[Paper Review] Ancilla-free synthesis of large reversible functions using binary decision diagrams

Mathias Soeken, Laura Tague+2 more|arXiv (Cornell University)|Aug 18, 2014

Quantum Computing Algorithms and Architecture18 references4 citations

TL;DR

This paper presents an ancilla-free synthesis method for large reversible functions using symbolic Boolean representations via binary decision diagrams (BDDs). By leveraging BDDs to efficiently compute control functions in a decomposition based on Young subgroups, the approach enables scalable synthesis of functions with more than 15 variables—previously infeasible with truth table-based methods—resulting in significantly smaller circuits with reduced gate count and quantum cost.

ABSTRACT

The synthesis of reversible functions has been an intensively studied research area in the last decade. Since almost all proposed approaches rely on representations of exponential size (such as truth tables and permutations), they cannot be applied efficiently to reversible functions with more than 15 variables. In this paper, we propose an ancilla-free synthesis approach based on Young subgroups using symbolic function representations that can efficiently be implemented with binary decision diagrams (BDDs). As a result, the algorithm not only allows to synthesize large reversible functions without adding extra lines, called ancilla, but also leads to significantly smaller circuits compared to existing approaches.

Motivation & Objective

• To address the scalability limitation of existing ancilla-free reversible function synthesis methods that rely on exponential-size representations like truth tables.
• To enable synthesis of large reversible functions (beyond 15 variables) without adding ancilla lines.
• To reduce circuit size and quantum cost compared to state-of-the-art approaches by using symbolic manipulation via BDDs.
• To support partial functions and improve synthesis quality through variable ordering heuristics.
• To provide open-source implementations for both the proposed BDD-based method and the original truth table-based approach.

Proposed method

• The method uses characteristic functions of reversible functions represented as BDDs to enable symbolic manipulation.
• It applies a decomposition technique based on co-factors of the characteristic function, derived from the Young subgroup structure, to express the function as a sequence of single-target gates.
• Control functions l and r for each gate are computed symbolically using BDD operations such as existential quantification and conjunction.
• Variable ordering heuristics (e.g., greedy, Hamming distance) are applied to minimize gate count and circuit cost.
• The approach supports partial functions by extending the characteristic function representation to handle unspecified input-output mappings.
• Circuits are synthesized using single-target gates, which can be mapped to Toffoli gates, with the final circuit being ancilla-free and linear in gate count.

Experimental results

Research questions

• RQ1 Can BDD-based symbolic manipulation enable ancilla-free synthesis of large reversible functions beyond the 15-variable limit of truth table-based methods?
• RQ2 Does the use of BDDs for computing control functions lead to smaller circuits compared to traditional truth table-based synthesis?
• RQ3 How do different variable ordering heuristics affect the size and cost of the resulting reversible circuits?
• RQ4 Can the proposed method be extended to handle partial reversible functions, where not all input-output mappings are specified?
• RQ5 What is the scalability of the BDD-based approach in terms of runtime and circuit size for large reversible functions?

Key findings

• The proposed BDD-based method successfully synthesizes reversible functions with up to 30 variables, significantly exceeding the 15-variable limit of truth table-based approaches.
• The method produces circuits with substantially fewer Toffoli gates and lower quantum cost compared to existing state-of-the-art synthesis techniques.
• The run-time increases with function complexity, but the approach remains feasible for large functions due to the compact BDD representation.
• The greedy variable ordering heuristic achieves the best trade-off between circuit size and runtime, outperforming the Hamming-based approach in most cases.
• The algorithm supports partial functions by extending the characteristic function representation, enabling synthesis where input-output mappings are incomplete.
• The experimental evaluation confirms that the method scales effectively, with runtime growing polynomially in practice despite theoretical exponential complexity in worst-case scenarios.