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 operator, and algorithms for the hidden subgroup problem. The quantum Fourier transform was discovered by Don Coppersmith. With small modifications to the QFT, it can also be used for performing fast integer arithmetic operations such as addition and multiplication. The quantum Fourier transform can be performed efficiently on a quantum computer with a decomposition into the product of simpler unitary matrices. The discrete Fourier transform on 2 n {\displaystyle 2^{n}} amplitudes can be implemented as a quantum circuit consisting of only O ( n 2 ) {\displaystyle O(n^{2})} Hadamard gates and controlled phase shift gates, where n {\displaystyle n} is the number of qubits. This can be compared with the classical discrete Fourier transform, which takes O ( n 2 n ) {\displaystyle O(n2^{n})} gates (where n {\displaystyle n} is the number of bits), which is exponentially more than O ( n 2 ) {\displaystyle O(n^{2})} . The quantum Fourier transform acts on a quantum state vector (a quantum register), and the classical discrete Fourier transform acts on a vector. Both types of vectors can be written as lists of complex numbers. In the classical case, the vector can be represented with e.g. an array of floating-point numbers, and in the quantum case it is a sequence of probability amplitudes for all the possible outcomes upon measurement (the outcomes are the basis states, or eigenstates). Because measurement collapses the quantum state to a single basis state, not every task that uses the classical Fourier transform can take advantage of the quantum Fourier transform's exponential speedup. The best quantum Fourier transform algorithms known (as of late 2000) require only O ( n log n ) {\displaystyle O(n\log n)} gates to achieve an efficient approximation, provided that a controlled phase gate is implemented as a native operation.
Definition The quantum Fourier transform is the classical discrete Fourier transform applied to the vector of amplitudes of a quantum state, which has length N = 2 n {\displaystyle N=2^{n}} if it is applied to a register of n {\displaystyle n} qubits. The classical Fourier transform acts on a vector ( x 0 , x 1 , … , x N − 1 ) ∈ C N {\displaystyle (x_{0},x_{1},\ldots ,x_{N-1})\in \mathbb {C} ^{N}} and maps it to the vector
( y 0 , y 1 , … , y N − 1 ) ∈ C N {\displaystyle (y_{0},y_{1},\ldots ,y_{N-1})\in \mathbb {C} ^{N}} according to the formula
y k = 1 N ∑ j = 0 N − 1 x j ω N − j k , k = 0 , 1 , 2 , … , N − 1 , {\displaystyle y_{k}={\frac {1}{\sqrt {N}}}\sum _{j=0}^{N-1}x_{j}\omega _{N}^{-jk},\quad k=0,1,2,\ldots ,N-1,}
where ω N = e 2 π i N {\displaystyle \omega _{N}=e^{\frac {2\pi i}{N}}} is an N-th root of unity. Similarly, the quantum Fourier transform acts on a quantum state | x ⟩ = ∑ j = 0 N − 1 x j | j ⟩ {\textstyle |x\rangle =\sum _{j=0}^{N-1}x_{j}|j\rangle } and maps it to a quantum state ∑ j = 0 N − 1 y j | j ⟩ {\textstyle \sum _{j=0}^{N-1}y_{j}|j\rangle } according to the formula
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