Asymptotically Optimal Quantum Circuits for Comparators and Incrementers

• URL: http://arxiv.org/abs/2603.12917v1
• Date: Fri, 13 Mar 2026 11:41:14 GMT
• Title: Asymptotically Optimal Quantum Circuits for Comparators and Incrementers
• Abstract summary: We present quantum circuits for comparison and increment operations that achieve anally optimal gate count of $(n)$ and depth of $(log n)$ over the Clifford+Toffoli gate set.<n>We extend these results to classical-quantum comparators, yielding an improved classical-quantum adder with an optimal qubit count.
• Score: 2.488003578430483
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We present quantum circuits for comparison and increment operations that achieve an asymptotically optimal gate count of $Θ(n)$ and depth of $Θ(\log n)$ over the Clifford+Toffoli gate set, while using a provably minimal number of qubits. We extend these results to classical-quantum comparators, yielding an improved classical-quantum adder with an optimal qubit count. Given the ubiquity of these operations as algorithmic building blocks, our constructions translate directly into reduced circuit complexity for many quantum algorithms. As a notable example, they can be used to improve a space-efficient circuit for Shor's factoring algorithm, reducing circuit depth from $\mathcal{O}(n^3)$ to $\mathcal{O}(n^2 \log^2 n)$ without increasing either the qubit count or the asymptotic gate complexity. Underpinning these results is a general theorem demonstrating how to trade ancilla qubits for control qubits with low overhead in both depth and gate count, providing a broadly applicable tool for quantum circuit design.

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