A complete set of transformation rules for reversible circuits

TL;DR

This paper introduces the first complete set of transformation rules for reversible circuits, enabling any equivalent circuits to be transformed into each other, which advances reversible logic synthesis in quantum electronic design automation.

Contribution

The paper presents a complete set of five transformation rules for reversible circuits and proves their completeness using a canonical circuit representation.

Findings

Any two equivalent reversible circuits can be transformed into each other using the rules.

Every reversible function has a unique canonical circuit form.

Any reversible circuit can be transformed into its canonical form using the rules.

Abstract

Reversible logic synthesis is a crucial component in quantum electronic design automation. While rule-based methodologies have gained prominence in reversible circuit optimization, the completeness of the transformation rule systems is a longstanding problem in this domain. In this work, we propose the first complete set of transformation rules for reversible circuits, comprising five fundamental rules: any two equivalent reversible circuits can be transformed into each other using the rules. To prove the completeness, a canonical circuit representation for reversible functions is introduced, and we show that every reversible function is computed by a unique reversible circuit in the canonical form, and any reversible circuit can be transformed into its canonical form by applying the rules.