A complete set of transformation rules for reversible circuits

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.