Asymptotically Optimal Quantum Circuits for Comparators and Incrementers

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.

Figures (12)

• Figure 1: Naive implementations of the $\mathcal{F}_1^{(5)}$, $\mathcal{L}_2^{(5)}$, and $\mathcal{V}_2^{(5)}$ operators.
• Figure 2: Decomposing a multi-controlled $U$ gate to take advantage of promise gates.
• Figure 3: Using controlled conjugation simplification with promise gates.
• Figure 4: Ancilla-free circuit for quantum addition over $5$-bit registers, where $s = a + b$Takahashi_2010.
• Figure 5: Ancilla-free circuit for the quantum--quantum comparator over $5$-bit registers.

Theorems & Definitions (44)

• Definition 2.1
• Lemma 1: Nie et al. Nie_2024
• Definition 2.2
• Lemma 2
• Definition 2.3
• Lemma 3
• Lemma 4
• Corollary 1
• Definition 2.4
• Definition 3.1