A Comprehensive Study of Quantum Arithmetic Circuits

TL;DR

The paper addresses the need for a systematic, up-to-date survey of quantum arithmetic circuits, which are central to many quantum algorithms. It analyzes two mainstream architectures—Clifford+T gate-based designs and QFT-based designs—across addition, subtraction, multiplication, division, and modular exponentiation, comparing resource costs with standardized metrics. Key contributions include a structured taxonomy of adders, subtractors, and multipliers (including advanced designs like Carry-Lookahead, Karatsuba, Toom-Cook, and Schönhage-Strassen variants), as well as discussions of data formats and modular exponentiation strategies, complemented by practical applications and directions for future work. The survey highlights practical impact for Shor’s algorithm, linear systems solvers, and quantum finance, and points to promising research avenues in fixed-point/floating-point arithmetic, constant arithmetic, and ternary quantum bases, underlining the potential gains from systematic design-space exploration and optimized fault-tolerant implementations.

Abstract

In recent decades, the field of quantum computing has experienced remarkable progress. This progress is marked by the superior performance of many quantum algorithms compared to their classical counterparts, with Shor's algorithm serving as a prominent illustration. Quantum arithmetic circuits, which are the fundamental building blocks in numerous quantum algorithms, have attracted much attention. Despite extensive exploration of various designs in the existing literature, researchers remain keen on developing novel designs and improving existing ones. In this review article, we aim to provide a systematically organized and easily comprehensible overview of the current state-of-the-art in quantum arithmetic circuits. Specifically, this study covers fundamental operations such as addition, subtraction, multiplication, division and modular exponentiation. We delve into the detailed quantum implementations of these prominent designs and evaluate their efficiency considering various objectives. We also discuss potential applications of presented arithmetic circuits and suggest future research directions.

Figures (15)

• Figure 1: Clifford+T gate set.
• Figure 2: Quantum Fourier Transformation to a $n$ qubit quantum state $\ket{a}:=\ket{a_{n-1}a_{n-2}\cdots a_1a_0}$.
• Figure 3: Architecture of QFT-based arithmetic circuits.
• Figure 4: Three main aspects of quantum arithmetic designs discussed in this paper.
• Figure 5: IEEE-754 Floating-Point standard.