Generalization of the Bernstein–Vazirani algorithm beyond qubit systems

Here, we generalize the Bernstein–Vazirani algorithm beyond qubit systems. First, we review the Bernstein–Vazirani algorithm for determining a bit string. Second, we discuss the generalized Bernstein–Vazirani algorithm for determining a natural number string. The speed of determining the strings is shown to outperform the best classical case by a factor of the number of the systems in every cases.