#nqubitquantumstate

The Quantum Fourier Transform, its history, how it works, architecture, types, features, advantages, disadvantages, applications, and challenges are covered in this article.

Quantum Fourier Transform

The Quantum Fourier Transform (QFT) is a fundamental linear transformation in quantum computing and the quantum version of the DFT, a major digital signal processing tool. The classical DFT maps time and frequency representations to analyse periodic functions, while the QFT does the same to a quantum state. A quantum state's computational basis is replaced with a Fourier basis, a superposition of basis states weighted by Fourier coefficients.

QFT overview:

History

Jean-Baptiste Joseph Fourier invented the classical Fourier transform in the early 19th century. In his 1994 breakthrough paper on factoring large numbers, Peter Shor introduced the quantum version. Don Coppersmith also helped develop it. Since its discovery, quantum technology has shown its promise to improve computers, thanks to QFT.

Its Function and Architecture

The QFT is a quantum circuit that uses an n-qubit quantum state. The general procedure:

When a Hadamard gate is used on the first qubit, the |0⟩ and |1⟩ states are superimposed at the start of the operation.

Next, controlled phase shift gates are used. These gates rotate a qubit's phase based on the state of a “control” qubit and the circuit's qubit placement.

The Hadamard gate and controlled phase shift gate sequence is repeated for all n qubits.

After all gates are applied, qubits are swapped to reverse order to obtain the correct output ordering. PennyLane and others do this. Individual qubit states encode the frequency components of the initial quantum state, while the output state is a tensor product of single-qubit states.

Types and Transformations

Discrete Fourier Transform (DFT): A mathematical method called the DFT may transform a finite sequence of equally spaced data points (like signals) from time to frequency. Showing data's numerous frequency components allows signal analysis, filtering, and compression. A frequent digital signal processing application.

Since the QFT is reversible, its inverse, the Hermitian adjoint of the QFT matrix, can be effectively performed by reversing the QFT circuit. Due to an exponent sign convention, DFT and inverse QFT are mathematically identical.

The Fourier transform is the Hadamard transform for n-qubit quantum registers indexed by the Boolean group if a Hadamard gate is applied to each qubit in parallel. A QFT and initial Hadamard transform are employed in Shor's algorithm.

The Fourier transform can be extended to the quantum environment for groups other than the cyclic group, such as the symmetric group or over a finite field. This is Other Groups/Finite Fields QFT.

Features

The QFT converts quantum state amplitudes from computational to Fourier bases, a key feature. QFT is likely to measure states corresponding to frequency multiples of the inverse period when applied to a periodic function, making it useful for periodic structure problems. This allows frequency spectrum inference from time-domain sequences.

Advantages

For some workloads, the QFT is exponentially faster than the Fast Fourier Transform (FFT). A QFT can be implemented in O(n^2) operations on n qubits, while the traditional FFT requires O(N log N) steps (where N = 2^n).

It is a fundamental subroutine and building block for many quantum algorithms, making them faster.

QFT is a unitary transformation, therefore it can be reversed and keep inner products.

Disadvantages

Measurement Issue: The QFT does not immediately yield all Fourier coefficients. Single output state measurements yield only one potential frequency component. To get all results or find the dominant frequency, the process must be performed numerous times.