Rapid progress in the design of scalable, robust quantum computing necessitates efficient quantum circuit implementation for algorithms with practical relevance. For several algorithms, arithmetic kernels, in particular, division plays an important role. In this manuscript, we focus on enhancing the performance of quantum slow dividers by exploring the design choices of its sub-blocks, such as, adders. Through comprehensive design space exploration of state-of-the-art quantum addition building blocks, our work have resulted in an impressive achievement: a reduction in Toffoli Depth of up to 93.90%, accompanied by substantial reductions in both Toffoli and Qubit Count of up to 92.12% and 99.38%, respectively. This paper offers crucial perspectives on efficient design of quantum dividers, and emphasizes the importance of adopting a systematic design space exploration approach.
Rapid progress in the design of scalable, robust quantum computing necessitates efficient quantum circuit implementation for algorithms with practical relevance. For several algorithms, arithmetic kernels, in particular, division plays an important role. In this manuscript, we focus on enhancing the performance of quantum slow dividers by exploring the design choices of its sub-blocks, such as, adders. Through comprehensive design space exploration of state-of-the-art quantum addition building blocks, our work have resulted in an impressive achievement: a reduction in Toffoli Depth of up to 93.90%, accompanied by substantial reductions in both Toffoli and Qubit Count of up to 92.12% and 99.38%, respectively. This paper offers crucial perspectives on efficient design of quantum dividers, and emphasizes the importance of adopting a systematic design space exploration approach.
In recent decades, quantum computing has witnessed remarkable progress, achieved through various advancements across the technology stack, such as quantum error-correcting codes and quantum algorithms. It is noteworthy that several promising algorithms incorporate quantum arithmetic circuits as fundamental building blocks. However, the inherent properties of quantum circuits challenge the efficiency of quantum arithmetic circuits, prompting creative solutions. While several quantum arithmetic circuits have been proposed, there is still a substantial gap in quantum arithmetic circuits optimization. For example, methods for integrating classical techniques such as high radix strategy into quantum designs still remain underexplored, highlighting a significant space for further advancement. This thesis aims to improve the performance of quantum arithmetic designs by integrating classical arithmetic circuit design techniques. Specifically, this thesis contains the following research: (1) The utilization of classical higher radix strategy to enhance quantum addition, reducing overall cost by lowering T-depth and T-count. The proposed design outperforms existing quantum carry-lookahead adders. (2) The application of Ling structure, achieving a reduced Toffoli-depth of 4log(n/2). (3) The exploration of the Kogge-Stone architecture, resulting in the lowest Toffoli-depth of log(n) for quantum adders. (4) The development of an efficient modular addition framework using carry-save architecture, demonstrating a 33.3% improvement in Toffoli depth compared to existing designs. (5) The design of an optimized quantum non-restoring divider framework. (6) A comprehensive overview of current advancements in quantum arithmetic designs, offering insights into potential research directions. These proposed designs are mapped to real-world quantum hardware architectures, including Noisy Intermediate-Scale Quantum systems, to validate their functionality and evaluate efficiency. Their applications in quantum machine learning and cryptography further underscore the broader impact of these improvements. Overall, this thesis makes significant advancements in reducing quantum circuit depth and improving resource efficiency, contributing to the development of scalable and efficient large-scale quantum circuits. By adapting classical design strategies to quantum circuits, it not only enhances various quantum circuit designs but also fosters further exploration at the intersection of classical and quantum computing.
Shor's factorization algorithm, as one of the most significant achievements in quantum computing, exhibits an exponential speedup compared to the corresponding classical algorithm. In Shor's factorization algorithm, modular exponentiation is one of the most computationally intensive components, which relies on the modular addition building block. This work aims to explore novel designs for enhancing the efficiency of quantum modular addition. In particular, we introduce a novel quantum modular addition framework based on carry-save architecture, which facilitates the conversion of multiple 2-addend quantum operations within modular addition into a single 3-addend operation, thereby reducing the computational depth. Compared to the most efficient existing quantum modular addition, our design has achieved an impressive result - a reduction in Toffoli Depth by up to 33.33%, while maintaining comparable Toffoli Count and Qubit Count. This research underscores the potential of carry-save architecture as a promising technique for accelerating quantum modular arithmetic as well as advancing the development of quantum computing in general.
Efficient quantum arithmetic circuits are commonly found in numerous quantum algorithms of practical significance. To date, the logarithmic-depth quantum adders include a constant coefficient k ≥ 2 while achieving the Toffoli-Depth of k log n + 𝒪(1). In this work, 160 alternative compositions of the carry-propagation structure are comprehensively explored to determine the optimal depth structure for a quantum adder. By extensively studying these structures, it is shown that an exact Toffoli-Depth of log n + 𝒪(1) is achievable. This presents a reduction of Toffoli-Depth by almost 50% compared to the best known quantum adder circuits presented to date. We demonstrate a further possible design by incorporating a different expansion of propagate and generate forms, as well as an extension of the modular framework. Our article elaborates on these designs, supported by detailed theoretical analyses and simulation-based studies, firmly substantiating our claims of optimality within all possible configurations outlined in this work. The results also mirror similar improvements, recently reported in classical adder circuit complexity.
Quantum computing has been considered as an emerging approach in addressing problems which are not easily solvable using classical computers. In parallel to the physical implementation of quantum processors, quantum algorithms have been actively developed for real-life applications to show quantum advantages, many of which benefit from quantum arithmetic algorithms and their efficient implementations. As one of the most important operations, quantum addition has been adopted in Shor's algorithm and quantum linear algebra algorithms. Although various least-significant digit-first quantum adders have been introduced in previous work, interest in investigating the efficient implementation of most-significant digit-first addition is growing. In this work, we propose a novel design method for most-significant digit-first addition with several quantum circuit optimisations to reduce the number of quantum bits (i.e. qubits), quantum gates, and circuit depth. An open-source library of different arithmetic operators based on our proposed method is presented, where all circuits are implemented on IBM Qiskit SDK. Extensive experiments demonstrate that our proposed design, together with the optimisation techniques, reduces T-depth by up-to 4.0×, T-count by 3.5×, and qubit consumption by 1.2×.
Quantum computing has advanced rapidly in recent decades, driven by developments across the technology stack, including quantum error-correcting codes and efficient quantum algorithms. Among these, quantum arithmetic circuits serve as fundamental building blocks for various promising algorithms. Despite their crucial role, the design of quantum arithmetic circuits faces challenges arising from the no-cloning theorem, qubit limitations, and circuit depth constraints, which significantly impact the efficiency of large-scale quantum computing. We provide an overview of quantum arithmetic circuits in the context of public-key cryptanalysis, with particular emphasis on optimization strategies such as measurement-based uncomputation and conditionally clean ancilla. We review state-of-the-art designs for essential arithmetic operations in public-key cryptanalysis such as addition, multiplication, and modular exponentiation. We also present an overview of the techniques used for fault-tolerant runtime and resource estimation in quantum cryptanalysis. In brief, this chapter emphasizes strategies for designing resource-efficient quantum arithmetic circuits, providing a basis for realistic evaluations of quantum cryptanalytic capabilities.
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 with 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 to develop novel designs and improve 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 the presented arithmetic circuits and suggest future research directions.This article is part of the theme issue 'Emerging technologies for future secure computing platforms'.