[Paper Review] Ancilla-free Reversible Logic Synthesis via Sorting

Anupam Chattopadhyay, Sharif Md Khairul Hossain|arXiv (Cornell University)|May 19, 2016

Quantum Computing Algorithms and Architecture10 references3 citations

TL;DR

This paper proposes RevCol, a novel ancilla-free reversible logic synthesis method inspired by radix sort, which transforms reversible logic synthesis into a sorting problem. By leveraging column-wise sorting principles, RevCol achieves a 54% improvement in logical depth over the MMD method, with a hybrid framework further enhancing performance on benchmark functions.

ABSTRACT

Reversible logic synthesis is emerging as a major research component for post-CMOS computing devices, in particular Quantum computing. In this work, we link the reversible logic synthesis problem to sorting algorithms. Based on our analysis, an alternative derivation of the worst-case complexity of generated reversible circuits is provided. Furthermore, a novel column-wise reversible logic synthesis method, termed RevCol, is designed with inspiration from radix sort. Extending the principles of RevCol, we present a hybrid reversible logic synthesis framework. The theoretical and experimental results are presented. The results are extensively benchmarked with state-of-the-art ancilla-free reversible logic synthesis methods.

Motivation & Objective

• To establish a theoretical and practical link between sorting algorithms and reversible logic synthesis.
• To design an ancilla-free synthesis method that minimizes logical depth and quantum cost.
• To develop a scalable hybrid framework combining RevCol with permutation optimization for improved circuit performance.
• To benchmark RevCol and its hybrid variant against state-of-the-art ancilla-free methods, including MMD and BDD-based approaches.

Proposed method

• RevCol uses a column-wise sorting approach inspired by radix sort to synthesize reversible circuits from truth table representations.
• The method maps each column of the truth table to a sorting operation, using generalized Toffoli gates to perform bit-level swaps.
• A hybrid synthesis framework combines RevCol with input/output permutation techniques to further reduce gate and quantum costs.
• The approach avoids ancilla lines entirely, ensuring ancilla-free synthesis while maintaining low logical depth.
• Theoretical analysis derives worst-case circuit complexity based on sorting algorithm behavior.
• The hybrid method dynamically selects synthesis strategies (e.g., cycle-based or RevCol) based on function patterns to optimize performance.

Experimental results

Research questions

• RQ1 Can sorting algorithms be systematically applied to reversible logic synthesis to improve circuit depth and cost?
• RQ2 How does a radix-sort-inspired column-wise approach compare to existing ancilla-free methods in terms of gate count, quantum cost, and logical depth?
• RQ3 What is the theoretical worst-case complexity of circuits generated by this sorting-based synthesis method?
• RQ4 Can a hybrid framework combining RevCol with permutation techniques outperform standalone RevCol and other state-of-the-art methods?
• RQ5 To what extent do function-specific patterns influence the effectiveness of the RevCol and hybrid synthesis approaches?

Key findings

• RevCol achieves a 54% improvement in logical depth compared to the MMD method on benchmark functions.
• The hybrid RevCol framework reduces quantum cost by 15% on average compared to MMD, with significant improvements on specific functions like rand4_4 and rand5_4.
• For 4-bit functions, RevCol matches or exceeds optimal gate counts in 7 out of 10 benchmarks, with only minor increases in gate count.
• The hybrid method outperforms BDD-based synthesis in both gate count and quantum cost, with fewer lines and lower QC on all tested small functions.
• On larger functions like hwb9 and urf4, the hybrid method achieves lower quantum cost than ancilla-free BDD synthesis, despite higher gate counts in some cases.
• The performance gains are attributed to function-specific patterns and effective synthesis strategy selection within the hybrid framework.