Quantum Algorithm Uncertainty Principles

Abstract

Previously, Bennet and Feynman asked if Heisenberg's uncertainty principle puts a limitation on a quantum computer (Quantum Mechanical Computers, Richard P. Feynman, Foundations of Physics, Vol. 16, No. 6, p597-531, 1986). Feynman's answer was negative. In this paper, we will revisit the same question for the discrete time Fourier transform uncertainty principle. We will show that the discrete time Fourier transform uncertainty principle plays a fundamental role in showing that Shor's type of quantum algorithms has efficient running time and conclude that the discrete time uncertainty principle is an aid in our current formulation and understanding of Shor's type of quantum algorithms. It turns out that for these algorithms, the probability of measuring an element in some set $T$ (at the end of the algorithm) can be written in terms of the time-limiting and band-limiting operators from finite Fourier analysis. Associated with these operators is the finite Fourier transform uncertainty principle. The uncertainty principle provides a lower bound for the above probability. We will derive lower bounds for these types of probabilities in general. We will call these lower bounds quantum algorithm uncertainty principles or QAUP. QAUP are important because they give us some sense of the probability of measuring something desirable. We will use these lower bounds to derive Shor's factoring and discrete log algorithms.