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Spectral Decomposition of Quantum Fourier Transform

青木, 隆;AOKI, Takashi, Yusuke Maesaka, Mikio Nakahara

Published Sep 1, 2006• 0 citations
OpenAlex

Abstract

The spectral decomposition of an n-qubit quantum Fourier transform (QFT_n) is derived exactly. It is then applied to find the logarithm of a QFT_n gate. Hamiltonian which generates QFT_n in a single step is suggested.

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Their explicit forms are In this article we present a concrete form of the spectral decomposition of the quantum Fourier transform (QFT). QFT is one of the most useful mathematical tools in applied mathematics and mathematical sciences such as signal processing and quantum computing') . For example, in the famous quantum algorithm of factorization of a large integer due to Shor2}, QFT plays an essential role. Thus studying QFT is an important subject in various fields of applied mathematics. Spectral decomposition of QFT provides us with a very convenient way of theoretical treatment of QFT. As an application, we compute the logarithm of QFT.
For classical simulation, the quantum Fourier transform (QFT) requires very large matrix operations. Previous work has shown that the semi-classical quantum Fourier transform (SCQFT) can use these individual coefficients to perform the QFT using only single-quantum bit (qubit) unitary gates and measurement operators. However, the SCQFT requires these individual decomposed qubits of the quantum system as input to the algorithm. We devise two methods to find separable approximations of quantum systems to serve as inputs to the SCQFT. We introduce an application of the approach on classical radio frequency signals represented through a quantum model. The resulting decomposition and QFT are computed on several simulated results, and an example is given using an experimental signal.
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.
Operator-Schmidt decompositions of the quantum Fourier transform on ℂ N 1 ⊗ ℂ N 2 are computed for all N 1 , N 2 ≥ 2. The decomposition is shown to be completely degenerate when N 1 is a factor of N 2 and when N 1 > N 2 . The first known special case, N 1 = N 2 = 2 n , was computed by Nielsen in his study of the communication cost of computing the quantum Fourier transform of a collection of qubits equally distributed between two parties (M A Nielsen 1998 PhD Thesis University of New Mexico ch 6 Preprint quant-ph/0011036). More generally, the special case N 1 = 2 n 1 ⩽ 2 n 2 = N 2 was computed by Nielsen et al in their study of strength measures of quantum operations (M A Nielsen et al 2002 Preprint quant-ph/0208077 (2003 Phys. Rev. A at press)). Given the Schmidt decompositions presented here, it follows that in all cases the bipartite communication cost of exact computation of the quantum Fourier transform is maximal.
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.