Asymptotically Optimal Quantum Circuits for Comparators and Incrementers

TL;DR

This paper introduces quantum circuits for comparison and increment operations that are asymptotically optimal in gate count and depth, with minimal qubits, significantly improving the efficiency of quantum algorithms like Shor's factoring.

Contribution

The authors develop asymptotically optimal quantum circuits for comparison and increment, extend these to classical-quantum comparators, and demonstrate their application in reducing circuit complexity in quantum algorithms.

Findings

Optimal gate count of Θ(n) for comparison and increment circuits

Reduced circuit depth for Shor's algorithm from O(n^3) to O(n^2 log^2 n)

A general theorem for trading ancilla qubits for control qubits with low overhead

Abstract

We present quantum circuits for comparison and increment operations that achieve an asymptotically optimal gate count of $\Theta(n)$ and depth of $\Theta(\log n)$ 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.