Quantum Communication Based on an Algorithm of Determining a Matrix

Abstract

Let us consider the following game; Bob has a N × 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 × M queries. In the quantum case, she needs just a query. We propose an algorithm for determining the N × 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 {a1, a2, a3,…, aN} and a special function g, we determine N values of the function g(a1), g(a2), g(a3),…, g(aN) 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; C1 = (g(a1), g(a2), g(a3),…, g(aN)) = (a11, a21,..., an1). By using M parallel quantum systems, we have M columns simultaneously, C1, C2,..., CM. The speed of obtaining the M columns (the matrix) is shown to outperform the classical case by a factor of N × M. This implies she needs just a query.