The Quantum Fourier Transform for Continuous Variables

TL;DR

This paper extends the quantum Fourier transform to continuous-variable systems, defining it as a rotation operator related to the DFT, and demonstrates its application to Gaussian states with reduced implementation complexity.

Contribution

It introduces a continuous-variable quantum Fourier transform based on rotation operators and adapts FFT algorithms for efficient implementation, applying it to Gaussian states.

Findings

cvQFT transforms displacement vectors via DFT

cvQFT simplifies the squeeze matrix transformation

Efficient implementation using FFT algorithms

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.