The Quantum Fourier Transform for Continuous Variables

TL;DR

The paper extends the quantum Fourier transform to continuous variables by defining the cvQFT as a rotation in the $N$-mode Hilbert space with a rotation matrix equal to the DFT matrix $W_N$, and analyzes both theoretical structure and practical implementation. It develops a modified Murnaghan decomposition for constructing the cvQFT from primitive components, and introduces an FFT-inspired time-decimation approach to reduce complexity to an $O(N\log N)$-like scaling, including explicit results for $N=4$. The work then details how cvQFT interacts with Gaussian unitaries and Gaussian states, deriving exact transformations for displacement vectors, squeeze matrices, and rotation parameters in both the Hilbert space and phase-space formalisms, with switching rules enabling rearrangements of the cvQFT with minimal parameter updates. The results demonstrate that cvQFT preserves Gaussian structure and yields simple, interpretable updates to state parameters, suggesting cvQFT as a natural building block for multimode Gaussian networks and continuous-variable signal-processing protocols. These insights offer practical pathways for implementing Fourier-type mode mixing in CV quantum information systems.

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.

Figures (14)

• Figure 1: The Hadamard and the controlled--phase gates.
• Figure 2: Implementation of the dvQFT for $N=4$ according to Ref. Niel2000. The coefficients $x_i$ and $y_k$ are related by the 4--DFT.
• Figure 3: Application of the cvQFT to a pure quantum state and to a mixed quantum state.
• Figure 4: Graphical representation of the two primitive components.
• Figure 5: The beam splitter with phase.

Theorems & Definitions (11)

• Definition 1
• Proposition 1
• Proposition 2
• Theorem 1
• Theorem 2
• Proposition 3
• Theorem 3
• Proposition 4
• Theorem 4
• Proposition 5