In this paper we describe an efficient involutive algorithm for constructing Gröbner bases of polynomial ideals. The algorithm is based on the concept of involutive monomial division which restricts the conventional division in a certain way. In the presented algorithm a reduced Gröbner basis is the internally fixed subset of an involutive basis, and having computed the later, the former can be output without any extra computational costs. We also discuss some accounts of experimental superiority of the involutive algorithm over Buchberger's algorithm.

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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.

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