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In quantum computing, the quantum Fourier transform (QFT) is a linear transformation on quantum bits, and is the quantum analogue of the discrete Fourier transform.The quantum Fourier transform is a part of many quantum algorithms, notably Shor's algorithm for factoring and computing the discrete logarithm, the quantum phase estimation algorithm for estimating the eigenvalues of a unitary ...
Quantum mechanics is a fundamental theory that describes the behavior of nature at and below the scale of atoms. [ 2 ]: 1.1 It is the foundation of all quantum physics, which includes quantum chemistry, quantum field theory, quantum technology, and quantum information science. Quantum mechanics can describe many systems that classical physics ...
The Fourier transform can be defined on domains other than the real line. The Fourier transform on Euclidean space and the Fourier transform on locally abelian groups are discussed later in the article. The Fourier transform can also be defined for tempered distributions, dual to the space of rapidly decreasing functions (Schwartz functions). A ...
The quantum Fourier transform is the quantum analogue of the discrete Fourier transform, and is used in several quantum algorithms. The Hadamard transform is also an example of a quantum Fourier transform over an n-dimensional vector space over the field F 2. The quantum Fourier transform can be efficiently implemented on a quantum computer ...
A Fourier series (/ ˈfʊrieɪ, - iər / [1]) is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series, but not all trigonometric series are Fourier series. [2] By expressing a function as a sum of sines and cosines, many problems involving the function become ...
A quantum algorithm for solving this problem exists. This algorithm is, like the factor-finding algorithm, due to Peter Shor and both are implemented by creating a superposition through using Hadamard gates, followed by implementing as a quantum transform, followed finally by a quantum Fourier transform. [3]
Quantum Fourier Transform is the quantum analogue of the classical discrete Fourier transform (DFT), as it takes quantum states represented as superpositions of basis states, and utilizes phase kickback to transform them into frequency-domain representation.
The Fourier transform takes functions in the above space to elements of l 2 (Z), the space of square summable functions Z → C. The latter space is a Hilbert space and the Fourier transform is an isomorphism of Hilbert spaces. [nb 10] Its basis is {e i, i ∈ Z} with e i (j) = δ ij, i, j ∈ Z.