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arXiv · 2507.23079

A Classical-Quantum Adder with Constant Workspace and Linear Gates

Craig Gidney
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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.

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Craig Gidney. 2025-07-30. A Classical-Quantum Adder with Constant Workspace and Linear Gates. https://arxiv.org/abs/2507.23079

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