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The quantum Fourier transform is the classical discrete Fourier transform applied to the vector of amplitudes of a quantum state, which has length if it is applied to a register of qubits. The classical Fourier transform acts on a vector and maps it to the vector according to the formula: where is an N -th root of unity.
The output of the transform is a complex -valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function.
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 quantum mechanical state can be fully represented in terms of either variables, and the transformation used to go between position and momentum spaces is, in each of the three cases, a variant of the Fourier transform. The table uses bra-ket notation as well as mathematical terminology describing Canonical commutation relations (CCR).
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.
Thus, this quantum-mechanical bound precludes a Wigner function which is a perfectly localized δ-function in phase space, as a reflection of the uncertainty principle. [6] 10. The Wigner transformation is simply the Fourier transform of the antidiagonals of the density matrix, when that matrix is expressed in a position basis. [7]
The Fourier–Plancherel transform defined by ^ = ¯ () extends to a C*-isomorphism from the group C*-algebra C*(G) of G and C 0 (G ^), i.e. the spectrum of C*(G) is precisely G ^. When G is the real line R , this is Stone's theorem characterizing one-parameter unitary groups.
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.