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A framework for reversible circuit complexity

Mathias Soeken, Nabila Abdessaied, Rolf Drechsler

Abstract

Reversible single-target gates are a generalization of Toffoli gates which are a helpful formal representation for the description of synthesis algorithms but are too general for an actual implementation based on some technology. There is an exponential lower bound on the number of Toffoli gates required to implement any reversible function, however, there is also a linear upper bound on the number of single-target gates which can be proven using a constructive proof based on a former presented synthesis algorithm. Since single-target gates can be mapped to a cascade of Toffoli gates, this synthesis algorithm provides an interesting framework for reversible circuit complexity. The paper motivates this framework and illustrates first possible applications based on it.

A framework for reversible circuit complexity

Abstract

Reversible single-target gates are a generalization of Toffoli gates which are a helpful formal representation for the description of synthesis algorithms but are too general for an actual implementation based on some technology. There is an exponential lower bound on the number of Toffoli gates required to implement any reversible function, however, there is also a linear upper bound on the number of single-target gates which can be proven using a constructive proof based on a former presented synthesis algorithm. Since single-target gates can be mapped to a cascade of Toffoli gates, this synthesis algorithm provides an interesting framework for reversible circuit complexity. The paper motivates this framework and illustrates first possible applications based on it.

Paper Structure

This paper contains 11 sections, 5 theorems, 9 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation and Definitions
  3. Boolean Functions
  4. Exclusive Sum of Products
  5. Reversible Circuits
  6. Upper bound for Single-Target Gate Circuits
  7. Lower Bound for Toffoli Gate Circuits
  8. Framework for Circuit Complexity Analysis
  9. Application: "Better than Optimal" Embedding
  10. Conclusions
  11. Acknowledgments.

Key Result

Theorem 1

Let $f\in\mathcal{B}_{n,n}$ be reversible and $x$ a variable in $f$. Then, $f$ can be decomposed as $f=g_2\circ f'\circ g_1$ into three functions $g_1,f',g_2\in\mathcal{B}_{n,n}$ such that $f'$ is a reversible function that does not change $x$, and $g_1$ and $g_2$ can each be realized as a single-ta

Figures (4)

  • Figure 1: Reversible gates
  • Figure 2: Reversible circuit
  • Figure 3: Synthesis based on Young subgroups
  • Figure 4: Reversible circuit complexity

Theorems & Definitions (8)

  • Definition 1: Reversible single-target gate
  • Definition 2: Toffoli gate
  • Example 1
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Lemma 1
  • Corollary 2