Exact Synthesis of 3-qubit Quantum Circuits from Non-binary Quantum Gates Using Multiple-Valued Logic and 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

Algebraic Characterization of CNOT-Based Quantum Circuits with its Applications on Logic Synthesis

The exponential speed up of quantum algorithms and the fundamental limits of current CMOS process for future design technology have directed attentions toward quantum circuits. In this paper, the matrix specification of a broad category of quantum circuits, i.e. CNOT-based circuits, are investigated. We prove that the matrix elements of CNOT-based circuits can only be zeros or ones. In addition, the columns or rows of such a matrix have exactly one element with the value of 1. Furthermore, we show that these specifications can be used to synthesize CNOT-based quantum circuits. In other words, a new scheme is introduced to convert the matrix representation into its SOP equivalent using a novel quantum-based Karnaugh map extension. We then apply a search-based method to transform the obtained SOP into a CNOT-based circuit. Experimental results prove the correctness of the proposed concept.Comment: 8 pages, 13 figures, 10Th EUROMICRO Conference on Digital System Design, Architectures, Methods and Tools, Germany, 200