06 04 00 1 v 2 2 7 F eb 2 00 8 Quantum Circuit Simplification and Level Compaction ∗

Quantum circuits are time dependent diagrams describing th e process of quantum computation. Usually, a quantum algorithm must be mapped into a quantum circuit. Optimal synthesis of quantum circuits is intractable and heuristic methods must be employed. With the use of heuristi cs, the optimality of circuits is no longer guaranteed. In this paper, we consider a local optimization technique ba sed on templates to simplify and reduce the depth of non-optimal quantum circuits. We present and analyze templ ates in the general case, and provide particular details for the circuits composed of NOT, CNOT and controlledsqrt-of-NOT gates. We apply templates to optimize various common circuits implementing multiple control Toffoli gat es and quantum Boolean arithmetic circuits. We also show how templates can be used to compact the number of levels of a q uantum circuit. The runtime of our implementation is small while the reduction in number of quantum gates and nu mber of levels is significant.