In quantum computing, the quantum Fourier transform 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 can be performed efficiently on a quantum computer, with a particular decomposition into a product of simpler unitary matrices. Using a simple decomposition, the discrete Fourier transform can be implemented as a quantum circuit consisting of only Hadamard gates and controlled phase shift gates, where is the number of qubits. This can be compared with the classical discrete Fourier transform, which takes gates (where is the number of bits), which is exponentially more than . However, the quantum Fourier transform acts on a quantum state, whereas the classical Fourier transform acts on a vector, so not every task that uses the classical Fourier transform can take advantage of this exponential speedup.
The best quantum Fourier transform algorithms known today require only gates to achieve an efficient approximation.
Read more about Quantum Fourier Transform: Definition, Circuit Implementation, Example
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