Reversible computing, Boolean algebra, Logic design, Group theory
We propose an approach to optimally synthesize quantum circuits from non-permutative quantum gates such as Controlled-Square-Root–of-Not (i.e. Controlled-V). Our approach reduces the synthesis problem to multiple-valued optimization and uses group theory. We devise a novel technique that transforms the quantum logic synthesis problem from a multi-valued constrained optimization problem to a permutable representation. The transformation enables us to utilize group theory to exploit the symmetric properties of the synthesis problem. Assuming a cost of one for each two-qubit gate, we found all reversible circuits with quantum costs of 4, 5, 6, etc, and give another algorithm to realize these reversible circuits with quantum gates. The approach can be used for both binary permutative deterministic circuits and probabilistic circuits such as controlled random number generators and hidden Markov models.
Guowu Yang, William Hung, Xiaoyu Song, and Marek Perkowski, “Exact synthesis of 3-qubit quantum circuits from non-binary quantum gates using multiple-valued logic and group theory,” submitted to International Journal of Electronics, 2004.