Shor's Algorithm Research Articles

Harmonic flow-field representations of quantum bits and gates

We describe a general procedure for mapping arbitrary n-qubit states to two-dimensional (2D) vector fields. The mappings use complex rational function representations of individual qubits, producing classical vector field configurations that can be interpreted in terms of 2D inviscid fluid flows or electric fields. Elementary qubits are identified with localized defects in 2D harmonic vector fields, and multiqubit states find natural field representations via complex superpositions of vector field products. In particular, separable states appear as highly symmetric flow configurations, making them both dynamically and visually distinct from entangled states. The resulting real-space representations of entangled qubit states enable an intuitive visualization of their transformations under quantum logic operations. We demonstrate this for the quantum Fourier transform and the period finding process underlying Shor's algorithm, along with other quantum algorithms. Due to its generic construction, the mapping procedure suggests the possibility of extending concepts such as entanglement or entanglement entropy to classical continuum systems, and thus may help guide new experimental approaches to information storage and nonstandard computation. Published by the American Physical Society 2024

Multi-objective simulated annealing-based quantum circuit cutting for distributed quantum computation

In the current noisy intermediate-scale quantum (NISQ) era, the number of qubits and the depth of quantum circuits in a quantum computer are limited because of complex operation among increasing number of qubits, low-fidelity quantum gates under noise, and short coherence time of physical qubits. However, with distributed quantum computation (DQC) in which multiple small-scale quantum computers cooperate, large-scale quantum circuits can be implemented. In DQC, it is a key step to decompose large-scale quantum circuits into several small-scale subcircuits equivalently. In this paper, we propose a quantum circuit cutting scheme for the circuits consisting of only single-qubit gates and two-qubit gates. In the scheme, the number of non-local gates and the rounds of subcircuits operation are minimized by using the multi-objective simulated annealing (MOSA) algorithm to cluster the gates and to choose the cutting positions whilst using non-local gates. A reconstruction process is also proposed to calculate the probability distribution of output states of the original circuit. As an example, the 7-qubit circuit of Shor algorithm factoring 15 is used to verify the algorithm. Five cutting schemes are recommended, which can be selected according to practical requirements. Compared with the results of the mixing integer programming (MIP) algorithm, the number of execution rounds is efficiently reduced by slightly increasing the number of nonlocal gates.

Applying Polynomials for Developing Post Quantum Cryptography Algorithms to Secure Online Information - An Initial Hypothesis

In the contemporary era, a vast array of applications employs encryption techniques to ensure the safeguarding and privacy of data. Quantum computers are expected to threaten conventional security methods and two existing approaches, namely Shor's and Grover's algorithms, are expediting the process of breaking both asymmetric and symmetric key classical algorithms. The objective of this article is to explore the possibilities of creating a new polynomial based encryption algorithm that can be both classically and quantum safe. Polynomial reconstruction problem is considered as a nondeterministic polynomial time hard problem (NP hard), and the degree of the polynomials provide the usage of scalable key lengths. The primary contribution of this study is the proposal of a novel encryption and decryption technique that employs polynomials and various polynomial interpolations, specifically designed for optimal performance in the context of a block cipher. This study also explores various root convergence techniques and provides algorithmic insights, working principles and the implementation of these techniques, which can potentially be utilized in the design of a proposed block-cipher symmetric cryptography algorithm. From the implementation, comparison and analysis of Durand Kernal, Laguerre and Aberth Ehrlich methods, it is evident that Laguerre method is performing better than other root finding approaches. The present study introduces a novel approach in the field of polynomial-based cryptography algorithms within the floating-point domain, thereby offering a promising solution for enhancing the security of future communication systems.

Investigating the Impact of Quantum Computing on Algorithmic Complexity

This paper investigated the transformation that quantum computing brought into algorithmic complexity in the theoretical setting of computer science. This involved the appearance of new paradigm changes brought forth by quantum technologies, imposing on a glance at the performance of quantum algorithms in respect to classical models of computation regarding their effectiveness. Our contributions encompassed key quantum algorithms, specifically Shor's and Grover's algorithms, underlining the performance metrics of those algorithms. The mathematical modeling was supported by simulations using python libraries like Qiskit, which helped analyze the complexities of both algorithm types. From our simulations, it emerged that for smaller sizes of input, classical algorithms executed faster and illustrated their established efficiency. In contrast, quantum computing-algorithms like Grover's and Shor's - performed so much better for larger input sizes, showing potential advantages that may change the face of computational limitations. While promising much, it seemed from our findings that quantum computing would not show a quantum advantage for all computational tasks, because classical algorithms still remained robust and effective for many scenarios. This nuanced understanding underlines how complementary both the classical and quantum approaches are. Therefore, the insights gained through this work provide the basis for investigating where boundaries of computational capabilities evolve and what practical implications are of the integration of quantum technologies into existing systems.

Quantum Computing and the Implications on Online Security

Over the past year, quantum computing has seen many advancements due to contributions from UC Berkeley in collaboration with IBM’s laboratories. Although these breakthroughs seem promising, they introduce new security concerns regarding the potential for quantum attacks to become a major threat in future data breaches. This research investigates the efficiency of many quantum cryptographic techniques. These algorithms include Quantum Key Distribution (QKD), Lattice-Based Cryptography, and Multivariate Cryptography. Utilizing IBM’s Qiskit Python extensions, we were able to test these methods against quantum attack algorithms, including Shor's Algorithm, Grover's Algorithm, and the BB84 protocol. Our results indicate that, although current cryptographic methods provide some level of defense against quantum attacks, there needs to be advancements to protect our digital infrastructure for the future. This study puts emphasis on the need for more development in cryptographic algorithms to ensure proper security in the era of quantum computing.

Chaotic roots of the modular multiplication dynamical system in Shor's algorithm

Shor's factoring algorithm, believed to provide an exponential speedup over classical computation, relies on finding the period of an exactly periodic quantum modular multiplication operator. This exact periodicity is the hallmark of an integrable system, which is paradoxical from the viewpoint of quantum chaos, given that the classical limit of the modular multiplication operator is a highly chaotic system that occupies the “maximally random” Bernoulli level of the classical ergodic hierarchy. In this work, we approach this apparent paradox from a quantum dynamical systems viewpoint, and consider whether signatures of ergodicity and chaos may indeed be encoded in such an “integrable” quantization of a chaotic system. We show that Shor's modular multiplication operator, in specific cases, can be written as a superposition of quantized A-baker's maps exhibiting more typical signatures of quantum chaos and ergodicity. This work suggests that the integrability of Shor's modular multiplication operator may stem from the interference of other “chaotic” quantizations of the same family of maps, and paves the way for deeper studies on the interplay of integrability, ergodicity, and chaos in and via quantum algorithms. Published by the American Physical Society 2024

Constrained Device Performance Benchmarking with the Implementation of Post-Quantum Cryptography

Advances in quantum computers may pose a significant threat to existing public-key encryption methods, which are crucial to the current infrastructure of cyber security. Both RSA and ECDSA, the two most widely used security algorithms today, may be (in principle) solved by the Shor algorithm in polynomial time due to its ability to efficiently solve the discrete logarithm problem, potentially making present infrastructures insecure against a quantum attack. The National Institute of Standards and Technology (NIST) reacted with the post-quantum cryptography (PQC) standardization process to develop and optimize a series of post-quantum algorithms (PQAs) based on difficult mathematical problems that are not susceptible to being solved by Shor’s algorithm. Whilst high-powered computers can run these PQAs efficiently, further work is needed to investigate and benchmark the performance of these algorithms on lower-powered (constrained) devices and the ease with which they may be integrated into existing protocols such as TLS. This paper provides quantitative benchmark and handshake performance data for the most recently selected PQAs from NIST, tested on a Raspberry Pi 4 device to simulate today’s IoT (Internet of Things) devices, and provides quantitative comparisons with previous benchmarking data on a range of constrained systems. CRYSTALS-Kyber and CRYSTALS-Dilithium are shown to be the most efficient PQAs in the key encapsulation and signature algorithms, respectively, with Falcon providing the optimal TLS handshake size.

A Quantum Leap in Factoring

The cryptography-threatening Shor algorithm for quantum computing sees its greatest theoretical advance in more than 25 years.

Efficient and quantum‐secure authenticated key exchange scheme for mobile satellite communication networks

SummaryThe mobile satellite communication (MSC) system is a vital communication method in which mobile users and network control centers connect via satellites. Since the satellite's communication is wireless, communication security is a critical factor to ensure accountable communication. Key exchange and authentication (KEA) mechanism is widely used to achieve data security for data transmitted in an insecure or open channel. Different authentication systems are suggested in the last many years to establish safe communication, whereas most existing security protocols are based on factorization or discrete logarithm. However, these systems are no longer reliable by Shor's algorithm as any discrete logarithm and factorization can be resolved in polynomial time on quantum computers. Thus, developing a quantum secure KEA protocol for MSC systems is necessary. In this direction, recently a ring learning with an error‐based KEA technique is proposed to ensure a quantum‐safe environment. This scheme is quantum‐safe and satisfies desirable security attributes but has low efficiency in computation and communication. Moreover, establishing a secure session requires six communications among involved entities. As a result, replay attack detection at an early stage is not possible for the central authority (server), which could delay the server response, and the adversary gets the advantage of drawing a denial of service scenario for authorized entities. We propose a lattice‐based KEA protocol for the MSC system to improve computation, communication efficiency, and early‐stage replay attack detection. The security analysis of the proposed scheme is presented in the random oracle model. Calculation of performance is also presented to observe advantages in computation and communication overhead.

Quantum optimization algorithms: Energetic implications

SummarySince the dawn of quantum computing (QC), theoretical developments like Shor's algorithm proved the conceptual superiority of QC over traditional computing. However, such quantum supremacy claims are difficult to achieve in practice because of the technical challenges of realizing noiseless qubits. In the near future, QC applications will need to rely on noisy quantum devices that offload part of their work to classical devices. One way to achieve this is by using parameterized quantum circuits in optimization or even in machine learning tasks. The energy requirements of quantum algorithms have not yet been studied extensively. In this article, we explore several optimization algorithms using both theoretical insights and numerical experiments to understand their impact on energy consumption. Specifically, we highlight why and how algorithms like quantum natural gradient descent, simultaneous perturbation stochastic approximations or circuit learning methods, are at least to more energy efficient than their classical counterparts; why feedback‐based quantum optimization is energy‐inefficient; and how techniques like Rosalin can improve the energy efficiency of other algorithms by a factor of 20. Finally, we use the NchooseK high‐level programming model to run optimization problems on both gate‐based quantum computers and quantum annealers. Empirical data indicate that these optimization problems run faster, have better success rates, and consume less energy on quantum annealers than on their gate‐based counterparts.

An in-principle super-polynomial quantum advantage for approximating combinatorial optimization problems via computational learning theory.

It is unclear to what extent quantum algorithms can outperform classical algorithms for problems of combinatorial optimization. In this work, by resorting to computational learning theory and cryptographic notions, we give a fully constructive proof that quantum computers feature a super-polynomial advantage over classical computers in approximating combinatorial optimization problems. Specifically, by building on seminal work by Kearns and Valiant, we provide special instances that are hard for classical computers to approximate up to polynomial factors. Simultaneously, we give a quantum algorithm that can efficiently approximate the optimal solution within a polynomial factor. The quantum advantage in this work is ultimately borrowed from Shor's quantum algorithm for factoring. We introduce an explicit and comprehensive end-to-end construction for the advantage bearing instances. For these instances, quantum computers have, in principle, the power to approximate combinatorial optimization solutions beyond the reach of classical efficient algorithms.

Entanglement Trajectory and its Boundary

In this article, we present a novel approach to investigating entanglement in the context of quantum computing. Our methodology involves analyzing reduced density matrices at different stages of a quantum algorithm's execution and representing the dominant eigenvalue and von Neumann entropy on a graph, creating an "entanglement trajectory." To establish the trajectory's boundaries, we employ random matrix theory. Through the examination of examples such as quantum adiabatic computation, the Grover algorithm, and the Shor algorithm, we demonstrate that the entanglement trajectory remains within the established boundaries, exhibiting unique characteristics for each example. Moreover, we show that these boundaries and features can be extended to trajectories defined by alternative entropy measures. The entanglement trajectory serves as an invariant property of a quantum system, maintaining consistency across varying situations and definitions of entanglement. Numerical simulations accompanying this research are available via open access.

Implementation of Shor's Algorithm and Its Demonstrated Quantum Efficiency

The unique principles of quantum physics introduce an element of uncertainty into our previously deterministic world. Quantum computing, derived from quantum physics, enables the representation of a qubit in both |0> and |1> states, thus encoding more information and unleashing a realm of new possibilities. This paper conducts an analysis of the classical RSA encryption protocol and its vulnerability to the quantum algorithm—Shor's algorithm. Furthermore, this paper rigorously establishes the mathematical theories required for the analysis. Finally, the algorithm is thoroughly explored, and the potential efficiencies of quantum computing are discussed. As the study of quantum computing deepens, its application is poised to surpass that of traditional computers in various fields, leaving a profound impact.

Exploring and reviewing the potential of quantum computing in enhancing cybersecurity encryption methods

As the landscape of cybersecurity continually evolves, traditional encryption methods face unprecedented challenges from the impending era of quantum computing. This paper undertakes a comprehensive exploration of the potential transformative impact that quantum computing could have on enhancing cybersecurity encryption methods. Commencing with an overview of quantum computing fundamentals, including the principles of quantum mechanics and key quantum properties, the paper delves into the disruptive power of Shor's algorithm. This algorithm, capable of exponentially faster factorization than classical counterparts, poses a significant threat to prevalent cryptographic techniques such as RSA and ECC. In response to the vulnerabilities exposed by quantum algorithms, the paper investigates the field of post-quantum cryptography, examining cryptographic algorithms designed to resist quantum attacks. Additionally, the study scrutinizes Quantum Key Distribution (QKD) as a potential solution for secure communication in a quantum environment, analyzing its strengths and limitations. The paper provides an updated survey of the current state of quantum computing, highlighting achievements, milestones, and a comparative analysis of existing quantum computing platforms. Subsequently, it assesses the potential impact of quantum computing on cybersecurity, addressing both its ability to fortify encryption and potential risks and vulnerabilities. Striking a balance between benefits and challenges, the research offers insights into the coexistence of quantum and classical cryptographic methods. Looking toward the future, the paper explores ongoing research and development in quantum computing, identifying challenges and ethical considerations. In conclusion, it synthesizes key findings, emphasizing the implications for the future of cybersecurity and advocating for continued research to ensure the development of resilient encryption methods in the quantum era.

Simulating the operation of a quantum computer in a dissipative environment.

The operations of current quantum computers are still significantly affected by decoherence caused by interaction with the environment. In this work, we employ the non-perturbative hierarchical equations of motion (HEOM) method to simulate the operation of model quantum computers and reveal the effects of dissipation on the entangled quantum states and on the performance of well-known quantum algorithms. Multi-qubit entangled states in Shor's factorizing algorithm are first generated and propagated using the HEOM. It is found that the failure of factorization is accompanied by a loss of fidelity and mutual information. An important challenge in using the HEOM to simulate quantum computers in a dissipative environment is how to efficiently treat systems with many qubits. We propose a two-dimensional tensor network scheme for this problem and demonstrate its capability by simulating a one-dimensional random circuit model with 21 qubits.

Quantum Computing in Cryptographic Systems

Quantum computing represents a transformative advancement in computational power, with profound implications for cryptographic systems. This paper explores the intersection of quantum computing and cryptography, examining the potential for quantum computers to both break traditional cryptographic algorithms and enable new forms of secure communication. Classical cryptographic systems, such as RSA and ECC, rely on the computational difficulty of problems like integer factorization and discrete logarithms. However, quantum algorithms, notably Shor's algorithm, can solve these problems exponentially faster than the best-known classical algorithms, posing a significant threat to the security of current cryptographic practices. This vulnerability necessitates the development and implementation of quantum-resistant algorithms to ensure data security in a post-quantum world. The paper reviews the state-of-the-art in quantum-resistant cryptographic algorithms, including lattice-based, hash-based, code-based, and multivariate polynomial cryptography. These algorithms are analyzed for their security properties, efficiency, and practicality for real-world deployment. Additionally, the paper discusses the ongoing standardization efforts led by organizations such as NIST, which are crucial for establishing widely accepted quantum-safe cryptographic protocols. Beyond the threat posed by quantum computing, this paper also highlights the potential benefits quantum computing can bring to cryptographic systems. Quantum Key Distribution (QKD) is one such advancement, offering theoretically unbreakable encryption by leveraging the principles of quantum mechanics. QKD enables secure communication channels that are immune to any computational advances, including those of quantum computers, by ensuring that any attempt at eavesdropping can be detected. The practical implementation challenges of QKD and other quantum cryptographic protocols are also addressed, including issues related to transmission distance, error rates, and integration with existing communication infrastructures. Experimental results from recent QKD trials are presented, showcasing significant progress in extending the viability of quantum-secure communication over long distances. Furthermore, the paper delves into the hybrid approaches combining classical and quantum cryptographic techniques. These approaches aim to leverage the strengths of both paradigms to enhance security and performance. For instance, hybrid protocols might use classical encryption for bulk data transfer, supplemented by quantum techniques for key exchange, thereby ensuring robust security even against future quantum adversaries. In conclusion, while quantum computing poses substantial risks to existing cryptographic systems, it also offers novel opportunities for creating highly secure communication protocols. The transition to quantum-safe cryptography is imperative to protect sensitive information in the quantum era. This paper underscores the urgency of advancing quantum-resistant cryptographic research, promoting international collaboration for standardization, and exploring innovative applications of quantum technologies in cryptography. Continued research and development in these areas are essential to stay ahead of the evolving threat landscape and harness the full potential of quantum computing for secure communications.

Unlocking New Computational Paradigms: The Role of Quantum Mechanics in Algorithm Development

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.

Algorithm efficiency and hybrid applications of quantum computing

With the development of science and technology, it is difficult for traditional computers to solve cutting-edge problems due to the lack of computing power, and the importance of quantum computers is increasing day by day. This article starts with the simple principle of quantum computing, introduces the most advanced quantum computing instruments and quantum computing algorithms, and points out the application prospects in medicine, chemistry and other fields. This paper explains the basic principles of quantum computing algorithms, their efficiency over traditional algorithms, and focuses on the Shor algorithm and its variations. In terms of applications, the quantum computer Zu Chongzhi and its contribution to the sampling problem of quantum random circuits are introduced. This paper makes a certain analysis of the limitations of quantum computing, and gives the future development goals in a targeted manner. Besides, we summarize popular quantum computing algorithms and applications, and make contributions to the promotion and development of quantum computing. Overall, these results shed light on guiding further exploration of how to improve the computational efficiency of quantum computers.

Quantum Computing Algorithms for Solving Complex Mathematical Problems

The power of quantum mechanics, that is too complex for conventional computers, can be solved by an innovative model of computing known as quantum computing. Quantum algorithms can provide exponential speedups for some types of problems, such as many difficult mathematical ones. In this paper, we review some of the most important quantum algorithms for hard mathematical problems. When factoring large numbers, Shor's algorithm is orders of magnitude faster than any other known classical algorithm. The Grover's algorithm, which searches unsorted databases much more quickly than conventional algorithms, is then discussed.

QSMAH: A novel quantum-based secure cryptosystem using mutual authentication for healthcare in the internet of things

Healthcare in IoT provides many benefits, such as real-time data transfer and decision-making based on the information received from the patient's Body Sensor Nodes (BSNs). Many healthcare centers have adopted IoT-based devices such as heart monitoring implants and Electrocardiograms (ECG) for continuous patient monitoring in and out of hospitals. However, due to open Internet connectivity and lack of authentication, these devices may lead to several life-threatening risks. Recent advancements in Quantum Computing threatened the security of classical cryptographic primitives such as the RSA algorithm. The security and privacy of patient data are of utmost importance and are to be protected from existing vulnerabilities and futuristic Quantum attacks. Motivated by the abovementioned issues, we proposed the QSMAH protocol which ensures secure Key Agreement (KA) and Mutual Authentication (MA) based on Quantum Cryptography. Unlike classical authentication schemes, in our protocol, Quantum Teleportation and Quantum Entanglement are effectively utilized for secure data transfer among entities. The security of the proposed QSMAH protocol relies on a Quantum Key and Greenberger–Horne–Zeilinger (GHZ) states to achieve strong authentication. An extensive formal security analysis using BAN logic is provided to prove the goal of our protocol. The simulation of the proposed protocol is provided using Automated Validation of Internet Security Protocols and Applications (AVISPA). The results reflect that the security of our protocol cannot be tampered with by Quantum Shor's and Grover's algorithms. The security analysis proves that QSMAH is also resistant to classical attacks and futuristic Quantum attacks on cryptographic schemes.