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

TL;DR

This paper provides an algebraic characterization of CNOT-based quantum circuits, proving their matrix properties and introducing a synthesis method using a quantum Karnaugh map extension for logic circuit design.

Contribution

It introduces a novel algebraic framework for CNOT-based circuits and a new synthesis scheme converting matrix representations into quantum circuits.

Findings

Matrix elements of CNOT-based circuits are only zeros or ones

Each row and column of the matrix has exactly one '1' element

The proposed synthesis method accurately converts matrices into CNOT circuits

Abstract

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.