Quantum Circuit Simplification Using Templates

The synthesis of quantum circuits is a complex problem even for circuits with a small number of lines. At present, optimal synthesis is intractable and heuristic methods must be employed. In this paper, we consider the use of templates to simplify a quantum circuit initially found by other means. The templates tool presented here is a general approach to reversible and quantum circuit simplification. We introduce and analyze templates in the general case, and then provide particular details for circuits composed of NOT, CNOT and controlled-sqrt gates. We provide a set of quantum templates for this set of gates and apply them to the circuit specifications generated by earlier heuristic Fredkin and Toffoli gate synthesis algorithms. While the number of templates is quite small, the reduction in quantum cost is often significant.