Quantum modular multipliers are a promising development in the field of quantum computing that aim to perform modular multiplication faster and more efficiently than classical algorithms. This is achieved by encoding the numbers being multiplied into quantum bits (qubits) and using quantum gates to manipulate them in a way that represents the calculation. Since quantum physics principles like superposition and entanglement dramatically speed up processing compared to conventional computing, many calculations may be run simultaneously. Quantum modular multipliers have the potential to have a significant impact in a range of fields, including cryptography and beyond, since they offer a faster and more efficient way to do modular multiplications. Several fields, including simulation, optimization, and cryptography, can benefit from the use of quantum computing.One of the crucial operations in many of these applications is modular multiplication, a sort of multiplication that requires obtaining the remainder of the product of two numbers when divided by a third number. In this paper, we provide a novel modular multiplication method based on a quantum algorithm. To examine its complexity and compare it to that of current adders, we employ a novel form of adder that is intended to not retain the final carry. This adder's decrease percentage in terms of gates and depth is roughly 70%. Our method takes advantage of the special qualities of qubits to complete the computation significantly more quickly than conventional algorithms. It is based on the fundamental ideas of quantum physics, such as superposition and entanglement.
Quantum modular multipliers are a promising development in the field of quantum computing that aim to perform modular multiplication faster and more efficiently than classical algorithms. This is achieved by encoding the numbers being multiplied into quantum bits (qubits) and using quantum gates to manipulate them in a way that represents the calculation. Since quantum physics principles like superposition and entanglement dramatically speed up processing compared to conventional computing, many calculations may be run simultaneously. Quantum modular multipliers have the potential to have a significant impact in a range of fields, including cryptography and beyond, since they offer a faster and more efficient way to do modular multiplications. Several fields, including simulation, optimization, and cryptography, can benefit from the use of quantum computing.One of the crucial operations in many of these applications is modular multiplication, a sort of multiplication that requires obtaining the remainder of the product of two numbers when divided by a third number. In this paper, we provide a novel modular multiplication method based on a quantum algorithm. To examine its complexity and compare it to that of current adders, we employ a novel form of adder that is intended to not retain the final carry. This adder's decrease percentage in terms of gates and depth is roughly 70%. Our method takes advantage of the special qualities of qubits to complete the computation significantly more quickly than conventional algorithms. It is based on the fundamental ideas of quantum physics, such as superposition and entanglement.
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.
Quantum modular multiplication circuit is one of the basic quantum computation circuits which are basic functions in quantum algorithms. However, since quantum-quantum modular multipliers require a high cost reversible modular inversion routine for modular multiplication, researchers have been unable to propose a feasible quantum-quantum modular multiplier. In this paper, we proposed efficient quantum-classical modular multipliers and the first quantum-quantum modular multipliers that do not require a reduction stage by transforming the partial product used in multiplication utilizing bit-shift operation. Then, we calculated quantum resource complexity and analyzed it compared to other quantum modular multipliers and utilized ETRI (Electronics and Telecommunications Research Institute) Qcrypton to analyze quantum resource complexity in the practical quantum computing situation. The proposed quantum modular multipliers show an improvement of 50% in terms of gates and circuit depth compared to the most recently proposed high-performance quantum modular multipliers.
Quantum multiplication is a fundamental operation in quantum computing. It is important to have a quantum multiplier with low complexity. In this paper, we propose the Quantum Multiplier Based on Exponent Adder (QMbead), a new approach that requires just log2(n) qubits to multiply two n-bit integer numbers, in addition to O(n) ancillary qubits used for quantum state preparation. The QMbead uses a so-called exponent encoding to respectively represent two multiplicands as two superposition states which are prepared by a quantum state preparation method, then employs a quantum adder to obtain the sum of these two superposition states, and subsequently measures the outputs of the quantum adder to calculate the product of the multiplicands. Different quantum adders can be used in the QMbead. The circuit depth and time complexity of the QMbead, using a logarithmic-depth quantum carry lookahead adder (QCLA) as adder, are O(logn) and O(nlogn), respectively. The gate complexity of the QMbead is O(n). The circuit depth and gate complexity of the QMbead is better than existing quantum multipliers such as the quantum Karatsuba multiplier and the QFT based multiplier. The time complexity of the QMbead is identical to that of the fastest classical multiplication algorithm, Harvey-Hoeven algorithm. Interestingly, the QMbead maintains an advantage over the Harvey-Hoeven algorithm, given that the latter is only suitable for excessively large numbers, whereas the QMbead is valid for both small and large numbers. The multiplicand can be either an integer or a decimal number. The QMbead has been implemented on quantum simulators to compute products with a bit length of up to 273 bits using only 17 qubits, excluding the ancillary qubits used for quantum state preparation. This establishes QMbead as an efficient solution for multiplying large integer or decimal numbers with many bits.
Quantum computing represents a revolutionary leap in computational power, leveraging the principles of quantum mechanics to process information in fundamentally new ways. Unlike classical computers, which use bits to represent data as 0s or 1s, quantum computers use quantum bits or qubits, enabling the phenomena of superposition and entanglement to perform complex calculations more efficiently (Nielsen & Chuang, 2010). This article explores the basic principles of quantum computing, the different models and algorithms that have been developed, and the current state of quantum hardware. The power of quantum computing lies in its ability to solve certain types of problems exponentially faster than classical computers. For instance, Shor's algorithm can factorize large integers in polynomial time, posing a significant threat to classical encryption methods (Shor, 1997). Similarly, Grover's algorithm provides a quadratic speedup for unstructured search problems, showcasing the potential for dramatic improvements in database search capabilities (Grover, 1996). These algorithms exemplify how quantum computing can revolutionize fields that rely on complex computations. Quantum hardware, however, is still in its nascent stage, with significant challenges to be overcome. Current quantum computers, such as those developed by IBM and Google, have demonstrated quantum supremacy but are limited by issues of qubit coherence and error rates (Arute et al., 2019). Quantum error correction and the development of more stable qubit technologies are critical areas of ongoing research (Preskill, 2018). Despite these challenges, recent advancements in quantum hardware and algorithms continue to drive the field forward, promising transformative impacts on industries such as cryptography, drug discovery, and artificial intelligence. As quantum computing continues to advance, it holds the promise of solving problems that are currently intractable for classical computers, paving the way for unprecedented technological and scientific advancements (Shor, 1997; Grover, 1996). This article provides a comprehensive overview of the current landscape of quantum computing, discussing its foundational principles, key algorithms, hardware challenges, and potential applications. By understanding these aspects, we can better appreciate the profound implications of quantum computing for the future of technology and science.
The advent of quantum computing has ushered in a new era of computational capabilities, promising to revolutionize the resolution of complex problems. Unlike classical computers, which rely on binary digits (bits) to process information, quantum computers utilize quantum bits (qubits) that can exist in multiple states simultaneously due to the principles of superposition and entanglement inherent in quantum mechanics. This fundamental difference allows quantum computers to perform certain calculations exponentially faster than their classical counterparts, offering profound implications for fields ranging from cryptography to material science. At the heart of this transformative potential lie quantum algorithms, specifically designed to harness the unique properties of quantum systems to tackle problems currently intractable for classical computers. Notable examples include Shor's algorithm for integer factorization, which threatens the security of widely used cryptographic schemes, and Grover's algorithm for unstructured search, which provides a quadratic speedup over classical approaches. Beyond these pioneering works, ongoing research continues to uncover new quantum algorithms capable of addressing a broad spectrum of applications. The proposed method leverages the principles of quantum mechanics to develop efficient algorithms that outperform classical approaches in solving complex computational problems. By harnessing the unique properties of quantum bits (qubits), such as superposition and entanglement, the method aims to achieve superior accuracy and efficiency in various applications. The performance of the proposed method was evaluated using several key metrics: an accuracy of 94.8%, a Root Mean Squared Error (RMSE) of 0.208, and a Mean Absolute Error (MAE) of 0.406. These metrics collectively underscore the robustness and reliability of the proposed quantum algorithms, highlighting their potential to significantly enhance computational efficiency and accuracy across various domains. This study delves into the development of quantum algorithms, exploring their theoretical foundations, practical implementations, and the challenges faced in their realization. By leveraging the principles of quantum mechanics, these algorithms hold the promise of not only solving existing problems more efficiently but also unlocking new computational paradigms previously deemed unattainable. Through a comprehensive analysis, this research aims to elucidate the current state of quantum algorithm research, highlight the advancements made, and identify the pathways forward in this rapidly evolving field.
Addition is the most basic operation of computing based on a bit system. There are various addition algorithms considering multiple number systems and hardware, and studies for a more efficient addition are still ongoing. Quantum computing based on qubits as the information unit asks for the design of a new addition because it is, physically, wholly different from the existing frequency-based computing in which the minimum information unit is a bit. In this paper, we propose an efficient quantum circuit of modular addition, which reduces the number of gates and the depth. The proposed modular addition is for the Galois Field GF(2n−1), which is important as a finite field basis in various domains, such as cryptography. Its design principle was from the ripple carry addition (RCA) algorithm, which is the most widely used in existing computers. However, unlike conventional RCA, the storage of the final carry is not needed due to modifying existing diminished-1 modulo 2n−1 adders. Our proposed adder can produce modulo sum within the range 0,2n−2 by fewer qubits and less depth. For comparison, we analyzed the proposed quantum addition circuit over GF(2n−1) and the previous quantum modular addition circuit for the performance of the number of qubits, the number of gates, and the depth, and simulated it with IBM’s simulator ProjectQ.