Quantum Algorithm for Determining a Complex Number String

Abstract

Here, we discuss the generalized Bernstein-Vazirani algorithm for determining a complex number string. The generalized algorithm presented here has the following structure. Given the set of complex values {a1, a2, a3,…, aN} and a special function $$ g:\mathbf{C}\to \mathbf{C} $$, we determine N real parts of values of the function l(a1), l(a2), l(a3),…, l(aN) and N imaginary parts of values of the function h(a1), h(a2), h(a3),…, h(aN) simultaneously. That is, we determine the N complex values g(aj) = l(aj) + ih(aj) simultaneously. We mention the two computing can be done in parallel computation method simultaneously. The speed of determining the string of complex values is shown to outperform the best classical case by a factor of N. Additionally, we propose a method for calculating many different matrices A, B, C,... into g(A), g(B), g(C),... simultaneously. The speed of solving the problem is shown to outperform the classical case by a factor of the number of the elements of them. We hope our discussions will give a first step to the quantum simulation problem.