A Study of Optimal 4-bit Reversible Toffoli Circuits and Their Synthesis

TL;DR

This paper introduces algorithms for optimal synthesis of 4-bit reversible Toffoli circuits, enabling efficient generation and analysis of all optimal solutions, with implications for quantum circuit design and optimization.

Contribution

The paper presents two algorithms for synthesizing optimal 4-bit reversible circuits and all their optimal implementations, addressing the large search space with novel techniques.

Findings

Synthesized all optimal 4-bit permutations.

Identified the hardest permutations to synthesize.

Analyzed distribution of optimal circuits.

Abstract

Optimal synthesis of reversible functions is a non-trivial problem. One of the major limiting factors in computing such circuits is the sheer number of reversible functions. Even restricting synthesis to 4-bit reversible functions results in a huge search space (16! {\approx} 2^{44} functions). The output of such a search alone, counting only the space required to list Toffoli gates for every function, would require over 100 terabytes of storage. In this paper, we present two algorithms: one, that synthesizes an optimal circuit for any 4-bit reversible specification, and another that synthesizes all optimal implementations. We employ several techniques to make the problem tractable. We report results from several experiments, including synthesis of all optimal 4-bit permutations, synthesis of random 4-bit permutations, optimal synthesis of all 4-bit linear reversible circuits, synthesis of existing benchmark functions; we compose a list of the hardest permutations to synthesize, and show distribution of optimal circuits. We further illustrate that our proposed approach may be extended to accommodate physical constraints via reporting LNN-optimal reversible circuits. Our results have important implications in the design and optimization of reversible and quantum circuits, testing circuit synthesis heuristics, and performing experiments in the area of quantum information processing.