Quantum Fourier Transform and Its Application in Shor’s Algorithm

Quantum Fourier transform (QFT) plays an eminent role in quantum computation.It creates a superposition of different quantum states, allowing simultaneous calculation, which would take many steps were the same program implemented on a classical computer.The computational speed of a quantum computer is thus boosted dramatically.In this article, we explain the application of QFT in Shor's algorithm, which was proposed by Peter Shor to factor large integers on quantum computers.Specifically, the principle and design of quantum Fourier transform are explained.We stress the subtle distinction between QFT and its inverse.Since former articles did not emphasize it, we hope it could be a supplement to former articles.Our next work is to verify the existing articles on executing Shor's algorithm by conducting two experiments on IBMQ of factoring N = 15 in two ways (a = 7 and a = 11).We find that the effect of quantum entanglement might be crucial to the speed boost of factoring large integers in Shor's algorithm. IntroductionFourier transform is one of the most powerful tools in math and physics, eluding almost no theoretical work from the theory of deferent and epicycles proposed in 3rd century BCE, to the latest theory of solid state physics, signal processing, and optical imaging.Based on the most simple but also most profound belief that everything can be decomposed into a superposition of waves of certain pure frequencies, Fourier transform enables us to perceive the unperceivable space to our naked eyes -the reciprocal space, which is also characterized by a four-vector (ω, , , ), as opposed to the four-vector (t, x, y, z) which characterizes real space.Owing to the mighty power of Fourier transform shown in myriad fields of math and physics, there is no surprise that its quantum version, or rather the discrete version, could also play an eminent role in quantum computation, such as in quantum phase estimation algorithm [1] for estimating the eigenvalues of a unitary operator, and in algorithms for hidden subgroup problem [2].One of the most prominent roles quantum Fourier transform has played in quantum computation is that in Shor's algorithm, proposed in 1994.Running on the quantum circuit, Shor's algorithm can factor large numbers into the multiplication of two integers in polynomial time O((log ) ) [3], much faster than the classical algorithm which takes up sub-exponential time, O( 1.9(log ) 2/3 (log log ) 1/3 ), at best to factor the same number [4].The contrast between the two algorithms will become strikingly stark as N goes up.For example, to factor a 200-digit number N, it will take a common PC 10 3 years.However, it would be nearly instantly broken down by Shor's algorithm were there an available quantum computer.The reason why factoring large numbers is crucial is that modern public key cryptography used in internet communication, such as the RSA scheme [5], is based on the assumption that large integer factorization is computationally intractable.However, this assumption holds for non-quantum computers, but not for quantum computers.Consequently, anyone with a quantum computer can pose an immediate threat to the privacy and security of the Internet nowadays.The efficiency of Shor's algorithm is attributed to two key