Optimizing Quantum Circuits via Letterplace Ideals

In this article, we discuss an algebraic approach for optimizing quantum circuits. This approach is based on the construction of a Gröbner basis (standard basis) on a set of equations describing the equivalences of gate operations. It enables us to derive additional equivalences up to a bound and to use generated equivalences as rewriting rules of quantum circuits. If such rules are generated sufficiently large, the ambiguity in the rewriting circuit, such as the occurrence of multiple different results, could be avoided.