Quantum Communication Based on an Algorithm of Determining a Matrix

Abstract

Let us consider the following game; Bob has a N x M matrix (N rows and M columns) but Alice does not know what matrix he has. The goal is of knowing the unknown matrix. How many queries does she need? In the classical case, she needs N x M queries. In the quantum case, she needs just a query. We propose an algorithm for determining the N x M matrix (N rows and M columns). First, we discuss an algorithm for determining an integer string. The algorithm presented here has the following structure. Given the set of real values {a(1), a(2), a(3),..., a(N)} and a special function g, we determine N values of the function g(a(1)), g(a(2)), g(a(3)),..., g(a(N)) simultaneously. The speed of determining the string is shown to outperform the best classical case by a factor of N. Next, we consider it as a column of the matrix; C-1 = (g(a(1)), g(a(2)), g(a(3)),..., g(a(N))) = (a(11), a(21),..., a(n1)). By using M parallel quantum systems, we have M columns simultaneously, C-1, C-2,..., C-M. The speed of obtaining the M columns (the matrix) is shown to outperform the classical case by a factor of N x M. This implies she needs just a query.