A complete set of transformation rules for reversible circuits
Shiguang Feng, Lvzhou Li
TL;DR
This work presents the first complete transformation-rule set $\mathcal{RC}$ for reversible circuits, coupling five fundamental rules with a canonical form derived from a Hamiltonian path on the $n$-dimensional hypercube. It proves that every $n$-bit reversible function has a unique canonical circuit and that any circuit can be transformed into this canonical form, thereby ensuring completeness: any two equivalent circuits can be connected through the canonical form. The canonical form is constructed via a universal gate set $\Delta_{\mathbb{H}}$ associated with a Hamiltonian path $\mathbb{H}$, with a constructive algorithm running in $O(n\cdot 4^n)$ time. While offering a strong theoretical guarantee for rule-based optimization, the paper also notes potential exponential blow-up in canonical form size and discusses scope limitations to ancilla-free circuits and directions for practical heuristic enhancements. These results provide a rigorous foundation for completeness in reversible circuit optimization within quantum EDA contexts.
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.