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 {a(1), a(2), a(3), horizontal ellipsis , a(N)} and a special function g:C, we determine N real parts of values of the function l(a(1)), l(a(2)), l(a(3)), horizontal ellipsis , l(a(N)) and N imaginary parts of values of the function h(a(1)), h(a(2)), h(a(3)), horizontal ellipsis , h(a(N)) simultaneously. That is, we determine the N complex values g(a(j)) = l(a(j)) + ih(a(j)) 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.