O ct 2 00 2 QUANTUM ALGORITHM UNCERTAINTY PRINCIPLES

Abstract. In this paper, we will be concerned with a class of quantum algorithms that contains Shor’s factoring and discrete log 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. The main goal of this paper is to derive lower bounds for these types of probabilities. 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. It turns out that these two applications are quite natural and elegant. We will also show that the application of Hallgren’s [Ph.D. Thesis, Quantum Fourier Sampling, the Hidden Subgroup Problems and Beyond (2000)] and Hales and Hallgren’s [Quantum Fourier Sampling Simplified, Thirty-First Annual ACM Symposium on Theory of Computing (STOC), May 1999] quantum Fourier sampling theorems (QFST) to the above class of quantum algorithms is an instance of QAUP.