The Quantum Fourier Transform (QFT) is the quantum counterpart to the classical Fourier transform and plays a fundamental role in various quantum algorithms. A classical Fourier transform takes in a function and outputs a function that describes the frequencies present in the original function. A Fourier transform is both this computation and output.
Mathematical properties of the Fourier transform allow for quantum computers to utilize properties of quantum physics and achieve an exponential computation speedup compared to a classical transform. This increase in efficiency is a cornerstone of quantum computing’s potential for solving previous problems deemed intractable for classical alternatives.
It has been known that the Fourier transform can be generalized to functions of variables on Euclidean space, making the spatial transform very natural in quantum mechanics. This allows for manipulation of the computation process and is responsible for speed-ups.
Utilizing the concept of quantum parallelism, multiple qubits being manipulated at once, along with the concepts of superposition, entanglement, and amplitude amplification are key for achieving an exponential speedup.
The QFT is executed as a quantum Fourier transform circuit, which consists of a sequence of Hadamard and controlled phase shift gates. An input quantum state is mapped into a superposition of basis states, encoding the Fourier coefficients. For a quantum register with a N-dimensional state x can be represented by this equation.
Shor’s Algorithm: A quantum algorithm for finding the prime factors of an integer. Shor also developed conceptually similar algorithms that solve the discrete logarithm and period-finding problems. It is one of the few known algorithms with a proven exponential speedup compared to classical solutions. The properties of the quantum fourier transform are responsible for this algorithm’s efficiency.
Quantum Phase Estimation: This algorithm estimates the phase (or eigenvalue) of a quantum system and is both useful and advantageous in terms of quantum computing. Phase estimation is crucial for the efficiency of many quantum algorithms, including Shor’s, by serving as a subroutine to it.
Quantum Machine Learning: Currently QFT is being used in quantum algorithms for feature extraction, data compression, and pattern recognition in high-dimensional quantum datasets. Although applications do not yet provide significant acceleration in the training of machine learning models, QFT provides mathematically proven exponential runtime benefits for aspects of data processing, key to machine learning. As quantum error correction and hardware continues to develop, these advantages will become real and will allow quantum powered models to surpass classically trained models.
The concept of the Fourier transform is from classical mathematics, attributed to the work of Jean-Baptiste Joseph Fourier in the early 19th century. Fourier introduced the idea of expressing functions in terms of sums of sin and cosine waves. This laid the groundwork for signal processing and various other fields of engineering.
The development of the quantum Fourier transform is closely tied to Shor’s algorithm, proposed by Peter Shor in 1994. Additionally, Don Coppersmith improved the QFT and is considered a key contributor to concepts originally suggested by Shor. Since then QFT has been used to provide an exponential speed up while solving problems including the hidden subgroup problem and estimating the eigenvalues of a unitary operator.
Although little has changed fundamentally since its discovery, QFT continues to be essential in proving the ability of quantum technologies to improve computing drastically. Researchers today still search for new applications of QFT to achieve advantages over classical alternatives and use the concept as a basis for algorithm development. The runtime advantage provided serves as a key piece of the puzzle in developing quantum solutions that may revolutionize energy use, cryptography, machine learning, and more in the future to come.
.svg){kind=small}
