An algorithmic approach to solving polynomial equations associated with quantum circuits

Abstract

In this paper we present two algorithms for reducing systems of multivariate polynomial equations over theˇniteˇeld F2 to the canonical triangular form called lexicographical Gré obner basis. This triangular form is the most appropriate forˇnding solutions of the system. On the other hand, the system of polynomials over F2 whose variables also take values in F2 (Boolean polynomials) completely describes the unitary matrix generated by a quantum circuit. In particular, the matrix itself can be computed by counting the number of solutions (roots) of the associated polynomial system. Thereby, efˇcient construction of the lexicographical Gré obner bases over F2 associated with quantum circuits gives a method for computing their circuit matrices that is alternative to the direct numerical method based on linear algebra. We compare our implementation of both algorithms with some other software packages available for computing Gré obner bases over F2.