Document Type


Publication Date



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.


Author's version of an article submitted to the International Journal of Electronics. An abbreviated version was subsequently published in Design, Automation and Test in Europe, Proceedings. Design, Automation and Test in Europe 2005, pp. 434-435, doi:10.1109/DATE.2005.145.

Persistent Identifier