Quantum Arithmetic Algorithms: Implementation, Resource Estimation, and Comparison
Dmytro Fedoriaka, Brian Goldsmith, Yingrong Chen
TL;DR
This work addresses the practical challenge of implementing and comparing quantum arithmetic, using Azure Quantum Resource Estimator to assess runtime, qubit cost, and space-time trade-offs across a broad library of algorithms for addition, multiplication, division, and modular exponentiation. By examining design choices, parameter tuning, and tipping points, the authors deliver a KBD-style resource-performance map that guides algorithm selection for given hardware constraints and error budgets. Key contributions include a thorough implementation in Q# of multiple arithmetic primitives, comprehensive resource-estimation results, and actionable insights on uncomputation, parallelization, and problem-size scaling, including asymptotic analyses such as $O(n)$ for adders, $O(n^2)$ for many primitives, and $O(n^3)$ for ModExp with optimized windowing. The practical impact is a ready-to-use library and decision-support framework that helps researchers and engineers optimize quantum arithmetic for near- and long-term fault-tolerant architectures.
Abstract
As quantum computing technology advances, the need for optimized arithmetic circuits continues to grow. This paper presents the implementation and resource estimation of a library of quantum arithmetic algorithms, including addition, multiplication, division, and modular exponentiation. Using the Azure Quantum Resource Estimator, we evaluate runtime, qubit usage, and space-time trade-offs and identify the best-performing algorithm for each arithmetic operation. We explore the design space for division, optimize windowed modular exponentiation, and identify the tipping point between multipliers, demonstrating effective applications of resource estimation in quantum research. Additionally, we highlight the impact of parallelization, reset operations, and uncomputation techniques on implementation and resource estimation. Our findings provide both a practical library and a valuable knowledge base for selecting and optimizing quantum arithmetic algorithms in real-world applications.