The Quantum Fourier Transform for Continuous Variables

• URL: http://arxiv.org/abs/2512.12771v1
• Date: Sun, 14 Dec 2025 17:23:17 GMT
• Title: The Quantum Fourier Transform for Continuous Variables
• Authors: Gianfranco Cariolaro, Edi Ruffa, Amir Mohammad Yaghoobianzadeh, Jawad A. Salehi,
• Abstract summary: The quantum Fourier transform for discrete variable (dvQFT) is an efficient algorithm for several applications.<n>We show that cvQFT has the simple effect of transforming the displacement vector by a one-dimensional DFT, the squeeze matrix by a two-dimensional DFT, and the rotation matrix by a Fourier-like similarity transform.
• Score: 0.9974630621313313
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The quantum Fourier transform for discrete variable (dvQFT) is an efficient algorithm for several applications. It is usually considered for the processing of quantum bits (qubits) and its efficient implementation is obtained with two elementary components: the Hadamard gate and the controlled--phase gate. In this paper, the quantum Fourier transform operating with continuous variables (cvQFT) is considered. Thus, the environment becomes the Hilbert space, where the natural definition of the cvQFT will be related to rotation operators, which in the $N$--mode are completely specified by unitary matrices of order $N$. Then the cvQFT is defined as the rotation operator whose rotation matrix is given by the discrete Fourier transform (DFT) matrix. For the implementation of rotation operators with primitive components (single--mode rotations and beam splitters), we follow the well known Murnaghan procedure, with appropriate modifications. Moreover, algorithms related to the fast Fourier transform (FFT) are applied to reduce drastically the implementation complexity. The final part is concerned with the application of the cvQFT to general Gaussian states. In particular, we show that cvQFT has the simple effect of transforming the displacement vector by a one-dimensional DFT, the squeeze matrix by a two-dimensional DFT, and the rotation matrix by a Fourier-like similarity transform.

Related papers

• WUSH: Near-Optimal Adaptive Transforms for LLM Quantization [52.77441224845925]
  Quantization to low bitwidth is a standard approach for deploying large language models.<n>A few extreme weights and activations stretch the dynamic range and reduce the effective resolution of the quantizer.<n>We derive, for the first time, closed-form optimal linear blockwise transforms for joint weight-activation quantization.
  arXiv  Detail & Related papers  (2025-11-30T16:17:34Z)
• LFA applied to CNNs: Efficient Singular Value Decomposition of Convolutional Mappings by Local Fourier Analysis [4.69726714177332]
  singular values of convolutional mappings encode interesting spectral properties.<n> computation of singular values is typically very resource-intensive.<n>We propose an approach of complexity O(N) based on local Fourier analysis.
  arXiv  Detail & Related papers  (2025-06-05T22:10:01Z)
• Inverse nonlinear fast Fourier transform on SU(2) with applications to quantum signal processing [10.193028145901248]
  We investigate the inverse NLFT on $mathrmSU(2)$ and establish the numerical stability of the layer stripping algorithm for the first time.<n>We develop a fast and numerically stable algorithm, called inverse nonlinear fast Fourier transform, for performing inverse NLFT with near-linear complexity.
  arXiv  Detail & Related papers  (2025-05-19T01:57:04Z)
• Variable-size Symmetry-based Graph Fourier Transforms for image compression [65.7352685872625]
  We propose a new family of Symmetry-based Graph Fourier Transforms of variable sizes into a coding framework. Our proposed algorithm generates symmetric graphs on the grid by adding specific symmetrical connections between nodes. Experiments show that SBGFTs outperform the primary transforms integrated in the explicit Multiple Transform Selection.
  arXiv  Detail & Related papers  (2024-11-24T13:00:44Z)
• Quantum Inverse Fast Fourier Transform [0.0]
  An algorithm for Quantum Inverse Fast Fourier Transform (QIFFT) is developed to work for quantum data. We have included the complete formulation of QIFFT algorithm from the classical model and have included butterfly diagram.
  arXiv  Detail & Related papers  (2024-09-12T12:27:47Z)
• Direct interpolative construction of the discrete Fourier transform as a matrix product operator [0.0]
  We present a simple closed-form construction of the QFT MPO using the interpolative decomposition. This construction can speed up the application of the QFT and the DFT, respectively, in quantum circuit simulations and QTT applications.
  arXiv  Detail & Related papers  (2024-04-04T03:42:17Z)
• Deep Fourier Up-Sampling [100.59885545206744]
  Up-sampling in the Fourier domain is more challenging as it does not follow such a local property. We propose a theoretically sound Deep Fourier Up-Sampling (FourierUp) to solve these issues.
  arXiv  Detail & Related papers  (2022-10-11T06:17:31Z)
• Quantum Fourier Addition, Simplified to Toffoli Addition [92.18777020401484]
  We present the first systematic translation of the QFT-addition circuit into a Toffoli-based adder. Instead of using approximate decompositions of the gates from the QFT circuit, it is more efficient to merge gates.
  arXiv  Detail & Related papers  (2022-09-30T02:36:42Z)
• Softmax-free Linear Transformers [90.83157268265654]
  Vision transformers (ViTs) have pushed the state-of-the-art for visual perception tasks. Existing methods are either theoretically flawed or empirically ineffective for visual recognition. We propose a family of Softmax-Free Transformers (SOFT)
  arXiv  Detail & Related papers  (2022-07-05T03:08:27Z)
• Learning Set Functions that are Sparse in Non-Orthogonal Fourier Bases [73.53227696624306]
  We present a new family of algorithms for learning Fourier-sparse set functions. In contrast to other work that focused on the Walsh-Hadamard transform, our novel algorithms operate with recently introduced non-orthogonal Fourier transforms. We demonstrate effectiveness on several real-world applications.
  arXiv  Detail & Related papers  (2020-10-01T14:31:59Z)
• Quantum Fourier Transform Revisited [3.23402688755667]
  We show that the quantum Fourier transform (QFT) can be derived by further decomposing the diagonal factors of the FFT matrix decomposition into products of matrices with Kronecker product structure. We explain why such a structure can take advantage of an important quantum computer feature that enables the QFT algorithm to attain an exponential speedup on a quantum computer over the FFT algorithm on a classical computer.
  arXiv  Detail & Related papers  (2020-03-06T02:46:58Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.