Synthesizing reversible functions via bounded model checking

We consider the problem of synthesizing optimized circuits for reversible functions using NOT, CNOT and Toffoli gates (NCT), which has important applications in circuit design and quantum cryptography. By defining the state space and state transitions from NCT gates, we transform the synthesis problem into a reachability problem of finding the shortest path connecting the identity function and the targeted function that is to be synthesized. Using Bounded Model Checking (BMC), we propose an algorithm for synthesizing the optimized circuit for reversible functions. For the synthesis of linear reversible functions, we design another method by directly parametrizing the linear invertible matrix. Our methods can produce optimal synthesis of reversible circuits when the number of bits is small, and also can be used as a subroutine of heuristic methods for large bits. Experimental results show the effectiveness of our methods.