Reversible addition circuit using one ancillary bit with application to quantum computing

TL;DR

This paper presents a quantum addition circuit using only one ancillary bit and Toffoli gates, showing that quantum-specific operations are not necessary for reversible addition, with a trade-off in circuit depth.

Contribution

It introduces a reversible in-place addition circuit with a single ancillary bit using classical reversible computing techniques, avoiding quantum rotation gates.

Findings

Uses O(n^3) Toffoli gates for addition

Requires only one ancillary bit

Demonstrates classical techniques suffice for quantum addition

Abstract

Most of the work on implementing arithmetic on a quantum computer has borrowed from results in classical reversible computing (e.g. [VBE95], [BBF02], [DKR04]). These quantum networks are inherently classical, as they can be implemented with only the Toffoli gate. Draper [D00] has proposed an inherently "quantum" network for addition based on the quantum Fourier transform. His approach has the advantage that it requires no carry qubits (the previous approaches required O(n) carry qubits). The network in [D00] uses quantum rotation gates, which must either be implemented with exponential precision, or else be approximated. In this paper I give a network of O(n^3) Toffoli gates for reversibly performing in-place addition with only a single ancillary bit, demonstrating that inherently quantum techniques are not required to achieve this goal (provided we are willing to sacrifice quadratic circuit depth). After posting the original version of this note it was pointed out to me by C. Zalka that essentially the same technique for addition was used in [BCD+96]. The scenario in that paper was different, but it is clear how the technique they described generalizes to that in this paper.