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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. 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 ...
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).
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 ...
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.
Linearity. The Schrödinger equation is a linear differential equation, meaning that if two state vectors and are solutions, then so is any linear combination of the two state vectors where a and b are any complex numbers. [13]: 25 Moreover, the sum can be extended for any number of state vectors.
On a quantum computer, to factor an integer , Shor's algorithm runs in polynomial time, meaning the time taken is polynomial in . [6] It takes quantum gates of order using fast multiplication, [7] or even utilizing the asymptotically fastest multiplication algorithm currently known due to Harvey and Van Der Hoven, [8] thus demonstrating that ...
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.