Strong equivalence of reversible circuits is coNP-complete

TL;DR

Deciding strong equivalence of reversible circuits with ancilla bits, especially those built from Fredkin or Toffoli gates, is coNP-complete, extending known complexity results for circuit equivalence problems.

Contribution

The paper proves that strong equivalence of reversible circuits with ancilla bits is coNP-complete, using Barrington's theorem, for various universal gate sets.

Findings

Strong equivalence decision problem is coNP-complete.

This complexity holds for circuits with Fredkin, Toffoli, or similar gates.

The result extends known complexity classifications to reversible circuits with ancilla bits.

Abstract

It is well-known that deciding equivalence of logic circuits is a coNP-complete problem. As a corollary, the problem of deciding weak equivalence of reversible circuits, i.e. ignoring the ancilla bits, is also coNP-complete. The complexity of deciding strong equivalence, including the ancilla bits, is less obvious and may depend on gate set. Here we use Barrington's theorem to show that deciding strong equivalence of reversible circuits built from the Fredkin gate is coNP-complete. This implies coNP-completeness of deciding strong equivalence for other commonly used universal reversible gate sets, including any gate set that includes the Toffoli or Fredkin gate.