Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum adders: on the structural link between the ripple-carry and carry-lookahead techniques

Maxime Remaud

TL;DR

The paper addresses the lack of a unified framework for quantum adders by introducing the Toffoli ladder as a common subroutine that drives both ripple-carry and carry-lookahead designs. It analyzes three ladder implementations (linear, polylogarithmic, logarithmic) and two adder structures (original and space-optimized), yielding six adders of which two are novel; notably, a new carry-lookahead variant achieves improved performance. The work clarifies how existing adders are interrelated and demonstrates that selecting ladder depth together with the underlying adder structure yields trade-offs in ancilla usage, depth, and gate count, including no-ancilla constructions with sublinear depth. This unifying perspective advances quantum arithmetic circuit design by enabling targeted optimizations across depth, ancilla, and connectivity constraints, with potential impact on cryptography, simulation, and quantum algorithms requiring efficient addition.

Abstract

This paper is motivated by two key observations. First, Toffoli ladders can be implemented in three distinct ways: with linear or polylogarithmic depth using no ancilla, or with logarithmic depth using ancilla qubits. Second, two fundamental structural approaches to designing addition algorithms can be identified in several well-known quantum adders. At their core is the Toffoli ladder, and both provide a clear and simple connection between ripple-carry and carry-lookahead adder designs. Combining these two structures with the three Toffoli ladder implementations yields six quantum adders: four are well-known and two novel. Notably, one of the novel designs is a carry-lookahead adder that outperforms previous approaches.

Quantum adders: on the structural link between the ripple-carry and carry-lookahead techniques

TL;DR

The paper addresses the lack of a unified framework for quantum adders by introducing the Toffoli ladder as a common subroutine that drives both ripple-carry and carry-lookahead designs. It analyzes three ladder implementations (linear, polylogarithmic, logarithmic) and two adder structures (original and space-optimized), yielding six adders of which two are novel; notably, a new carry-lookahead variant achieves improved performance. The work clarifies how existing adders are interrelated and demonstrates that selecting ladder depth together with the underlying adder structure yields trade-offs in ancilla usage, depth, and gate count, including no-ancilla constructions with sublinear depth. This unifying perspective advances quantum arithmetic circuit design by enabling targeted optimizations across depth, ancilla, and connectivity constraints, with potential impact on cryptography, simulation, and quantum algorithms requiring efficient addition.

Abstract

This paper is motivated by two key observations. First, Toffoli ladders can be implemented in three distinct ways: with linear or polylogarithmic depth using no ancilla, or with logarithmic depth using ancilla qubits. Second, two fundamental structural approaches to designing addition algorithms can be identified in several well-known quantum adders. At their core is the Toffoli ladder, and both provide a clear and simple connection between ripple-carry and carry-lookahead adder designs. Combining these two structures with the three Toffoli ladder implementations yields six quantum adders: four are well-known and two novel. Notably, one of the novel designs is a carry-lookahead adder that outperforms previous approaches.

Paper Structure

This paper contains 13 sections, 5 theorems, 4 equations, 7 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ladders
  4. Addition
  5. Notation
  6. Implementation of the Toffoli ladder
  7. Linear depth
  8. Polylogarithmic depth
  9. Logarithmic depth
  10. New Quantum Carry-Lookahead Adder
  11. The "original" structure
  12. The space optimized structure
  13. Conclusion

Key Result

Lemma 1

There exists a Toffoli circuit that implements $\mathsf{L}_2^{(n)}$ with a Toffoli-depth of $n$ and a Toffoli-count of $n$, without any ancilla qubit.

Figures (7)

  • Figure 1: Venn diagram of in-place quantum reversible adders with classical logic only.
  • Figure 2: Linear depth implementation of the operator $\mathsf{L}_2^{(7)}$.
  • Figure 3: Polylogarithmic depth implementation of the operator $\mathsf{L}_2^{(7)}$RV25.
  • Figure 4: Logarithmic depth implementation of the operator $\mathsf{L}_2^{(7)}$DKR06.
  • Figure 5: On the left, the subroutine used in Vedral $et \; al.$'s paper. On the right, the equivalent circuit used in the structure described in Algorithm \ref{['alg:main1']}.
  • ...and 2 more figures

Theorems & Definitions (10)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Lemma 2: Lemma 4 in RV25
  • Lemma 3: Section 3 in DKR06
  • proof
  • Remark 1
  • Lemma 4: Lemma 2 in RV25
  • Theorem 1
  • proof