Realizing 4-input Functions with the Minimum Toffoli Gate Count

This paper shows that the minimum number of Toffoli gates to realize any 4-input function is at most five for the first time as far as we know. This result should be not only theoretically interesting but also practically useful because sub-circuits to calculate 4-input functions are used in the practical quantum Boolean circuit design framework based on LUT-network synthesis, and the state-of-the-art heuristic method requires an average of 9.29 Toffoli gates to realize 4-input functions. Because previous SAT-based exact design methods cannot find the minimum number of Toffoli gates to implement a target function, we introduce a novel idea, complex Toffoli gates, which are very useful to significantly reduce the search space to find a quantum Boolean circuit with the minimum number of Toffoli gates; we can successfully formulate the problem as an SMT problem by using complex Toffoli gates, which should be interesting on their own.