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

TL;DR

This paper introduces an optimal method for synthesizing 3-qubit quantum circuits from non-binary gates by transforming the problem into a group permutation problem using multiple-valued logic and group theory.

Contribution

It presents a novel transformation technique that reduces quantum logic synthesis to a group permutation problem, enabling efficient synthesis of circuits from non-permutative gates.

Findings

Identified all reversible circuits with quantum costs of 4, 5, 6, etc.

Developed an algorithm to realize these circuits with quantum gates.

Utilized group theory to exploit properties of the synthesis problem.

Abstract

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 group permutation problem. The transformation enables us 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 reversible circuits with quantum costs of 4, 5, 6, etc, and give another algorithm to realize these reversible circuits with quantum gates.