A Classical-Quantum Adder with Constant Workspace and Linear Gates

TL;DR

This paper introduces new classical-quantum adders that operate with constant workspace and linear gate complexity, advancing the efficiency of quantum arithmetic circuits for modular operations.

Contribution

It constructs classical-quantum adders with minimal ancillae and linear Toffoli gate complexity, addressing open questions about asymptotic efficiency.

Findings

Adders with 3 clean ancillae and $4n \\pm O(1)$ Toffoli gates for classical offset addition.

Adders with $3n \\pm O(1)$ Toffoli gates using 2 clean and some dirty ancillae.

Conditioned adders require no extra workspace or Toffoli gates.

Abstract

In 2004, Cuccaro et al found a quantum-quantum adder with $O(n)$ gate cost and $O(1)$ ancilla qubits. Since then, it's been an open question whether classical-quantum adders can achieve the same asymptotic complexity. These costs are particularly relevant to modular arithmetic circuits, which often offset by the classically known modulus. In this paper, I construct an adder that uses 3 clean ancillae and $4n \pm O(1)$ Toffoli gates to add a classical offset into a quantum register. I also present an adder with a Toffoli cost of $3n \pm O(1)$ that uses 2 clean ancillae and $n-2$ dirty ancillae. I further show that applying the presented adders conditioned on a control qubit requires no additional workspace or Toffolis.