Asymptotically Optimal Quantum Circuits for Comparators and Incrementers
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866914390980165632 |
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| author | Vandaele, Vivien |
| author_facet | Vandaele, Vivien |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_12917 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotically Optimal Quantum Circuits for Comparators and Incrementers Vandaele, Vivien Quantum Physics 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. |
| title | Asymptotically Optimal Quantum Circuits for Comparators and Incrementers |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2603.12917 |