Phase-matching approach to eliminate the dynamical phase error in Shor's factoring algorithm
Wei Li, Xiao Li, Xuedong Hu, Franco Nori · arXiv (Cornell University) · 2003
Ideal quantum algorithms usually assume that quantum computing is performed continuously by a sequence of unitary transformations. However, there always exist finite time intervals of idling between consecutive operations in a realistic quantum computing process. During these delays, the interaction Hamiltonian is zero, and only single-qubit rotations occur. Therefore, 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 an acute and severe sensitivity of Shor's factorization algorithm to the presence of delay times between successive unitary transformations. Specifically, in the presence of these delays, the probability of obtaining the correct answer decreases exponentially with the number of qubits of the work register. Moreover, we prove that the probability of obtaining correct answers for Shor's factoring algorithm decreases fast as the total effective delay time between successive unitary transformations increases. A particularly simple phase-matching approach is proposed in this paper to eliminate or suppress these phase errors when using Shor's algorithm to factorize integers. The robustness of this phase-matching condition is evaluated numerically for several integers: 15,21, and 33.