As we venture into the Intermediate-Scale Quantum (ISQ) era, the proficiency of modular arithmetic operations becomes pivotal for advancing quantum cryptographic algorithms. This study presents an array of quantum circuits, each precision-engineered for modular arithmetic functions critical to cryptographic applications. Central to our exposition are quantum modular adders, multipliers, and exponential operators, whose designs are rigorously optimized for ISQ devices. We provide a theoretical framework and practical implementations in the PennyLane quantum software, bridging the gap between conceptual and applied quantum computing. Our simulations validate the efficacy of these methodologies, offering a strategic compass for developing quantum algorithms that align with the rapid progression of quantum technology.
As we venture into the Intermediate-Scale Quantum (ISQ) era, the proficiency of modular arithmetic operations becomes pivotal for advancing quantum cryptographic algorithms. This study presents an array of quantum circuits, each precision-engineered for modular arithmetic functions critical to cryptographic applications. Central to our exposition are quantum modular adders, multipliers, and exponential operators, whose designs are rigorously optimized for ISQ devices. We provide a theoretical framework and practical implementations in the PennyLane quantum software, bridging the gap between conceptual and applied quantum computing. Our simulations validate the efficacy of these methodologies, offering a strategic compass for developing quantum algorithms that align with the rapid progression of quantum technology.
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.
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.
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.
Quantum computing (Q-computing) is a revolutionary technology that applies quantum mechanics (QM) to implement calculations significantly quicker than traditional computers. Q-computing employs Q-bits (qubits) rather than classical bits, which can take on a superposition state, making it possible to represent both 0 and 1 at once. This distinctive property allows qubits to perform multiple calculations simultaneously, making Q computers exponentially more powerful as more qubits are added, compared to the linear scaling of classical computers. However, challenges such as qubit instability and error rates remain significant obstacles to fully realizing this potential. A vital aspect of this technology is the development of Q arithmetic circuits, which act as the foundational blocks for efficiently executing Q algorithms. Q half-adder and full-adder circuits are integral to performing fundamental arithmetic operations within Q computers. In contrast to classical adders, Q adders rely on Q gates such as Controlled NOT (CNOT) and Toffoli gates. These gates exploit Q phenomena like entanglement and superposition, allowing for more efficient computations. The Q half-adder consists of two qubits and adds two single bits, producing a carry and sum output. It utilizes the CNOT gate to compute the sum and the Toffoli gate (controlled-controlled-NOT [CCNOT]) to compute the carry. The Q full-adder builds on this, adding three qubits to produce a sum and a carry output, further increasing computational capacity. Through the entanglement of qubits, these circuits allow for more parallelism and speed compared to classical adders. Q carry-look-ahead adders (QCLAs) and Q ripple-carry adders (QRCAs) are used to perform more complex arithmetic operations based on these half-adder and full-adder structures. Q arithmetic circuits have broad applications, boosting secure key generation in cryptography, speeding up matrix operations for faster model training in machine learning, and enabling more precise, efficient image transformations in Q image processing, benefiting fields like medical imaging and computer vision.
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'.
The transition from classical to quantum computation marks a shift from deterministic binary logic to a probabilistic, entanglement‑driven framework able to address problems beyond classical reach. Today’s Noisy Intermediate ‑ Scale Quantum era features devices that lack full error correction yet already point toward larger, modular systems designed for fault‑tolerant operation. This paper examines the evolving paradigm shift from classical and NISQ ‑ era machines toward practical fault‑tolerant quantum computing, focusing on how advances in physical qubit technologies, control architectures, and error‑correction strategies are converging to bridge noisy prototypes and scalable, useful machines. We synthesize insights from historical semiconductor scaling trends, contemporary roadmaps from leading institutions and industry, and emerging modular designs with real‑time error decoding. Our analysis shows that while challenges around decoherence, connectivity, and classical co‑processing remain, coordinated progress in hardware, control, and software is laying a coherent foundation for future quantum platforms. The work concludes that quantum computing is maturing into a strategically significant paradigm, poised to redefine the limits of computation in simulation, optimization, and secure communication.