Nicolas Heurtel

We introduce the first complete equational theory for quantum circuits. More precisely, we introduce a set of circuit equations that we prove to be sound and complete: two circuits represent the same quantum evolution if and only if they can be transformed one into the other using the equations. The proof is based on the properties of multi-controlled gates – that are defined using elementary gates – together with an encoding of quantum circuits into linear optical circuits, for which we introduce a complete axiomatisation. This completeness result lays the formal foundation for the development of compiling tasks like circuit optimisation, hardware constraint satisfaction, and circuit verification.