An algorithm for constructing polynomial systems whose solution space characterizes quantum circuits

An algorithm and its first implementation in C# are presented for assembling arbitrary quantum circuits on the base of Hadamard and Toffoli gates and for constructing multivariate polynomial systems over the finite field Z2 arising when applying the Feynman's sum-over-paths approach to quantum circuits. The matrix elements determined by a circuit can be computed by counting the number of common roots in Z2 for the polynomial system associated with the circuit. To determine the number of solutions in Z2 for the output polynomial system, one can use the Grobner bases method and the relevant algorithms for computing Grobner bases.