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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 : C → 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. [ABSTRACT FROM AUTHOR]

Two quantum algorithms of finding the roots of a polynomial function f(x) = xm + am− 1xm− 1 +... + a1x + a0 are discussed by using the Bernstein-Vazirani algorithm. One algorithm is presented in the modulo 2. The other algorithm is presented in the modulo d. Here all the roots are in the integers Z. The speed of solving the problem is shown to outperform the best classical case by a factor of m in both cases. [ABSTRACT FROM AUTHOR]

We present a new method of more speedily calculating a multiplication by using the generalized Bernstein-Vazirani algorithm and many parallel quantum systems. Given the set of real values {a1,a2,a3,…,aN} and a function g:R→{0,1}, we shall determine the following values {g(a1),g(a2),g(a3),…,g(aN)} simultaneously. The speed of determining the values is shown to outperform the classical case by a factor of N. Next, we consider it as a number in binary representation; M1 = (g(a1),g(a2),g(a3),…,g(aN)). By using M parallel quantum systems, we have M numbers in binary representation, simultaneously. The speed of obtaining the M numbers is shown to outperform the classical case by a factor of M. Finally, we calculate the product; M1×M2×⋯×MM. The speed of obtaining the product is shown to outperform the classical case by a factor of N × M. [ABSTRACT FROM AUTHOR]