Effects of dynamical phases in Shor's factoring algorithm with operational delays

Abstract

Ideal quantum algorithms usually assume that quantum computing is performed continuously by a sequence of unitary transformations. However, there always exist idle finite time intervals between consecutive operations in a realistic quantum computing process. During these delays, coherent "errors" will accumulate from the dynamical phases of the superposed wave functions. Here we explore the sensitivity of Shor's quantum factoring algorithm to such errors. Our results clearly show a severe sensitivity of Shor's factorization algorithm to the presence of delay times between successive unitary transformations. Specifically, in the presence of these {\it coherent "errors"}, the probability of obtaining the correct answer decreases exponentially with the number of qubits of the work register. A particularly simple phase-matching approach is proposed in this paper to {\it avoid} or suppress these {\it coherent errors} when using Shor's algorithm to factorize integers. The robustness of this phase-matching condition is evaluated analytically or numerically for the factorization of several integers: $4, 15, 21$, and 33.