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arXiv:1911.03055 · Quantum Information Processing

Quantum circuit for the fast Fourier transform

Ryo Asaka, Kazumitsu Sakai, Ryoko Yahagi

Published Aug 1, 2020• 31 citations
arXivHTMLPDF

Abstract

Abstract We propose an implementation of the algorithm for the fast Fourier transform (FFT) as a quantum circuit consisting of a combination of some quantum gates. In our implementation, a data sequence is expressed by a tensor product of vector spaces. Namely, our FFT is defined as a transformation of the tensor product of quantum states. It is essentially different from the so-called quantum Fourier transform (QFT) defined to be a linear transformation of the amplitudes for the superposition of quantum states. The quantum circuit for the FFT consists of several circuits for elementary arithmetic operations such as a quantum adder, subtractor and shift operations, which are implemented as effectively as possible. Namely, our circuit does not generate any garbage bits. The advantages of our method compared to the QFT are its high versatility, and data storage efficiency in terms, for instance, of the quantum image processing.

DOI
10.1007/s11128-020-02776-5

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quant-pharXiv:2512.12771•Dec 16, 2025•0 citations
Abstract We propose an implementation of the algorithm for the fast Fourier transform (FFT) as a quantum circuit consisting of a combination of some quantum gates. In our implementation, a data sequence is expressed by a tensor product of vector spaces. Namely, our FFT is defined as a transformation of the tensor product of quantum states. It is essentially different from the so-called quantum Fourier transform (QFT) defined to be a linear transformation of the amplitudes for the superposition of quantum states. The quantum circuit for the FFT consists of several circuits for elementary arithmetic operations such as a quantum adder, subtractor and shift operations, which are implemented as effectively as possible. Namely, our circuit does not generate any garbage bits. The advantages of our method compared to the QFT are its high versatility, and data storage efficiency in terms, for instance, of the quantum image processing.
This survey presents a comprehensive analysis of the quantum Fourier transform (QFT), a fundamental tool in quantum computing, in comparison to its classical counterpart, the fast Fourier transform (FFT). The study begins with an introduction to the classical Fourier transform, which is widely used in different disciplines such as signal and image processing. The article introduces the QFT as a quantum version of the Fourier transform, detailing how it leverages quantum parallelism and superposition to reduce the time complexity of the operation, and highlighting its crucial role in quantum algorithms like Shor’s algorithm for integer factorization. The analysis also addresses the mathematical foundations of the QFT, its implementation in quantum circuits, and the key advantages and challenges associated with its use, such as measurement precision and quantum decoherence. Finally, it concludes with an exploration of the current and potential applications of QFT in quantum computing.
The fast Fourier transform (FFT) is one of the most successful numerical algorithms of the 20th century and has found numerous applications in many branches of computational science and engineering. The FFT algorithm can be derived from a particular matrix decomposition of the discrete Fourier transform (DFT) matrix. In this paper, 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 analyze the implication of this Kronecker product structure on the discrete Fourier transform of rank-1 tensors on a classical computer. We also 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. Further, the connection between the matrix decomposition of the DFT matrix and a quantum circuit is made. We also discuss a natural extension of a radix-2 QFT decomposition to a radix-d QFT decomposition. No prior knowledge of quantum computing is required to understand what is presented in this paper. Yet, we believe this paper may help readers to gain some rudimentary understanding of the nature of quantum computing from a matrix computation point of view.
Because of the difficulty of building a high-dimensional quantum register, this paper presents an implementation of the high-dimensional quantum Fourier transform (QFT) based on a low-dimensional quantum register. First, we define the t-bit semiclassical quantum Fourier transform. In terms of probability amplitude, we prove that the transform can realize quantum Fourier transformation, illustrate that the requirement for the two-qubit gate reduces obviously, and further design a quantum circuit of the transform. Combining the classical fixed-window method and the implementation of Shor’s quantum factorization algorithm, we then redesign a circuit for Shor’s algorithm, whose required computation resource is approximately equal to that of Parker’s. The requirement for elementary quantum gates for Parker’s algorithm is O(⌈ log N⌉3, and the quantum register for our circuit requires t−1 more dimensions than Parker’s. However, our circuit is t 2 times as fast as Parker’s, where t is the width of the window.
Summary The fast Fourier transform (FFT) is one of the most successful numerical algorithms of the 20th century and has found numerous applications in many branches of computational science and engineering. The FFT algorithm can be derived from a particular matrix decomposition of the discrete Fourier transform (DFT) matrix. In this paper, 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 analyze the implication of this Kronecker product structure on the discrete Fourier transform of rank‐1 tensors on a classical computer. We also 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. Further, the connection between the matrix decomposition of the DFT matrix and a quantum circuit is made. We also discuss a natural extension of a radix‐2 QFT decomposition to a radix‐d QFT decomposition. No prior knowledge of quantum computing is required to understand what is presented in this paper. Yet, we believe this paper may help readers to gain some rudimentary understanding of the nature of quantum computing from a matrix computation point of view.
We introduce QFT→\rightarrow→FFT, a family of HPC FFT libraries that compute the discrete Fourier transform by executing a quantum Fourier transform (QFT) circuit on classical quantum computer simulators. Input arrays are mapped directly to state amplitudes with explicit normalization/indexing, making QFT a drop-in replacement for FFT primitives. A backend-agnostic planner builds a fused-gate schedule and memory layout adapters to increase arithmetic intensity and reduce memory data movement. We implement this design on top of Google's C++ \texttt{qsim} and evaluate OpenMP, AVX, and CUDA backends. On an AMD EPYC Zen2 processor, our AVX performance is on par with that of multithreaded FFTW, utilizing 64 threads. On an NVIDIA A100, the CUDA backend achieves more than 4×4\times4× lower time than both AVX and FFTW on AMD EPYC Zen2 at larger sizes. We also employ an approximate QFT (AQFT) that truncates small-angle controlled rotations beyond a cutoff kkk, reducing circuit depth and runtime while preserving accuracy.
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 NNN--mode are completely specified by unitary matrices of order NNN. 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.