量子电路压缩的MPO实用构造方案尚未达到最优效率。本文提出一种基于插值分解的QFT MPO闭式构造方法，在给定秩条件下保证近乎最优的压缩误差。该方法可分别加速量子电路模拟与QTT应用中的QFT及DFT运算。此外，通过证明近似量子傅里叶变换（AQFT）可视为采用不同插值方案构造的MPO，我们将所提出的插值构造与AQFT建立了联系。