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We propose an approach to optimally synthesize quantum circuits fromnon-permutative quantum gates such as Controlled-Square-Root-of-Not (i.e.Controlled-V). Our approach reduces the synthesis problem to multiple-valuedoptimization and uses group theory. We devise a novel technique that transformsthe quantum logic synthesis problem from a multi-valued constrainedoptimization problem to a group permutation problem. The transformation enablesus to utilize group theory to exploit the properties of the synthesis problem.Assuming a cost of one for each two-qubit gate, we found all reversiblecircuits with quantum costs of 4, 5, 6, etc, and give another algorithm torealize these reversible circuits with quantum gates.

Exact Synthesis of 3-Qubit Quantum Circuits from Non-Binary Quantum Gates Using Multiple-Valued Logic and Group Theory