CAPACITIVE COMPLEXITY OF DETERMINING GCD IN THE SHOR S ALGORITHM

Abstract

The article analyzes the results of finding the period r of the function y = axmodM (a is a random number) which is used in the Shor's factorization algorithm for quantum computers. The module M is the product of two primes p and q. The article analyzes the solutions r obtained for various a, for which the capacitive complexity H of finding the greatest common divisor GCD(ar/2 + 1, M) is the least. A digital quantum computer is a classic processor and its digital quantum coprocessor. A digital quantum coprocessor with hundreds and thousands of digital qubits can be implemented in one programmable logic integrated circuit FPGA. In the Shor’s algorithm, the factorization problem of the number M reduces to the problem of determining the period r of the function y. It is known that GCD(ar/2 + 1, M) can be a divisor of the number M The task of the quantum coprocessor in implementing the Shor’s algorithm is to find the period r. After that it is necessary to find the GCD. Since for random a the problem of finding the period r has many solutions, these solutions can be compared by the value of one of the arguments when finding the GCD - the number ar / 2 . In this case, H = (rlog2a)/2 is taken for analysis. It approximately represents the bit depth of binary codes that a classic computer will have to process when determining the GCD. H can vary over a wide range from tens to thousands of bits even for small values of M. In this research the period r, which ensures the least complexity of the subsequent task of finding the GCD, is most often a solution for a = 3 and a = 2, but it can also occur often with other values of a. To clarify the revealed patterns, especially for large M, it is necessary to conduct additional research.