A complete transformation rule set and a minimal equation set for CNOT-based 3-qubit quantum circuits (Draft)

TL;DR

This paper presents a comprehensive set of transformation rules and a minimal equation set for CNOT-based 3-qubit quantum circuits, enabling efficient circuit equivalence checking through normal form comparison.

Contribution

It introduces a complete, terminating, and confluent transformation rule set and identifies a minimal equation set for simplifying CNOT-based quantum circuits.

Findings

Normal forms enable easy circuit equivalence checking.

114 transformation rules form a complete set.

A minimal equation set was identified.

Abstract

We introduce a complete transformation rule set and a minimal equation set for controlled-NOT (CNOT)-based quantum circuits. Using these rules, quantum circuits that compute the same Boolean function are reduced to the same normal form. We can thus easily check the equivalence of circuits by comparing their normal forms. By applying the Knuth-Bendix completion algorithm to a set of modified 18 equations introduced by Iwama et al. 2002, we obtain a complete transformation rule set (i.e., a set of transformation rules with the properties of `termination' and `confluence'). Our transformation rule set consists of 114 rules. Moreover, we discovered a minimal combination of equations for the initial equation set.