Quantum Computing: A Taxonomy, Systematic Review and Future Directions
Sukhpal Singh Gill, Adarsh Kumar, Harvinder Singh, Manmeet Singh, Kamalpreet Kaur, Muhammad Usman, Rajkumar Buyya
1 Quantum Computing: A Taxonomy, Systematic Review and Future Directions
SUKHPAL SINGH GILL, Queen Mary University of London, UK ADARSH KUMAR and HARVINDER SINGH, University of Petroleum and Energy Studies, Dehradun, India MANMEET SINGH, Indian Institute of Tropical Meteorology, Pune, India and Indian Institute of Technology (IIT) Bombay, India KAMALPREET KAUR, QualiteSoft, Chandigarh, India MUHAMMAD USMAN and RAJKUMAR BUYYA, The University of Melbourne, Australia
Quantum computing is an emerging paradigm with the potential to offer significant computational advantage over conventional classical computing by exploiting quantum-mechanical principles such as entanglement and superposition. It is anticipated that this computational advantage of quantum computing will help to solve many complex and computationally intractable problems in several areas of research such as drug design, data science, clean energy, finance, industrial chemical development, secure communications, and quantum chemistry, among others. In recent years, tremendous progress in both quantum hardware development and quantum software/algorithm have brought quantum computing much closer to reality. Indeed, the demonstration of quantum supremacy marks a significant milestone in the Noisy Intermediate Scale Quantum (NISQ) era – the next logical step being the quantum advantage whereby quantum computers solve a real-world problems much more efficiently than classical computing. As the quantum devices are expected to steadily scale up in the next few years, quantum decoherence and qubit interconnectivity are two of the major challenges to achieve quantum advantage in the NISQ era. Quantum computing is a highly topical and fast-moving field of research with significant ongoing progress in all facets. A systematic review of the existing literature on quantum computing will be invaluable to understand the current status of this emerging field and identify open challenges for the quantum computing community in the coming years. This review article presents a comprehensive review of quantum computing literature, and taxonomy of quantum computing. Further, the proposed taxonomy is used to map various related studies to identify the research gaps. A detailed overview of quantum software tools and technologies, post-quantum cryptography and quantum computer hardware development to document the current state-of-the-art in the respective areas. We finish the article by highlighting various open challenges and promising future directions for research. Categories and Subject Descriptors:" A.1 [ General Literature ]: Introductory and Survey; C.0 [
General ]: Systems Architectures; C.2.4 [
Computer-Communication Networks ]: Distributed Systems; D.4.1 [
Process Management ]: Scheduling; H.3.4 [
Systems and Software ]: Distributed Systems; J.7 [
Distributed Parallel and Cluster Computing ]; K.6.2 [
Management of Computing and Information Systems ]: Installation Management General Terms: Documentation, Quantum Computing, Methodical Analysis, Conceptual Model, Focus of Study, Research Challenges, Theory, Reference Architecture, Trade-off, Future Directions, Management Additional Key Words and Phrases: Computing, Quantum Computing, Classical Computing, Quantum computer, Robotics
ACM Reference Format:
Sukhpal Singh Gill, Adarsh Kumar, Harvinder Singh, Manmeet Singh, Kamalpreet Kaur, Muhammad Usman, Rajkumar Buyya. 2020. Quantum Computing: A Taxonomy, Systematic Review and Future Directions.
ACM Computing Surveys.
YYYY, YY (YYYY 2020), xx pages.
DOI: https://doi.org/12456
1. INTRODUCTION
In his famous lecture in 1982, Richard Feynman envisioned a quantum machine which can simulate quantum physics, and in many ways, this is considered one of the initial conceptions of quantum computing [1]. He postulated that nature is not classical and therefore to simulate natural phenomena, one would need a computing device which works on quantum mechanical principles. Indeed, quantum computers offer such possibilities, where computing can exploit quantum mechanical properties such as entanglement and superposition to offer tremendous computational capabilities necessary for simulations of complex quantum systems. The initial progress towards developing quantum computer hardware was relatively slow, because the proposed quantum mechanical properties are only observed at the very fundamental scale of nature (i.e. electron spins or photon polarization), which were
Author's address: S. S. Gill, School of Electronic Engineering and Computer Science, Queen Mary University of London, UK; email: [email protected]; A. Kumar, Department of Systemics, School of Computer Science, University of Petroleum and Energy Studies, Dehradun, India; email: [email protected]; H. Singh, Department of Virtualization, School of Computer Science, University of Petroleum and Energy Studies, Dehradun, India; email: [email protected]; M. Singh, Centre for Climate Change Research, Indian Institute of Tropical Meteorology (IITM), Pune, India and Interdisciplinary Programme (IDP) in Climate Studies, Indian Institute of Technology (IIT), Bombay, India; email: [email protected]; K. Kaur, QualiteSoft, Chandigarh, India; email: [email protected];. M. Usman, School of Computing and Information Systems, Melbourne School of Engineering, The University of Melbourne, Parkville, 3010, Victoria Australia; email: [email protected]; and R. Buyya, Cloud Computing and Distributed Systems (CLOUDS) Laboratory, School of Computing and Information Systems, The University of Melbourne, Australia; email: [email protected]. very challenging to manipulate due to technological limitations. However, in recent years, the field of quantum computing has rapidly progressed and emerged as one of the highly topical areas of research. Quantum computing has the potential to offer computational capabilities which will surpass any existing supercomputer, and this has sparked huge interest from both industry and academia to build a world’s first quantum machine . Today, many big industry names such as IBM, Google, Microsoft, and Intel, as well as many ambitious start-up companies such as Rigetti and IonQ are actively perusing the race to develop a first large scale universal quantum computer. In parallel to quantum hardware development, the area of quantum software and quantum algorithm development has also seen tremendous progress in the last few years. It is well known that in conventional classical digital computing, the information is stored and processed as bits which can take a definite binary value (‘0’ or ‘1’). The equivalent in quantum computing is known as quantum bit, or just qubit, which by the virtue of quantum mechanics could take values of ‘0’, ‘1’, or any superpositions of ‘0’ and ‘1’ (effectively bein g in both 0 and 1 states at once!). Quantum computers, therefore, can access an exponentially large Hilbert space (or computational space), where ‘n’ qubits could be in a superposition state of 2 n possible outcomes at any given time. This may allow quantum computers to tackle many data intensive problems such as analysis of chemical systems, searchers in large databases, finding solutions of intractable optimisation problems, and cracking cryptography by factorization of large numbers. Developing a large-scale quantum computer has its own challenges. One of the major challenges in quantum hardware development arises from decoherence of qubits, whereby qubits lose their coherent properties via interaction with environment. This implies that qubits in a superposition state will decohere to classical bits and therefore any quantum advantage will diminish. Indeed, the efficiency of the current generation of quantum computers suffer from noise or errors arising from decoherence mechanisms, and therefore these are known as “Noisy Intermediate Scale Quantum” (NISQ) machines [2] [3]. Much of the ongoing research efforts in quantum computing are focused on overcoming errors in NISQ devices by developing efficient error correction protocols. A second major challenge is related to connectivity of qubits in today’s quantum devices. Due to relatively sparse connectivity of qubits in the current generation of quantum machines, it is non-trivial task to map a quantum circuit (for a quantum algorithm) on the real quantum hardware devices. Despite technical challenges, NISQ quantum computers are already offering glimpses of computational capabilities which are expected from such quantum devices. The recent demonstration of quantum supremacy by Google team is a significant milestone in the area of quantum computing [10]. An intense global race is now ongoing to achieve a first quantum computing application which solves a useful real-world problem that is intractable on classical computers – also known as ‘ quantum advantage ’ . To achieve this feat, significant progress in both quantum hardware and quantum algorithm development would be required in the coming years. Quantum algorithms are already being developed and benchmarked on NISQ devices at a rapid pace. In early 90’s, there were only a few notable quantum algorithms such as Grover’s and Shor’s; however today hundreds of new quantum algorithms have been developed [254]. Among these, the most widely used quantum algorithm is Variation Quantum Algorithm (VQE) [11] which are based on a combination of quantum and classical components. VQE algorithms have shown excellent results on NISQ devices for problems in quantum chemistry and quantum machine learning fields. A few other major categories of quantum algorithms include Algebraic (such as discrete log or verifying matrix products), search (such as Grover and amplitude amplification), and variational (such as quantum approximate optimisation). The full potential of quantum computing for real-world applications can only be realized when a large-scale fault-tolerant universal quantum computer will be available which may require several years of further development; however, the quantum speed-up on the existing NISQ era devices is already being accessed for prototype applications exhibiting promising results. Among these, variational quantum algorithms and quantum machine learning are two of the most active areas of research for NISQ devices. Quantum machine learning promises to speed up machine learning algorithms for analyzing the classical data. There have already been proposals for quantum principle component analysis, quantum support vector machine and quantum neural network. It is not yet fully established if quantum machine learning would offer superior computational efficiency when compared to classical machine learning implementations; however, recent work has shown promising results [12] [13]. Quantum computers consume less energy, therefore processing data intensive problems by quantum machine learning algorithms may cut down energy cost, and the dependency on fossil-fuels could decrease [14]. For the implementation of quantum algorithms on NISQ devices, there are several well-known models such as the quantum circuit model, quantum Turing machine, adiabatic quantum computer and one-way quantum computer . Quantum computer’s architecture is simpler as compared to classical computers. Also, they do not contain any memory or processor, rather consisting of qubits and gates allowing them to perform computational tasks [2]. Representation of data through qubits is done efficiently as compared to classical bits. Quantum computing does not yet have its own high-level programming language. Thus, it takes the assistance of very specific algorithms that can be encoded in quantum circuits by systematically applying the available set of quantum gates or operations. However, classical computing incorporates standardized languages such as Java, SQL and Python [3]. Quantum Computing Language (QCL) is the most proficient quantum programming language. QCL syntax and data types are similar to the C language’s syntax and data types [4]. Another highly active area of research in the field of quantum computing is post-quantum secure communication. Cryptography is a technique which is used for hiding information from any unintended recipient [5]. Although quantum cryptography expresses with the help of quantum phenomenon which is at the centre of a safety plan, post-quantum cryptography refers to cryptographic algorithms that are thought to be protected against an assault by a quantum computer [6]. A secure communication way which provides a protection layer for confidential data and executes a cryptographic protocol having components of quantum mechanics is known as Quantum Key Distribution (QKD) [7]. QKD shares a random secret key between two parties, which is known only to them; further, it can be used for encrypting and decrypting messages. For post-quantum cryptography, lattice-based cryptography is imperative as it provides more security to data and prevents threats [8]. Error-correcting code is used by algorithmic primitive in code-based cryptographic systems [9]. Asymmetric cryptographic primitives are based on multivariate polynomials over a finite field are known as Multivariate cryptography.
Quantum computing is a rapidly progressing field of research with major developments happening all over the world towards many different aspects such as hardware development, software/algorithm development, error correction on NISQ devices, and applications. This review article is expected to provide a comprehensive and timely report on the recent progress and future directions, which will be beneficial for researchers as well as industry engineers working on a broad range of topics. Quantum computing brings various advantages for the applications, application developers, and several industries by distributing the primary functions as shown in Figure 1.
The key motivation behind this comprehensive survey is to conduct a review of the existing literature on quantum computing. It encompasses the definition of quantum computing, its background, taxonomy, comparison of related studies based on taxonomy, quantum software tools and technologies, post-quantum cryptography and scalable quantum computer hardware. There is a need to identify open challenges and future research directions within the field of quantum computing.
There have been only a few surveys conducted on quantum computing in the existing literature. Savchuk and Fesenko [58] presented a general overview of quantum computing research, while Laszlo and Sandor [42] discussed the fundamentals of quantum mechanics, such as quantum entanglement and quantum superposition. Dagmar et al. [60] presented a survey on quantum cryptography until the year 2007 but a lot of advanced research has been carried out in this field after that survey. Nour et al. [61] discussed advances in the quantum-theoretical approach to image processing applications only. An introduction to quantum computing for non-physicists was presented by Eleanor et al. [62]. Hamid et al. [63] proposed a survey on post-quantum lattice-based cryptography implementations, which is just one type of post-quantum cryptography. There is a need for a fresh systematic review which discusses everything from the definition of quantum computing to open challenges. Therefore, this study offers a systematic review of quantum computing literature, its taxonomy and maps the related studies based on this taxonomy. Further, detailed discussion on quantum software tools and technologies, post cryptography and industrial quantum computers is presented along with possible future directions. Table 1 shows the comparison of our survey with the existing surveys. Figure 1: Quantum brings various advantages for the applications, application developers, and several industries by distributing the main functions
Figure 2 describes the organization of the rest of the paper. Section 2 presents the building blocks and state of the art techniques for quantum computing. The taxonomy of quantum computing is proposed in Section 3. Section 4 offers the mapping of the taxonomy. Section 5 presents quantum software tools and technologies. The quantum and post-quantum cryptography are presented in Section 6. Section 7 presents the scalable quantum computer hardware. Section 8 highlights the future research directions. Section 9 concludes the paper.
Appendix A describes the list of abbreviations.
Table 1: Comparison of our survey with existing surveys
Survey General Overview Basics of Quantum Computing State of the art techniques Taxonomy Quantum Cryptography Mapping of the Taxonomy Quantum Software Tools and Technologies Post-Quantum Cryptography Scalable Quantum Computer Hardware Future Research Directions ✔ (*) ✔ (*) ✔ (*) ✔ (*) ✔ ✔ (+) ✔ (+) ✔ (+) ✔ (+) ✔ (*) ✔ (+) ✔ (+) ✔ (+) ✔ (*) ✔ (*) ✔ (*) ✔ (*) ✔ (*) ✔ (*) ✔ (*) ✔ (*)
1: Savchuk and Fesenko [58], 2: Laszlo and Sandor [42], 3: Dagmar et al. [60], 4: Nour et al. [61], 5: Eleanor et al. [62], 6: Hamid et al. [63] and 7: Our Survey (This Paper). Note: (*: Means Comprehensive Discussion, +: Means Just an Overview)
Industries use Quantum:
Google, transportation, telecommunication, energy, weather, cyber security, and healthcare for fast radiotherapies, diagnosis and good trails, national laboratories for solving complex problems, automobiles for driverless cars, finance for fast estimating risks, trading, market instability, portfolio development, design automation,
Quantum benefits:
Medications, machine learning, efficient calculations, weather forecasting, biology, and, speech and image recognition, fault tolerant, quantum programming languages and system, quantum security and privacy
Quantum help devices: solve, determine, schedule, diagnostic, prediction
Quantum distributes core functions: keys, computation, algorithm, and communication, secure data computation, secure data transfer, attack detection, networking, distributed quantum computing Figure 2: The Organization of this Survey
2. BUILDING BLOCKS
Quantum mechanics concepts such as quantum interference, no-cloning theorem, quantum entanglement and quantum superposition are the underpinning principles of Quantum Computing (QC). In this section, we review the most recent literature on the technologies related to QC. The QC technologies offer many promising ways to solve computation problems which otherwise are challenging when traditional computing techniques are used. There are also many differences between the speed of solving computational problems using QC and traditional computing. In terms of the size of physical quantum devices, the quantum techniques are still in the phase of incubation.
The basic building blocks of the quantum computers, as shown in Figure 3, consists of quantum central processing units, quantum gates, quantum controlling and measurement, quantum error detection and correction tools, quantum memories. The function of quantum gates in a quantum computer is to perform the operations that are unitary in nature [218]. Further, quantum error detection and correction codes are used to locate and correct the errors that exist during the operations of the quantum gates. Quantum gates can be deployed in various arrangements such as shallow circuits [36], and instantaneous quantum polynomial-time circuit [37] depending upon the applications.
Quantum Computing: A Taxonomy, Systematic Review and Future Directions Building Blocks Taxonomy Mapping of the Taxonomy Quantum Software Tools and Technologies Scalable Quantum Computer Hardware Post-Quantum
Cryptography
Background State-of-the-Art Quantum computers and present-day classical computers Industrial applications of quantum computers Hardware requirements of industrial quantum computers Challenges in scalable production of industrial quantum computers Present status, state of the art and future outlook in industrial quantum computers Hybrid Schemes Lattice-based Cryptosystem Code-based Cryptosystem Multivariate Cryptosystem Isogenies on super-singular-based Cryptosystem
Engineering/Design Challenges Reliable Quantum Computing
Quantum- assisted Machine Learning Energy Management Quantum Internet Robotics
Simulations for Complex Quantum Experiments Post-quantum Cryptography
Future Research Directions Quantum and Post-quantum Cryptography Quantum Key Distribution
Numerical Weather Prediction Figure 3: Basic building blocks of quantum computers
The collection of multiple quantum states in various arrangements constitutes quantum memories. Quantum memories use quantum registers to save the quantum states of the quantum circuit. In the recent past, quantum memories have been realized using arrays of quantum states to form a stable quantum system [40]. The quantum central processing unit is an integral part of the quantum computer. It uses the quantum bus for communication amongst various other units of the quantum computer [52]. Quantum controlling and measurement mechanism is required in quantum computers for the proper monitoring of various manipulations of the quantum states and quantum computations while handling the error correction and detection processes [51]. The error in quantum computers can be identified by getting the information from quantum states. The nature of errors detected in quantum computers is entirely different as compared to traditional computing systems [217]. The optimal quantum error correction method can suggest additional ways to satisfy the requirements of high fidelity of the quantum computers [49]. The core concept of quantum computing like quantum logic gates, reversible computation idea of Fredkin and
Bennet, quantum registers, qubits, Shor’s factorization algorithm [228], quantum complexity and quantum entanglement have been discussed by Hey et al. [44]. Further, the experimental status has been reviewed to get a better understanding of the quantum computer’s physical implementation. Han et al. [43] proposed an algorithm that is evolutionary in nature and is inspired by the principles of quantum computing known as QEA i.e. “Quantum -i nspired Evolutionary Algorithm”. The concept of the quantum bits (Q -bit) and quantum gates (Q-gate) have been applied enabling the algorithm to reach out to an optimal solution. For validation of QEA algorithm, its applicability for solving the knapsack problem is demonstrated, and the results have been compared with the traditional genetic algorithm. Buhrman et al. [38] performed a detailed survey of quantum computing techniques and explored its various applications in the distributed network framework. Gay et al. [41] studied and reviewed the concept of quantum programming languages. Further, the design of quantum programming languages including their syntax, semantics and compilers for quantum computing have been discussed, and future research directions quoted. Rotteler et al. [57] provided an overview of quantum algorithms. They stated that quantum algorithms could be classified into three major categories namely amplitude amplification type algorithms, hidden subgroup type algorithms and the quantum alg orithm that doesn’t fall in the given two categories as the third category. All the three classifications have been studied to prove how quantum algorithms are different from traditional algorithms and how the computation speed will grow faster by using these quantum algorithms. For programming a quantum computer, a quantum programmer requires a programming language tools. Sofge et al. [166] analyzed various programming language tools that are present in the market for quantum programmers. The detailed comparative analysis among multiple tools available has been done and quoted for the future reference. Zhang et al. [171] inked and revealed the concept of Quantum-inspired Evolutionary Algorithm (QEA) by merging two recent buzzwords, i.e. evolutionary algorithms and quantum computing [222]. The basic architecture and system model of QEA has been explained and reviewed. The comparative analysis of various QEA has been done and quoted as future research directions. Menon et al. [50] pointed out the protocols required to provide the error-free translation of the abilities of the traditional computing system in contrast to the quantum computing system and vice versa. The existing simulators for quantum computing utilizing its capabilities to the fullest extent have been studied. To increase the computing performance of the classical computing system, various architectures of quantum computer that exist in the literature have been explored by S Jain et al. [45]. The tristate state logic (0, 1, 0 or 1) of qubits in a quantum computer has introduced the decoherence problem. The architecture of quantum computer should be such that it should resolve this decoherence problem by proper management of errors that occur when performing quantum arithmetic computations. Kumar et al. [47] discussed various components of quantum computing like qubits and quantum superposition. Quantum computers are studied in terms of their efficiency and power. They picked two organizations from the list of organizations dealing with quantum computing and explored for their recent contributions in the field. Further, the research and development challenges related to quantum computing faced by these two organizations have been highlighted as future research challenges. Shaikh et al. [165] demonstrate the significance of big data analytics in quantum computing. Quantum computers have been most suitable for processing a large amount of data and information that too in a really quick time. Quantum machine learning algorithms that will scale up the processing speed of quantum processors by applying quantum walk in quantum Artificial Neural Networks (ANN) have been discussed. Yan et al. [170] coined the concept of QIMP i.e. “Quantum Image Processing” which refers to the process of performing all kinds of manipulations on the quantum images for achieving multiple objectives. Further, different QIRs i.e. “Quantum Image Representation” which is the logical representation of the quantum images have been explored, and their applicability reviewed. Kaiser et al. [46] documented the lecture notes which have been delivered by him to the audience who were eager to learn but have significantly less knowledge about quantum computing. The basic idea was to provide the introduction of the fundamental concepts of quantum computing and explain how quantum topology enters the computation field. Roetteler et al. [56] stressed on the quantum security mechanisms and protocols involved in the various processes of a quantum computer. The comparative analysis of the cryptographic applications based on the quantum computing system and classical computing system has been done. Gyongyosi et al. [42] reviewed the most recent work done in the field of quantum computing. The experimental results of different quantum computing technologies have been demonstrated, and the problems related to it addressed. Savchuk et al. [58] emphasized on the concept of quantum computing which should be scalable. The existing quantum computer has been analyzed in detail for understanding its implementation. Further, it has been concluded that sufficient stress has not been laid by scientists and researchers on developing the scalable quantum computer. Li et al. [48] discussed merging the elements of quantum mechanics with the intelligent nature-inspired algorithms to mark the new era of computing in the making. Quantum optimization and quantum learning are two classifications based on which the existing quantum algorithms were studied. Further, it was concluded that the nature-inspired quantum algorithms possess high potential when compared with classical-quantum computing algorithms. Nobel laureate Richard Feynman was the first to postulate the idea of a quantum computer. The properties of quantum mechanics are leveraged by quantum computers and these properties form the basis of quantum computers. The quantum algorithms have come a long way starting from the simulations of quantum physics to a variety of applications in computer science. An industrial scale quantum computer will be a prized progress in achieving the processing power of its kind, which would have implications in various fields such as cybersecurity and others. The first quantum algorithm to find speed greater than that of a classical algorithm was proposed by Daniel Simon. Table 2 shows the comparison of quantum computing algorithms. [21, 22, 55, 33, 65, 69, 71, 100, 228, 247, 248]. Table 2: Summary of Quantum Computing Algorithms
Name Year Type Objective
Deutsch – Jozsa Algorithm 1992 Based on Quantum Fourier Transform Problems requiring exponential queries Bernstein – Vazirani Algorithm 1992 Efficient solutions of black-box problem Simon's Algorithm 1994 Faster computation, speedup Shor's Algorithm 1994 Integer factorization and discrete logarithm problems Grover's Algorithm 1996 Based on Amplitude Amplification Searching unstructured database for marked entry Quantum Counting 1998 Generalized search Quantum Approximate Optimization Algorithm 2014 Hybrid quantum/classical algorithm Solution of graph theory problems
3. TAXONOMY
In this section, quantum computing technologies are classified based on different types of features and operations it exhibits. The various components of the quantum computing taxonomy are a) Basic characteristics, b) Algorithmic characteristics, c) Time and gate characteristics and d) Other characteristics. The diagrammatic representation of the taxonomy of quantum computing technology is shown in Figure 4. For better understanding, the brief description of every element of quantum computing taxonomy is given in the adjoining subsection.
The basic characteristics of quantum computing include elements like qubit implementation, classification based on quantum computing technology and performance metrics. The basic features of quantum computing are to explore how qubits can be implemented and represented. Qubit representation can be done either in stationary, flying or mobile ways. The stationary method is similar to traditional programming, whereas mobile approach resembles designing conventional circuits. Further, the ensemble computing and singleton computing is another classification based on the choice of quantum computing technology. An ensemble computing system is a group of quantum computers that are identical in specifications and performs the same set of functions. In contrast, the singleton computing system consists of a single quantum computer performing designated operations. The performance metrics form the base of another classification of quantum computing techniques which has mechanical vibrations, fluorescence and concurrency as its attributes.
Quantum computing techniques can be realized by implementing quantum algorithms on the classical computing infrastructure. In lieu of this, it is essential to discuss and categorize the quantum computing technologies on the basis of characteristics represented by quantum algorithms. The algorithmic elements of quantum computing technologies such as parallelism, aggregate count of qubits available, topologies, techniques for locating the qubits and qubit operations. Parallelism is the central feature because the parallel implementation of quantum gates is required to either prevent or minimize the qubits decoherence. The aggregate count of qubits available is another feature that helps in realizing the reliability and scalability of the quantum computer. The various possible arrangements of different physical devices in the architecture of the quantum computer is termed as its topologies. Architecture optimization is the primary concern because it enables the smooth flow of data and information among different physical units of the system. The addressing scheme for locating an individual qubit is logically very complex. This feature enables to explore the qubit states more specifically as far as the quantum computer physical implementation is concerned. Further, for performing any operation on qubits, they have to be moved from the address where they are stored to the location where the qubit gates are performing the action or execution on them. The quantum computing technologies can be further classified on the basis of time and gate characteristics which include components like the decoherence time and measurement time. The decoherence time is given by the time until which a qubit can be kept in a specific state. The decoherence time is the topic of research in the field of quantum computing nowadays. Another essential characteristic of classification is the measurement time which is the time required to measure the qubit state precisely.
There are few other characteristics like scalability, timing and control of gate-level qubits on which the classifications of quantum computing technologies have been done. All of the above-discussed features contribute towards the scaling of qubits to larger numbers. Meanwhile, it is recommended to use multiple qubits so that it will not always represent a single ion or photon. The changes in the qubit states are a continuous process with respect to time, hence computing an accurate timing of gates is critical. The arrival times of the qubits should be precisely adjusted while placing multiple qubits in their relative phases at the same time.
Figure 4: Taxonomy of quantum computing technology
4. MAPPING OF THE TAXONOMY
In this section, various quantum computing technologies are mapped based on taxonomy identified and framed in the previous section. The taxonomy-based mapping of quantum computing techniques is shown in Table 3. The quantum computing technologies considered for mapping are chosen based on the following criteria. 1.
It represents the most recent and significant research work done in the field of quantum computing. 2.
It should exhibit the fundamental characteristics defined in the taxonomy of quantum computing which forms the basis of mapping. Table 3:
Taxonomies-based mapping of quantum computing techniques
Work A B C D E F G H I J K L
C Weitenberg et al. [169] ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ M Tomza et al. [167] ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ J O’ Gorman et al. [54] ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ X G Qiang et al. [55] ✓ ✓ ✓ ✓ ✓ ✓ E Compagno et al. [39] ✓ ✓ ✓ ✓ ✓ ✓ S Schaal et al. [59] ✓ ✓ ✓ ✓ ✓ ✓ ✓ F A Zwanenburg et al. [172] ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ M Veldhorst et al. [168] ✓ ✓ ✓ ✓ ✓ ✓ ✓ R Mizuta et al. [53] ✓ ✓ ✓ ✓ A C C de Albornoz et al. [35] ✓ ✓ ✓ ✓ ✓ ✓ Abbreviations - A: Qubit Implementation, B: Classification based on quantum computing technology, C: Performance Metrics, D: Parallelism, E: Aggregate count of qubits available, F: Topologies, G: Techniques for Locating qubits, H: Qubit operations, I: Decoherence time, J: Measurement time, K: Timing and control of Gate level qubits, L: Scalability.
5. QUANTUM SOFTWARE TOOLS AND TECHNOLOGIES
Quantum computers are expected to perform tasks more efficiently and quickly compared to classical computers. The experiments and results in quantum computation have created a great interest in the research community to develop tools and techniques that automate the quantum world practices. Over more than a decade, many quantum simulators have been designed to exploit the capabilities of experimental quantum computing infrastructure in academic and industry research [230]. Table 3 shows the comparative analysis of quantum tools developed. In this analysis, the parameters used for comparative analysis are briefly explained as follows (i) a library is considered as a collection of functions or classes designed for quantum information and similar computations, (ii) a tool is a piece of software that can simulate quantum computing or associated calculations, (iii) the quantum computing libraries, tools, or techniques are found to be either open-source , commercial or freeware , (iv) Graphical User Interface (GUI)-based quantum tools are available that ease the job of circuit designing, programming and displaying the results for users, (v) many GUI-based tools can display the results either in two or three-dimensions , (vi) additionally, many tools have command-line usage where instructions are predefined to connect the gates, design the inputs and observe the outputs, (vii) quantum gates are analogous to conventional logic gates for quantum computers. Few examples of quantum gates include Hadamard, phase shifter, controlled, uncontrolled, and CNOT (Controlled NOT gate). (viii) In this work, a review of most of the quantum tools available for academic or research work, used for simulation rather than real-implementation , is performed (ix) while exploring the quantum tools, it has been observed that many of such tools provide an existing implementation of quantum algorithms. Few of these algorithms include Shor, Deutsch-Jozsa, Simon, Quantum phase estimation, Hidden subgroup, and Grover algorithms [55] [100]. These algorithms are classified in one of the following categories: quantum Fourier transform, amplitude amplification, quantum walks, bounded-error quantum polynomial time (BQP)-complete, and hybrid quantum/classical, (x) gates scheduling, and parallelism is vital for circuit designing that is analogous to quantum computer operations. This concept speed-up the operations in quantum computing, and (xi) most of the quantum gates require matrix operations for their computations. This matrix and associated operations are incorporated in many tools. Table 4 shows the comparative analysis of quantum tools with the above parameters. This comparative analysis shows the programming languages used in the tools as well. Table 4: Comparative analysis of Software Tools and Technologies Tool/Technique Name A B C D E F G H I J K L M N O Programming Language
QuEST [96] ✔ ✔ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ✔ ✔ ❌ ❌ ❌ ❌ C Staq [97] ✔ ✔ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ✔ ✔ ❌ ❌ ❌ ❌ C++ Scaffold/ScaffCC [98] ✔ ✔ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ❌ ✔ ❌ ✔ ❌ ❌ Scaffold
Qrack [99] ✔ ✔ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ❌ ✔ ❌ ❌ ❌ ❌ C++ QX Simulator [100] ✔ ✔ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ❌ ✔ ❌ ❌ ✔ ❌ Quantum Code Quantum++ [101] ✔ ✔ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ❌ ✔ ❌ ❌ ❌ ❌ C++ QMDD [102] ✔ ❌ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ❌ ✔ ❌ ❌ ❌ ✔ C++ CHP [103] ✔ ❌ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ✔ ✔ ❌ ✔ ❌ ❌ C Eqcs [104] ✔ ❌ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ✔ ✔ ❌ ❌ ❌ ❌ C LanQ [105] ✔ ❌ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ✔ ✔ ❌ ✔ ✔ ❌ LanQ libquantum (C) [106] / (C++)[107] ✔ ❌ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ✔ ✔ ❌ ✔ ✔ ❌ C, C++ Open Qubit [108] ✔ ❌ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ✔ ✔ ❌ ✔ ❌ ❌ C++ Quantum Programming Studio [109] ❌ ✔ ✔ ❌ ❌ ❌ ❌ ✔ ❌ ✔ ✔ ❌ ✔ ✔ ✔ Javascript Qubit Workbench [110] ❌ ✔ ❌ ✔ ❌ ✔ ❌ ✔ ❌ ✔ ✔ ❌ ✔ ❌ ✔ -- Linear AI [111] ✔ ❌ ❌ ❌ ✔ ✔ ❌ ❌ ✔ ✔ ✔ ❌ ✔ ❌ ✔ Mathematica QCAD [112] ❌ ✔ ❌ ❌ ✔ ✔ ✔ ✔ ❌ ✔ ✔ ❌ ❌ ❌ ✔ -- qsims [113] ✔ ✔ ✔ ✔ ❌ ❌ ❌ ❌ ✔ ✔ ✔ ❌ ✔ ✔ ❌ C++ Q-gol [120] ✔ ✔ ✔ ❌ ❌ ✔ ✔ ✔ ❌ ✔ ✔ ❌ ✔ ❌ ❌ CaML QOCS [121] ✔ ✔ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ✔ ✔ ❌ ✔ ❌ ❌ OCaML Q++ [122] ✔ ❌ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ✔ ✔ ❌ ✔ ❌ ✔ C++ Qinf [123] ✔ ✔ ✔ ❌ ❌ ❌ ❌ ❌ ❌ ✔ ✔ ❌ ❌ ❌ ✔ Maxima Quantum Fog [124] ✔ ✔ ✔ ❌ ❌ ✔ ❌ ✔ ❌ ✔ ✔ ❌ ✔ ✔ ✔ -- SimQubit [125] ✔ ✔ ✔ ❌ ❌ ✔ ❌ ✔ ❌ ✔ ✔ ❌ ✔ ❌ ✔ C++ Q-Kit [126] ❌ ✔ ❌ ❌ ✔ ✔ ❌ ✔ ❌ ✔ ✔ ❌ ✔ ✔ ✔ -- Bloch Sphere [127] ✔ ✔ ✔ ❌ ❌ ✔ ✔ ✔ ❌ ✔ ✔ ❌ ✔ ❌ ✔ Java BackupBrain [128] ✔ ✔ ✔ ❌ ❌ ✔ ❌ ❌ ❌ ✔ ✔ ❌ ✔ ❌ ✔ Javascript Quantum Circuit [129] ✔ ✔ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ✔ ✔ ❌ ✔ ❌ ✔ Javascript Jsquis [130] ✔ ✔ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ✔ ✔ ❌ ❌ ❌ ❌ Javascript QSWalk.jl [131] ✔ ✔ ✔ ❌ ❌ ✔ ❌ ❌ ✔ ✔ ✔ ❌ ❌ ❌ ❌ Julia QuantumOptics.jl [132] ✔ ✔ ✔ ❌ ❌ ✔ ✔ ❌ ✔ ❌ ✔ ❌ ✔ ❌ ✔ Julia QuantumWalk,jl [133] ✔ ✔ ✔ ❌ ❌ ✔ ❌ ❌ ✔ ✔ ✔ ❌ ✔ ❌ ❌ Julia Feynman [134] ✔ ✔ ✔ ❌ ❌ ✔ ❌ ❌ ✔ ✔ ✔ ❌ ✔ ✔ ❌ Maple OpenQUACS [135] ✔ ✔ ✔ ❌ ❌ ✔ ❌ ❌ ✔ ✔ ✔ ❌ ✔ ✔ ❌ Maple Quantavo [136] ✔ ✔ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ✔ ✔ ❌ ✔ ✔ ❌ Maple QDENSITY [137] ✔ ✔ ✔ ❌ ❌ ✔ ❌ ❌ ✔ ✔ ✔ ❌ ❌ ❌ ✔ Mathematica Quantum [138] ✔ ✔ ❌ ❌ ✔ ✔ ❌ ❌ ✔ ✔ ❌ ❌ ✔ ✔ ✔ Mathematica QuantumUtils [139] ✔ ✔ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ✔ ✔ ❌ ❌ ✔ ✔ Mathematica Qi [140] ✔ ✔ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ✔ ✔ ✔ ❌ ❌ ❌ Mathematica M-fun [141] ✔ ❌ ✔ ❌ ❌ ✔ ❌ ❌ ✔ ✔ ✔ ❌ ❌ ❌ ❌ MATLAB/Octave Quantencomputer [142] ✔ ✔ ❌ ❌ ❌ ❌ ❌ ❌ ✔ ✔ ✔ ❌ ✔ ❌ ❌ MATLAB Drqubit [143] ❌ ❌ ✔ ❌ ✔ ❌ ❌ ❌ ✔ ✔ ✔ ❌ ❌ ✔ ❌ MATLAB Qubit4Matlab [144] ✔ ❌ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ✔ ✔ ❌ ✔ ✔ ✔ MATLAB QuIDE [145] ❌ ✔ ✔ ❌ ❌ ✔ ❌ ✔ ❌ ✔ ✔ ❌ ✔ ✔ ✔ .NET Quantum.NET [146] ✔ ✔ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ✔ ✔ ❌ ❌ ❌ ❌ .NET Qubit Workbench [147] ❌ ✔ ❌ ❌ ✔ ✔ ❌ ✔ ❌ ✔ ✔ ❌ ❌ ❌ ❌ _ Cirq [148] ✔ ✔ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ✔ ✔ ❌ ✔ ✔ ✔ Python ProjectQ [149] ✔ ✔ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ✔ ✔ ❌ ✔ ✔ ✔ Python QCircuits [150] ✔ ✔ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ✔ ✔ ❌ ✔ ✔ ✔ Python Qiskit [151] ✔ ✔ ✔ ❌ ❌ ✔ ❌ ❌ ✔ ✔ ✔ ❌ ✔ ✔ ✔ Python OpenQasm [152] ✔ ✔ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ✔ ✔ ❌ ❌ ❌ ✔ QASM QCGPU [153] ✔ ✔ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ✔ ✔ ❌ ❌ ✔ ❌ Rust & OpenCL QIO [154] ✔ ❌ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ✔ ✔ ❌ ❌ ❌ ✔ Qio + Haskell Qchas [155] ✔ ❌ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ✔ ✔ ❌ ❌ ❌ ✔ Haskell Quantum User Interface [255] ❌ ✔ ✔ ❌ ❌ ✔ ❌ ✔ ❌ ✔ ✔ ❌ ❌ ❌ ❌ Protobuf
A: Library, B: Toolkit, C: Open Source, D: Commercial, E: Freeware, F: GUI-based, G: 3D visualization, H: Drag & Drop Support, I: Command-Line Usage J: Support for Quantum Gates, K: Simulation, L: Real-Implementation. M: Built-in quantum algorithm support, N: Gates scheduling & Parallelism, O: Diagram or Matrix support
6. QUANTUM AND POST-QUANTUM CRYPTOGRAPHY
Quantum cryptography is defined as quantum mechanical properties for cryptography tasks such as quantum key distribution, encryption/decryption, signature, authentication and hashing [61-64]. The major advantages of quantum cryptography include the usage of fundamental laws of physics rather than mathematics-based algorithms which are simple to use but unbreakable, consume fewer resources. Copying quantum state’s encoded data is not feasible, and this reduces the chances of attack and increases the probability of eavesdropping detection, better performance as compared to traditional cryptography etc. Table 5 shows a comparative analysis of quantum cryptography approaches designed and experimented in recent times. These approaches are classified based on various parameters including: designed or experimented for communication protocols, implementation, simulation, quantum-based authentication mechanisms, quantum-based encryption/decryption operations, quantum key distribution, quantum attack detection & analysis, short survey, long survey, approaches using programming for quantum operations, long or short-distance entanglement, attacks (efficiency-mismatch, detector-blinding, detector dead-time, beam-splitter, spatial-mode, eavesdropping), and approaches where data analysis is performed either using machine learning. Table 5: Quantum Cryptography Approaches Author Year A B C D E F G H I J K L M N O P Q R S
Deutsch et al. [75] 1996 ✔ ❌ ❌ ❌ ❌ ❌ ✔ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ✔ ❌ Naik et al. [88] 2000 ✔ ❌ ✔ ❌ ❌ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ❌ Elboukhari et al. [78] 2010 ✔ ❌ ❌ ❌ ❌ ✔ ❌ ✔ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ Bugge et al. [95] 2014 ✔ ✔ ❌ ❌ ❌ ✔ ✔ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ✔ ❌ ✔ ❌ Jain et al. [93] 2014 ✔ ✔ ❌ ❌ ❌ ✔ ✔ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ✔ ❌ Bruss et al. [77] 2017 ✔ ❌ ❌ ✔ ✔ ✔ ✔ ❌ ✔ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ✔ ✔ ❌ Li et al. [79] 2018 ❌ ❌ ❌ ✔ ✔ ✔ ✔ ✔ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ✔ ✔ ❌ Bennett and Brassard [72] 2020 ✔ ❌ ❌ ❌ ❌ ✔ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ Bhusal et al. [83] 2020 ✔ ❌ ✔ ❌ ✔ ❌ ❌ ✔ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ✔ ✔ ✔ Brassard et al. [76] 2000 ✔ ❌ ❌ ❌ ❌ ✔ ✔ ✔ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ✔ ❌ Durak and Jam [90] 2020 ✔ ✔ ❌ ✔ ❌ ✔ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ❌ ❌ ❌ ❌ ❌ ✔ Gras et al. [85] 2020 ✔ ✔ ❌ ❌ ❌ ✔ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ✔ ✔ ❌ ❌ ❌ ❌ Guo et al. [81] 2020 ✔ ❌ ✔ ❌ ❌ ✔ ✔ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ✔ ✔ ✔ ❌ Huang et al. [94] 2020 ✔ ✔ ❌ ❌ ❌ ✔ ✔ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ✔ ❌ Melhem et al. [92] ❌ ❌ ❌ ❌ ✔ ✔ ✔ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ Qi et. al. [82] 2020 ✔ ✔ ❌ ✔ ✔ ✔ ❌ ❌ ❌ ❌ ✔ ✔ ✔ ❌ ✔ ❌ ✔ ✔ ❌ Shang et al. [89] 2020 ✔ ❌ ❌ ✔ ❌ ✔ ✔ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ✔ ❌ Trushechkin et al. [86] 2020 ✔ ❌ ❌ ❌ ❌ ✔ ❌ ❌ ❌ ❌ ❌ ❌ ✔ ❌ ❌ ❌ ❌ ❌ ❌ Vybornyi et al. [91] 2020 ✔ ❌ ❌ ❌ ✔ ✔ ✔ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ❌ ✔ ❌ Yin et al. [80] 2020 ❌ ✔ ❌ ✔ ✔ ✔ ✔ ❌ ❌ ❌ ✔ ❌ ✔ ✔ ✔ ✔ ✔ ✔ ❌ Zhang et al. [87] 2020 ✔ ❌ ✔ ❌ ❌ ✔ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ❌ ❌ ❌ ❌ ❌ ✔ Zhou et al. [84] 2020 ✔ ✔ ❌ ❌ ❌ ✔ ❌ ❌ ❌ ❌ ❌ ✔ ❌ ❌ ❌ ❌ ❌ ❌ ✔ A: Communication protocols, B: Implementation, C: Simulation, D: Quantum Authentication, E: Quantum Encryption, F: Quantum key distribution, G: Quantum Attack Detection & Analysis, H: Short survey (<10 pages), I: Long Survey (>10 pages), J: Quantum Programming, K: Long-distance entanglement, L: Short-distance entanglement, M: Efficiency-mismatch attack, N: Detector-blinding attack, O:Detector dead-time attack, P: Beam-splitter attack, Q: Spatial-mode attack, R: Eavesdropping attack, S: Data analysis/Machine Learning
Figure 5 shows the classification of quantum cryptography challenges. The challenges are categorized into four major categories, including security attacks and challenges, hardware challenges, performance and cost-related challenges, and quantum-related design challenges. The majority of security attacks and challenges considered various types of security attacks and their feasibility in the quantum world; hardware challenges include experimentation issues whose performance is affected with hardware used. Performance and cost-related challenges include reducing the cost while improving the performance parameters. Finally, design challenges include developing the novel quantum protocols, tools or techniques while addressing the challenges of existing real-time experimentations.
QKD is an effective way of protecting information security using quantum computers [70] [71]. As compared to traditional cryptography-based key distribution mechanisms, which are vulnerable to computational power-based scenarios, a quantum cryptography mechanism (like QKD) is secure against various attacks. In quantum cryptography, no-cloning theorem [114] in quantum mechanics states that it is impossible to make a perfect copy of the quantum system or its states. Thus, any eavesdropping attempt adds noise to the quantum transmission that is easily detectable by two parties (source and destination) [216]. QKD protocols can be classified based on the use of properties during transmission including applied modulation, encoding/decoding, quantum channel implementation etc. Likewise, there are various types of quantum key distribution approaches [73] [74]. Table 6 shows the comparative analysis of these approaches, and these approaches are briefly discussed as follows. Ghalaii et al. [114] discussed the Gaussian and non-Gaussian modulated continuous-variable QKD (CV-QKD) methods. Further, non-Gaussian CV-QKD protocol is extended with a discrete modulation approach for increasing the secret key rates. Besides, the proposed mechanism is found to support the discrete-modulation CV-QKD over CV quantum repeaters and to long-range system operation in-live. Valivarthi et al. [115] proposed a plug-and-play CV-QKD with Gaussian modulation quadratures. In experimentation, two independent fiber stands have been used for two narrow line-width lasers for quantum signal transmission.
Figure 5: Quantum Cryptography Challenges
This experimentation increases the secret key rate up to 0.88 Mb/s with different experimental setups and inputs. This experimentation is considered to be an effective mechanism in terms of low-cost deployment for metropolitan optical networks. The investigation is useful in terms of its design, use of Raleigh back-scattering mechanism to minimize noise, and integration of GG02 symmetric protocol with heterodyne detection [216]. The complete setup makes the proposed QKD faster and secure. Leverrier and Grangier presents a CV-QKD protocol combining discrete modulation and reverses reconciliation [116]. The protocol is tested experimentally, and it is observed that the proposed scheme can distribute the secret key over a long distance while ensuring security. Li et al. [117] proposed a discrete modulated CV-QKD scheme that improves the system performance and secure distance with machine-learning-based detectors. The proposed scheme is capable of processing the secret keys to improve the overall system performance. Lin et al. [118] applied numerical methods to analyze the security aspects in discrete modulated CV-QKD. The two proposed variants of discrete-modulated CV-QKD is capable of generating much high key rates for longer distances as compared to binary or ternary modulation schemes. Ruan et al. [119] analyzed optical absorption and scattering properties of discrete-modulated CV-QKD. It is observed that the performance of four and eight-state protocol in asymptotic and finite-size cases is dependent over seawater composition, i.e. if the composition is complex, then the performance of protocol decreases as well. The variation in optical modulation and minimizing the extra noise can improve the protocol’s performance. In another observation, it was found that the number of states improve performance. In this case, the performance of the eight-state protocol is better compared to the four-state protocol. In recommendations, CV-QKD is found to be significant over the seawater channel and provides a good medium to construct a secure communication network. Table 6:
Comparative analysis of QKD
Type of CV-QKD Pros Cons
Gaussian-modulated CV-QKD • Security analysis is much more advanced compared to discrete-modulated CV-QKD. • Distance limitation for secure QKD is a major concern. • The use of high-performance error-correcting code can improve security but reduces the distance coverage [117]. Discrete-modulated CV-QKD • More suitable for long-distance secure key transmission. • Simple experimentation setup. • Great potential for large-scale deployment in secure quantum networks. • The integration of post-selection strategies with reverse reconciliation can significantly improve the key rates. • Security analysis in this system is more challenging compared to Gaussian-modulated CV QKD because analysis relies on the linearity of the channels which is not an easy condition for verification. Coherent On-Way (COW) Quantum Key Distribution • Simple in experimentation • Reduce interference visibility • Avoid photon number splitting attack to a large extent • Falls in distributed-phase-reference QKD category • Empty pulses contain a light that can introduce noise. This can increase error rates. • Performance decreases with an increase in disturbances. Small disturbances do not affect performance. Differential Phase-Shift (DPS) Quantum Key Distribution • Falls in distributed-phase-reference QKD category. • Integration with randomness or improved transmitter can reduce the disturbances and improve the performance [163] [164]. • Chances of side-channel attacks are higher. Thus, techniques (e.g. attenuation) are required to be integrated for removing it. • Performance decreases with an increase in disturbances. Small disturbances do not affect performance. Six-State Quantum Key Distribution • Using this category of protocols, a high error-rate can be detected easily in the presence of any eavesdropping attack. • The speed of communication lies in the high-speed key distribution category. • The probability of interference, collective attack, and obtaining the secret is low. • Chances of obtaining the secret cannot be completely avoided. • Need to analyze the hidden variable models for protecting the protocols against attacks. • Multiple eavesdropping challenges to topple authenticated communication need to be addressed. Decoy-State Quantum Key Distribution • A high secure key rate can be generated using the decoy-state protocol. • The problem of lower secure key rate can be efficiently handled with inequality based statistical models. • Found to be an effective method in avoiding the photon-number splitting attack. • The secure key rate can be lower down to a significant level if parties' parameters are varied with different decoy states. • Usage of different decoy states is not yet experimented in realistic scenarios to confirm its adaptability with security. • Computational power challenge also reduces the secure key rate and can cause statistical fluctuations.
Stucki et al. [156] presented a COW-QKD protocol with weak coherent pulses. The simplicity of this experimentation increases the bit-rates and reduces interference visibility as well. This protocol achieves the high efficiency for secret bits per qubit generation while lowering the photon number splitting attacks. Mafu et al. [157] [158] realized the importance of the differential-phase-reference category of QKD protocols. In this category, there are mainly two types of QKD protocols, including COW and differential phase shift. Mafu et al. [158] formalized the COW-QKD protocol with non-computing Positive Operator-Value Measures (POVM). This formalization increases the chances to have unconditional security proof against general attacks. The POVM elements effective for generating security proofs against attack include measurement probabilities and positive operators, composite measures, all types of measurements distinguishing between two quantum states, and an informationally complete secure state. Mauf et al. [159] reanalyzed the necessary condition requirements with non-commuting POVM elements-based COW protocol. The major challenge considered in stating unconditional security proof is the class of protocol that uses coherent signals. These coherent signals are not symmetric as compared to qubits used in proof realization. Thus, there is a need to formalize the COW QKD protocol without disclosing the detailed working explanations and parties' confidentiality. It is observed that POVA elements can make this possible with high-security standards. Wonfor et al. [160] conducted a trial of COW-QKD protocol with a commercial-grade encrypted system. In experimentation, a link is launched for QKD with 500 Gbps encrypted data transmitted over a distance of 121 Km. In results, it is observed that QKD in O-band COW protocol with free detectors and C-band DWDM channels gives a stable performance for many weeks. Further, 25 DWDM channels with co-propagation can make the QKD process feasible while ensuring security proofs.
Alhussein and Inoue [161] realized the importance of side-channel attacks in the DPS-QKD system. DPS-QKD protocol is found to be another simple and efficient protocol because it works in cases when precise synchronization of signals between distant parties is not possible. The proposed scheme has avoided the control of blinding and controlling side-channel attacks. To detect a side-channel attack at Bob's side, a variable attenuator is added at random and occasional attenuation inserted. Further, the performance is analyzed, confirming the adaptability of the proposed approach. Collins et al. [162] experimented with the quantum digital signatures transmission over a long distance (90 Km) using the DPS-QKD protocol. The authors claimed that the transmission was aimed to be conducted for long-distance compared to previous works. Distribution of quantum digital signatures ensures message integrity as well as non-repudiation. Further, the performance of the proposed scheme is comparable to the BB84 protocol used for QKD with 1550 nm wavelength and similar experiment setting, including clock rate and transmission distance considered for the operation. Hatakeyama et al. [163] experimented with a round-robin DPS-QKD protocol to reduce the bit error rates. The experiment is conducted to take advantage of simple DPS-QKD functioning to increase the tolerance without compromising on the security issues. This work has extended with basic DPS-QKD protocol with randomness. The randomness and few additional delays increase the performance of the proposed protocol as compared to basic DPS-QKD protocol. The simulated experimentation and key generation rates are analyzed with different randomness patterns. It is observed that the performance of the proposed protocol can be significantly increased with a few parameter changes. Schrenk et al. [164] developed a low-complexity transmitter for DPS-QKD. This transmitter uses an integrated laser device with two electro-optic elements. This experimentation observed the quantum state preparation and chances of side-channel attacks with the proposed transmitter mechanism. A distributed environment with a centralized quantum receiver shows the performance of form-factor and successful deployment at a short-term distance. Overall, the performance of the whole system is found to be effective for QKD compared to generalize DPS-QKD. Sibson et al. [173] identified a low error rate; high speed clocked QKD operation of indium phosphide transmitter chip useful in the telecommunications industry. This configuration has experimented with three protocols, including BB84, coherent one way, and different phase shifts. Results show that the proposed approach gives better performance without impacting the security standards, and they are useful for any sort of communications in telecommunication networks.
In six-state quantum cryptography protocols, BB84 protocol is extended to use six-state polarization (|0>, |1>, |+i>, |-i>,|+>, |->) on three orthogonal bases. Further, the six-state protocol can tolerate a noisier channel and detect the higher rate errors during any eavesdropping attack. The six-state protocol can be implemented either using a quantum computer or optical technologies. For example, Lo [205] derived the proofs for unconditional security solutions in six-state quantum key distribution protocols. In this implementation, it has been observed that unconditional security could lie at a high bit error rate of 12.7% as compared to 11% in the BB84 protocol. The proposed technique has used
DiVincenzo, Shor, and Smolin’s quantum codes for bit-flip and phase error pattern analysis. It has been observed that bit-flip error syndromes entropy can be used for a phase error pattern that increases the security of the proposed protocol at a high error rate as well. Similarly, Azuma et al. [206] realized the security of the six-state quantum key distribution protocol against various attacks, including intercept/resend, collective, and eavesdropping.
Here, the probability of an attacker’s interference in legitimate user communication is noticed, and the chance of obtaining the secret is measured. In collective attacks observations, the security level is found to be high that can protect imposing looser constraints upon the attacker’s strategies. This work has considered the comparative analysis of proposed security -level detection with E91 protocol. Results show that the six-state protocol is comparatively secure against attacks if hidden variable theories are examined with a small disturbance of 1/3. Chau et al. [207] identified that four-dimensional qubits in quantum key distribution are possible, and it can have security equivalent to the six-state scheme with arbitrarily long raw key size. Here, the tolerance level is observed to be 21.6% using one-way classical communication with passive basis selection in decoy. Thus, an increase in security level with high key rate meets the requirements of current quantum key distribution.
The decoy-state quantum key distribution protocol is preferred over others because it provides better conditional or unconditional constraints over the gain and the error rate of single-photon states. In recent times, various amendments are made to improve the decoy-state quantum key distribution protocol. For example, Liu et al. [208] realized the importance of decoy-state QKD protocol and its capability to protect against photon-number splitting attacks. In this work, two-basis detector efficiency asymmetry was found to be existing in real experimentation. To improve the rate of QKD with asymmetric basis-detector efficiency asymmetry, this work has conducted the investigation over 4-intensity decoy-state optimization protocol to protect against attacks. In observation, it is found that X and Z basis efficiencies are not the same, and the practicality of decoy-state has high chances. Grasselli et al. [209] focused on twin-field (TF)-QKD protocol because of a secure secret-key mechanism. It has been observed in an analysis that the security of this protocol is associated with photon-number states using the decoy-state method. This work has derived analytical bounds on the parameters used by parties and concluded that either two, three, or four decoy intensity settings could be used for investigating the protocol’s performance. In further observations, the protocol is found to be robust against optical pulses’ fluctuations. Chau et al. [210] made various observations in the decoy-state protocol. In the first observation, it is found that a secure key rate can be seriously lowered-down with the deviation of single-photon. In their second observation, error-rate can also lower the secure key rate by bounding the yields and usage of the type of decoy. To improve the secure key rate in such conditions, McDiarmid inequality is found to be effective because it helps in computing the lower bound in centering sequence method. In results, it has been observed that the secure key rate can be doubled with the proposed approach for a realistic 100 km long quantum channel. This work has introduced a powerful inequality technique for handling problems beyond statistical data with the central limit theorem. Liu et al. [211] applied the chernof bound to passive decoy-state and improved the final key rate. In experimentation, it is claimed that the proposed approach can securely transmit the data over 205 km, which is close to an asymptotic limit of 212 km. This is found to be the highest key rate over a long distance compared to existing approaches. In conclusion, the majority of decoy-state protocols are used either to improve the secure key rate or its transmission over a long distance. The post-quantum cryptosystem is defined as the set of cryptography primitives and protocols that are secure against quantum computer attacks [65] [66] [67] [68] [69]. It is observed that the existing cryptography primitives and protocols rely on mathematical problems such as integer factorization, discrete logarithm, and elliptic-curve discrete logarithm. With the possibilities of quantum computers, it is theoretical proved that all of these mathematical problems could be solved in a short duration. Thus, post-quantum cryptography is widely discussed. The protocols in post-quantum cryptography are mainly classified into five categories: code-based, lattice-based, supersingular elliptic curve isogeny, multivariate, and hybrid, as shown in Figure 6.
Figure 6:
Post-quantum Cryptography Protocols
In mathematics, Lattice is an arrangement of regularly spaced points in a subgroup 𝑅 𝑛 that is isomorphic to another group 𝑍 𝑛 such that 𝑅 𝑛 is isomorphic to 𝑍 𝑛 i.e. all combinations of vectors in space lies in 𝑅 𝑛 . Ajtai [174] initiated the use of cryptography in the lattice-based system and derived the computationally hard problems on lattices. The computational hard problem provides security and is found to be useful in other cryptography primitives such as homomorphic encryption/decryption, attribute-based cryptography, and code-based cryptosystem. Various lattice- based cryptosystem approaches are summarized in Table 7. Table 8 shows the comparative analysis of lattice-based cryptography primitives Robert McEliece initiated the code-based cryptography based on NP-hardness of the Syndrome Decoding Problem (SDP) [185]. A code-based cryptosystem relies on secretly decoding the linear code having a predefined structure. McEliece scheme is based on binary Goppa codes (as linear code) with the Nicholas Patterson algorithm in the decoding process. Table 7: Lattice-based Cryptosystem Approaches
Lattice-Problem Variants Pros Cons
Small Integer Solution (SIS) and its Inhomogeneous Variants Ring-SIS [175], Bi-GISIS [176], Lattice-based Direct Anonymous Attestation (LDAA) [178], Certificateless signature (CLS) scheme on NTRU lattice [182]. Smaller storage and faster operations are preferred. Schemes, like LDAA, are secure against weak/strong deniability attacks, Weak/strong deniability is the least addressed. The SIS problem becomes solvable in polynomial time with various parameter variations. Learning With Errors (LWE) Ring-LWE [175] [176], MPSign[177], Decision-LWE [178], Decision-Ring-LWE [178], LDAA [178], Module-LWE [179], Module-Learning with Rounding (LWR) [180], NewHope [181], Kyber [181], R. EMBLEM [181], KCL [181], OKCN/AKCN-RLWE [181], AKCN-MLWE [181], ILWE [183], MPLWE [183]. Smaller storage and faster operations are preferred Polynomial-based LWE allows for secrets that are much smaller compared to modulus operations. In results, schemes are faster. There are equally likely chances of chi-square attack, cyclotomic vulnerabilities, inherent structure exploitability, and sensitive dependence to field parameters in the majority of existing schemes. Weak/strong deniability is the least addressed.
Table 8: Comparative Analysis of Lattice-based Cryptography Primitives
Author Major Observations Year A B C D E F G H
Banerjee et al. [179] Low-power crypto-processor is designed, configured, and tested to accelerate polynomial arithmetic operations. Lightweight cryptography primitives and protocols are combined with sampling techniques. This accelerates the polynomial sampling in discrete distribution parameters useful in lattice-based schemes. 2019 ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ Nejatollahi et al. [183] Conducted a survey over lattice-based cryptographic schemes, security challenges in software and hardware implementations, and technology adoption 2019 ✔ ✔ ✔ ✔ ❌ ✔ ✔ ✔ Akleylek et al. [176] An authentication key exchange-based scheme is designed using the Bi-GISIS problem. Comparative analysis with SIS and LWE problem is performed. Testing of the proposed approach with a security model is conducted. 2020 ✔ ✔ ✔ ✔ ❌ ❌ ✔ ✔ Bai et al. [177] Polynomial LWE-based digital signature scheme is proposed and found to be secure with a quantum-access random oracle model. This work has observed that efficient key-recovery attack against homogeneous polynomial SIS problem with small secrets. 2020 ✔ ❌ ✔ ✔ ❌ ❌ ✔ ✔ El Kassem [178] In this work, smart zero-knowledge proofs are designed and explained for lattice problems. 2020 ✔ ✔ ✔ ❌ ❌ ❌ ✔ ✔ Mera et al. [180] Designed and experimented with a polynomial multiplier using the Toom-Cook algorithm for cryptoprocessors in the lattice system. Usage of the proposed hardware-based system is tested for cryptography primitives especially public key protocol. 2020 ✔ ❌ ❌ ✔ ✔ ❌ ✔ ✔ Nejatollahi et al. [181] This work has explored the design space of a flexible and energy-efficient post-quantum cache-based hardware accelerator for five different submissions. 2020 ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ Xu et al. [182] Proposed quantum attack resilience certificateless signature scheme with difficulty of small integer solution on the NTRU lattice. 2020 ✔ ❌ ✔ ✔ ❌ ❌ ✔ ✔ A: Encryption/Decryption, B: Authentication, C: Digital Signature, D: Key Distribution, E: Cryptoprocessor design, F: Identification Scheme, G: Lattice-based approach for application, H: Protocol Design/Development/Implementation/Simulation,
McEliece cryptosystem is fast in its encryption and decryption operations. The major drawback of McEliece cryptosystem is the use of large key sizes that make this scheme infeasible for resource-constrained devices. In literature [184]-[189], various variants of McEliece scheme are proposed using different error-correcting codes such as Rank ECC, Gabidulin codes, Twisted Gabidulin codes, Twisted Reed-Solomon Codes, low-density parity-check (LDPC) codes, quasi-cycle codes, and quasi-cyclic low-rank parity -check (QC-LRPC). Among other code-based cryptosystems [189], Niederreiter and CFS (Courtois, Finiasz, Sendrier) cryptosystems are also very popular. The CFS system is found to be useful for Internet of Things (IoT) signature schemes with Fiat-Shamir transformation [223]. Both Niederreiter and CFS schemes generate small signatures that result in fast computations. Table 9 shows the analysis of a few recent contributions in code-based cryptosystems. Table 9:
Analysis of contributions in Code-based cryptosystems
Author Cryptosystem Error-Correcting Codes (ECC) Major Strengths
Jäämeri [184] McEliece, Gabidulin-Paramonov-Tretjakov (GPT) Rank ECC, Gabidulin codes, Twisted Gabidulin codes, Twisted Reed-Solomon Codes Protected from structural weaknesses and Overbeck's attack Singh [186] McEliece, Niederreiter, Classic McEliece, Linear codes, Goppa codes Strongly protected against brute force attacks. Lesser disclosure to secret information can protect the schemes from total break, global deduction, local deduction, information deduction, and distinguishing algorithms. Bardet et al. [187] McEliece Gabidulin codes, Reed-Solomon codes, Linear codes Identified an attack that is below the security level for all rank-based schemes available in NIST Post-Quantum processes. The proposed attack is useful for systems having small to medium scale parameters that require lesser memory compared to the best quantum attacks. Ezerman et al. [188] McEliece, Niederreiter Goppa codes A secure signature scheme is designed using a code-based cryptosystem. It is observed that the signature schemes in a code- based cryptosystem can be classified as “hash -and- sign” or “Fiat - Shamir”.
The proposed scheme is a group signature scheme that requires multi-layered operations for generating group signatures. Fernández-Caramés et al. [189] McEliece, Niederreiter Goppa codes, low-density parity-check (LDPC), moderate-DPC (MDPC), quasi cycle codes (QC-LDPC), quasi-cyclic low-rank parity-check (QC-LRPC), LRPC, LDPC Conducted an in-depth survey of various post-quantum cryptosystem approaches and their variants. The survey is focused on protecting the IoT systems using post-quantum computing. Further, IoT architectures and challenges are analyzed for providing guidelines to secure future post-quantum IoT systems.
In multivariate cryptosystem, NP-hard and NP-complete multivariate equations are considered. The efficiency of a multivariate cryptosystem is based on the difficult level in solving the systems of quadratic equations over a field.
The concept of “One - way functions” composes of multiple easily invertible maps that could result in a diff icult to invert function without much knowledge of individual sub-function in composition. Multivariate cryptosystem has many mature systems compared to other post-quantum cryptosystems because it started much earlier. The major advantage of multivariate cryptosystem includes fast processing, less computational and communicational resource requirements [253], and small signature generation in lesser polynomial time. Table 10 shows an analysis of multivariate cryptosystems. Table 10:
Analysis of multivariate cryptosystems
Cryptosystem Variants Major Strengths
Multivariate digital-signature schemes [190]-[192][194] Rainbow digital signature schemes, Tame Transformation Signature (TTS), Tractable Rational Map Signature (TRMS), GeMSS, LUOV, MQDSS, Oil and Vinegar, Unbalanced Oil and Vinegar, Rainbow, CyclicRainbow, RainbowLRS2, Circulant Rainbow, NC-Rainbow. Multivariate digital-signature schemes are comparatively more secure than multivariate encryption/decryption or public-key cryptosystem because short signatures are difficult to solve in polynomial time. Simple arithmetic operations (addition and multiplication) make the schemes much efficient, especially for low-cost devices. Multivariate schemes are considered to have very high security with small signature length. For example, the GeMSS scheme is found to achieve the NIST PQCSP level V security standard in the first round. Multivariate Encryption/decryption schemes [191]-[193] EFLASH, C* Toy, PFLASH, C*, SFLASH, Hidden Field Multivariate encryption/decryption schemes are considered to be secure if they are protected against differential techniques, MinRank and algebraic attacks. Equation (HFE), HFE - , ABC, SRP, EFC. Multivariate public-key cryptosystem [195][196] Multivariate Public Key Cryptosystem, Rainbow Signature Scheme In this system, public keys are a set of polynomial defined over a finite field. Infinite field, the degree of polynomial is often considered as 2. Thus, it is referred to as multivariate-quadratic cryptography as well. Most of the multivariate public-key cryptosystems are quantum-resistant because no quantum algorithm solves the multivariate quadratic problem in polynomial time. Three popular isogeny-based structures used in post-quantum cryptography include Ordinary Isogeny Diffie-Hellman (OIDH), supersingular isogeny DH (SIDH), and commutative SIDH (CSIDH) [197]. Using these structures, the protocols in isogenies on the super-singular cryptosystems are majorly classified as signature/encryption, key exchange, and hash function. Isogeny-based digital signature schemes ensure message integrity, nonrepudiation, and identity authentication. The core idea of ensuring these cryptography properties it to transform identification schemes into signature schemes with non-interactive zero-knowledge proofs. The challenges in the signature can be generated using hash functions. In key exchange protocols, public and private keys are used to generate session key that ensures confidentiality and integrity of subsequent communications. The hash function ensures collision-resistance and compression. The protocols in isogenies on the super-singular cryptosystem are briefly analyzed as shown in Table 11.
Table 11 : Analysis of isogenies on the super-singular cryptosystem
Category Variants Major Strengths
Isogeny-based signature/encryption algorithm [198][201]-[204] SeaSign, CSI-FiSh, Quantum-resistant undeniable blind signature scheme, isogeny-based designated verifier blind signature scheme. Lack of practices in the isogeny-based signature scheme makes this category of protocols weaker in post-quantum cryptography. Isogeny-based key exchange protocol [198] Longa, LeGrow, Galbraith, Authenticated Key Exchange (AKE)-SIDH-2, AKE-SIDH-3, SIDH-UM, biclique-SIDH. The major challenge in key exchange protocol is to design authenticated key exchange protocol and verify the security with well-known security models such as BR, CK CK + Isogeny-based Hash function [199][200] CGL, Very Smooth Hash (VSH), VSH-
DL, SWIFFT, Takashima’s hash function, Charles, Goren and Lauter’s hash function, High-speed isogeny-based Hash functions are protected from Pollard-rho, claw finding, preimage, and collision-resistant attacks. High-speed short messages-based Hash functions are useful to avoid quantum attacks of computational overhead that are used with novel solutions.
In hybrid schemes, different post-quantum cryptography primitives and protocols are integrated to achieve set goals. For example, Crockett et al. [212] proposed a hybrid key exchange and authentication mechanism in transport layer security (TLS) and secure shell (SSH) protocols. The adoption of post-quantum cryptography with these mechanisms is found to be dependent on the standard of communication and availability of infrastructure. The integration of post-quantum and hybrid key exchange and authentication lies over the negotiation of multiple algorithms in hybrid cryptography that combine multiple keys and other primitives and protocols. The hybrid approach is found to be possible with the different hybrid key exchanges such as TLS 1.2. TLS 1.3 and SSHv2. Campagna et al. [213] proposed the integration of independent key exchanges and feeding mechanisms with pseudorandom function (PRF) to drive a secret and secure exchange. In this work, a new hybrid key exchange mechanism is designed for TLS 1.2 protocol with elliptic curve Diffie-hellman protocol and post-quantum key encapsulation. Further, Bit Flipping Key Exchange (BIKE) and Supersingular Isogeny Key Exchange (SIKE) are combined with the key exchange in TLS 1.2 handshake mechanism. Overall, the integration is found to be effective, and desired goals are achievable with food performance measures. Qassim et al. [64] combined physical layer and cryptography security primitives for increasing the security standard and proposed a cross-layer key agreement scheme that is strongly protected against a man-in-the-middle attack. The proposed technique is found to be unbreakable and scalable to traditional cryptography primitives and protocols.
7. SCALABLE QUANTUM COMPUTER HARDWARE
Quantum computers use principles from quantum physics to perform computation and are theorized to solve several computational challenging problems such as encryption, much faster as compared to classical computers [250]. The core component of quantum computers equivalent to a bit in classical computers is known as a qubit. Qubits can carry both the values 0 and 1 at the same time and exist in superposition. Quantum computing is hypothesized to be so powerful that “quantum supremacy” is a terminology in the field, which means that a quantum machine can perform a task previously intractable by a computer in the classical world [2]. As a full-fledged field, experimental quantum computing started as early as the 1980s, however, until the late 1990s, the majority of the researcher’s envisaged industria l quantum computer as a distant reality [3]. Several contenders are being pursued as building blocks of a scalable quantum computer and are being developed independently by different academic researchers and industry engineers worldwide. Superconducting circuits having quantum properties such as entanglement, quantized energy levels and superposition and main component qubit can emerge as integral to industrial quantum computers [4]. In addition, there are front-runners which can emerge as the building blocks of industrial quantum computers and are based on trapped ions [5], optical lattices [6], spin [7] and spatial based [8] quantum dots [245], quantum wells [9], quantum wire [251], nuclear magnetic resonance (NMR) [252], solid-state NMR [231], molecular magnet [232], cavity quantum electrodynamics [220], linear optics [15], diamond [16], Bose-Einstein condensate [17], Rare-earth-metal-ion-doped inorganic crystal [18] and Metallic-like carbon nanospheres [19] amongst others. However, superconducting circuits have transpired as the most widely used. Two main approaches for the physical implementation of a quantum computer are currently available: analog and digital [215]. A significant challenge for the construction of error-free industrial quantum computers today is the maintenance of qubit state due to decoherence. Even with error rates achieved below 1%, the depth of quantum circuits required to solve real-world problems would be considerable, leading to detrimental cumulative error rates. Therefore, the area of quantum error correction is at present one of the most active areas of the research in the field of quantum computing. Google Quantum AI, in collaboration with NASA, reported a demonstration of quantum calculation which was shown to require several thousand years on any conventional classical computer on 23 October 2019. Although this work achieved an important milestone for the current generation of quantum computers, the solution of a practical real-world problem on a quantum computer is expected to require significant further development.
Notably, the work from IBM researchers showed that the efficiency of the same calculation on a classical supercomputer can be significantly improved [225].
Quantum computers and present-day classical computers are similar in the sense that they can solve the same problems in theory. However, quantum computers may solve the same tasks in significantly less time compared to classical computers for specific problems and therefore may demonstrate “quantum supremacy”.
Another important term commonly used in the quantum community is “quantum advantage”. While “quantum supremacy” implies solving a problem on a quantum computer which is intractable on any classical machine; “quantum advantage” is a more practical term which deals with solving a useful real-world problem which cannot be efficiently solved on a classical computer. It is yet an open area of research to find practical problems which can be efficiently solved on quantum computers. Although the quantum machines that have been engineered hitherto are bulky and offers limited computational power as they are made up of materials which have to be kept at superconducting temperatures, the potential of industrial quantum computers in future cannot be contested [215]. The motivation for these benefits of industrial quantum computers can be derived from the present-day success of classical computers and the way they took off in the 1950s. Similar to the practical state of quantum computers today, the first generation of classical computers used to be bulky and had to be cooled continuously. As the theory of Artificial Intelligence (AI) had started shaping from the early days of classical computers, albeit they were nowhere near the compute required for AI, powerful industrial quantum computers can be theorized to come to reality in near-future a nd achieve “quantum supremacy” [20].
Cryptanalysis is an inquiry into the information systems to determine the secret aspects of the system. It is used to circumvent the cryptographic safety mechanisms to access the contents of encrypted messages. An example is the RSA (Rivest – Shamir – Adleman) encryption which is widely used for encrypting data communication with banks and other nodes on the internet. Shor developed a quantum algorithm in 1994 which can, in principle break the operational RSA encryption if a large-scale error-corrected quantum computer can be developed. Hence, post-quantum encryption methods need to be formulated which can withstand an industrial quantum computer [21]. Searching efficiently and sorting through large data sets is now a high priority for many big enterprises. Grover developed an optimal quantum algorithm in 1996, which can speed up search through big data relative to the classical algorithms in query complexity. The present-day database software’s such as Oracle are not suitable enough for real- world search enough to run Grover’s algorithm ; hence software that does the work of oracle in the quantum world need to be developed [22]. A variety of areas in computational sciences such as numerical weather prediction, computational chemistry and others involve solving equations using approximate methods ignoring the fine details. An example is the parameterization techniques used to approximate the sub-grid scale processes in a weather/climate prediction model due to the computational constraints. These parameterizations have been known to propagate errors in the solutions to the system of equations, thus directly affecting the decision making in other fields. Industrial quantum computers offer hope in solving the equations in their exact form as they are written. Industrial quantum computers can be used to understand how different chemicals make fertilizers and improve upon the current high carbon footprint technique of manufacturing. Understanding chemistry, photosynthesis, superconductivity and magnetism, all being quantum mechanical phenomena can be better understood by industrial quantum computers. Although a scalable industrial quantum computer has still not been achieved, research into these fields has started using the available, much less powerful quantum computers. On a seven-qubit quantum processor, IBM recently simulated beryllium hydride molecule [20]. Various other applications which can harness the speed of industrial quantum computing such as patient diagnosis by quickly comparing the reports with a global database, modelling of live passenger and commercial traffic, the balance of energy supply and demand are expected to be significantly benefited. However, unless further upgrades are made, encryption, communications transactions, critical infrastructure, Blockchain and cryptocurrency are bound to become vulnerable by industrial quantum computers. International efforts on how to build, construct and monitor qubit systems by over 100 academic and government-affiliated labs are underway, and numerous developed, and start-up companies are now working on manufacturing industrial quantum computers built out of silicon, superconductive and ion qubits. Industrial quantum computers can efficiently deploy conventional computers for tasks which the classical computers excel at such as user interface, networks and data and in turn be also controlled by traditional computers for complex computations. The hardware requirements of industrial quantum computers can be divided into four layers based upon their functions, viz the “quantum data plane”, the “control and measurement plane”, the “control processor plane” and the “host processor”. The “quantum data plane” is the location where qubits are stores, operations and measurements are carried out by the “control and measurement plane”, the sequence of operations in algorithms are taken care of by the “control processor plane” , and the “host processor” carries out the user interface, networks and storage of large arrays.
In order to achieve functional industrial quantum computers, several issues have to be solved, the most important of which is the impact of noise or decoherence which causes errors in quantum computation and suppresses quantum advantage. An initial state of a qubit has to be set before it can be used in addition to developing circuits and gates. Photons remain coherent for a long time; however, creating quantum circuits out of them is a challenge. Superconductors possess quantum properties which can be harnessed to develop quantum circuits which are in use by IBM, Google, Rigetti and others to build their quantum computers. In 2016, IBM released a five-qubit processor free for everyone on the cloud, which can be used to construct a quantum circuit and run it as long as it uses five or fewer qubits [214]. At present, IBM offers cloud access to quantum computers consisting of up to 53 qubits and have recently announced a quantum computer with a record 64 quantum volume [233]. Table 12 shows five major candidate material systems for the development of an industrial quantum computer and the relevant metrics to measure their performance and the current state-of-the-art. Among these candidate systems, Trapped Ion and Superconducting qubits are the basis for the current generation of quantum machines available through cloud access. The other three material systems are still a subject of intense research and require significant further development to be available for quantum circuit simulations [246]. Although there has been much progress in designing smaller quantum computers, it is not yet possible to experimentally demonstrate a design for an industrial quantum computer which could be of the scale required to crack current cryptography and the existing implementations even if scaled up are not just enough. Scaling the qubits to achieve an industrial quantum computer has many challenges such as the quality of qubits when scaling up to industrial-scale quantum computers, wiring, refrigeration, packaging and others. Theoretically, silicon-based quantum computers have been predicted to offer the potential for scalability with error correction schemes. After the seminal work from Kane in 1998 [234], many surface-code quantum computer architectures have been proposed [235] [236] [237]. Remarkable advancements in silicon spin qubit design and characterization [238] [239] [240] [241] [242] [243] have been demonstrated in the recent years, which confirm the suitability of this material system as an attractive candidate for the construction of a scalable industrial quantum computer. The size of the industrial quantum computing market is expected to touch $ 1.9 bn by 2023 and $ 8 bn by 2027 [23]. Various computing giants such as IBM, Microsoft, Alibaba, Google, dedicated quantum enterprises such as D-Wave and others such as Rigetti Computing, NVision Imaging Technologies are testing quantum computers competing to launch the scalable industrial computer. Global research and development efforts are ongoing to commercialize industrial quantum computers with continuously increasingly leading contributions from US and other prominent efforts coming from The EU quantum technologies flagship, The UK national quantum technologies program, The Australian Centre for Quantum Computation and Communication Technology (CQC2T) and The Chinese quantum national laboratory for quantum information science.
Table 12: Major hardware candidates for industrial quantum computer and their properties.
Qubit Technologies Trapped Ion Qubits Superconducting Qubits Silicon Qubits Photonic Qubits Topological Qubits
Physical Qubits IonQ:79; AQT:20 IBM: 65 qubits; Google: 54 qubits; Rigetti: 30 2 6x3 In progress Coherence Times ~50 sec ~50-200 µsec ~1-10 sec ~150 µsec - Gate Fidelity ~99.9% ~99.4% ~90% ~98% Expected: ~99.9999% Gate Operation Time ~3-50 µsec ~10-50 nsec ~1 nsec ~1-10 nsec - Scalability Some potential Medium to high potential High potential High potential -
8. FUTURE DIRECTIONS
Figure 7 shows the hype cycle for quantum computing. We have identified various ongoing research areas for three different maturity levels (5 years, 5 to 10 years and more than 10 years) based on the current research. T represent Technology and A represent Application area in the Hype Cycle. As per Figure 7, post-quantum cryptography is at the peak while lot of research work has been done on simulations for complex quantum experiments. Research areas such Robotics, energy management, cybersecurity, distributed QC, complex computational chemistry, financial modelling and drug design are at the kickoff stage in their development under the domain of quantum computing. These use of quantum computing in these areas which are at their innovation trigger may take more than 10 years to mature. Traffic Optimization is also at its innovation trigger but is expected to top the hype cycle within the next 5-10 years. Post quantum cryptography, quantum control and adiabatic quantum computing have reached the peak of inflated expectations and it is expected that it would take less than five years for them under complete development under the purview of quantum computing. Quantum Internet, quantum-based satellite communication, quantum assisted machine learning, electronics material discovery and error corrected quantum computing have also reached the peak of inflated expectations but are anticipated to rapidly evolve in five to ten years. Quantum based portfolio-risk optimization and fraud detection and fault tolerant quantum computing presently have high expectations on the hype cycle and are blossom in more than ten years. Lot of research work has been done on quantum algorithms & complexity and quantum programming languages & systems, which could be active research areas for the next 5 to 10 years. Simulation software for quantum experiments and quantum simulators are at the slope of enlightenment and have a long way to fully develop in quantum computing. We have identified various open challenges and future research directions which are still a topic of active global research. Fragility is the main drawback of Quantum technology due to two following reasons [24]: 1) A very short coherence time of qubits because superconducting qubits forget its information very frequently (in nanoseconds). 2) There is unreliability in quantum operations due to relatively large error rates and it is challenging to develop a quantum computer with low error rates. Further, as compared to classical computing, error correction in quantum computing is quite challenging because a) errors are continuous (involve both amplitude and phase), b) cannot copy unidentified quantum states and c) measurement can collapse a quantum state and destroy the data saved in Qubits. To run a quantum algorithm efficiently, a large number of physical qubits are required, which need a close and continuous connection between the classical platform and quantum chip, and it forms a colossal control overhead [25]. Moreover, this interaction and overhead increases the complexity for quantum computing process in terms of run-time control, architecture and integration. Figure 7: Hype Cycle for Quantum Computing
It is very challenging to attain fault-tolerant and reliable quantum computations because the experimental implementation of quantum error correction is still an open problem [219]. Due to the fragile nature of quantum states, there is a need to operate bits at very low temperatures and fabrication should be highly accurate [26]. It is also challenging to measure complete quantum state accurately; therefore, verification is challenging. There is a large probability of errors during computation as compared to classical computing. So, there is a need for an effective error correction mechanism for quantum architectures to perform the operation as intended. There is also a need to redesign the architecture of quantum communication to increase the verification of precise fabrication constraints. On the other hand, qubits are very difficult to test after fabrication because tolerances are tight, and the use of incorrectly placed Qubits must be avoided to reduce the occurrence of error. There is a need to apply of error correction recursively to attain adequate fault tolerance to permit sustainable quantum computation [221]. Machine learning researchers are using principal component analysis, vector quantization, Gaussian models, regression and classification in routine [244]. To improve scalability and efficiency of machine learning algorithms quantum technology can be used in handling large datasets with large sizes of devices (100 – Energy management is an important challenge, where world’s powerful supercomputer consumes a lot of energy to solve different problems [224]. Quantum computers are expected to be more energy efficient as compared to supercomputers while doing a particular task [28]. On the other hand, a quantum computer has the capability to perform large calculations in a reliable manner with consuming less amount of energy, which further reduces the cost and carbon emissions. Classical computers use binary bits (0 or 1) for encoding information while quantum computing uses Qubits, which represent both 0s and 1s at the same time — this property of quantum computing to identify an optimal solution while consuming less energy. The reason for less energy consumption is that the quantum processors are working at extremely low temperature, and the processor is superconducting with no resistance, which means no production of heat [29]. Hybrid applications contain two portions: high-energy and low-energy [224]. Quantum computing executes the high-energy portion while classical computing executes low-energy portion using cloud [226]. To solve these kinds of problems, there is a need for hybrid computing comprises of quantum and classical computing to curb energy usage and costs dramatically. There is a need to do more work before implementing hybrid computing practically to solve today’s most challenging business problems. Quantum Internet enables distributed quantum computing by incorporating new communications and improving computing capabilities to a large extent. Quantum Internet has various challenges because it uses laws of quantum mechanics and the main constraints for network design are teleporting, entanglement, quantum measurement and no-cloning [30]. The error-control mechanism is an essential assumption of classical computing, which is no more valid in quantum computing. To design quantum Internet, there is a need for a major shift in network paradigm from classical to quantum specific. Further, when a qubit interacts with an environment, then it causes decoherence because Qubits are very fragile and losing information from Qubit to the environment with passage of time. Moreover, the long-distance entanglement distribution is also a challenge in quantum computing for effective transformation of data.
Robots use GPUs to solve intensive computational tasks such as drug discovery, logistics, cryptography, and finance, where quantum computing can be augmented to perform computations with a large speed. Quantum-powered robots can also utilize cloud-based quantum computing services to solve different types of problems [227]. Nowadays, quantum computing enhances robotic senses for manufacturing, such as the identification of several faults in a jet engine in a short period [31]. Further, Quantum image processing helps to understand the visual information efficiently and saves and manages image data effectively using two critical properties of quantum computing such as parallelism and entanglement. Artificial intelligence-based robotics are dealing with different kinds of problems using graph search to deduct new information, but complexity is increasing with the increase in data. Quantum computing can reduce complexity by using quantum random walks instead of graph search. Further, other important problems related to kinematics, such as the mechanical movement of robotics can be solved by quantum neural networks by enhancing the movement of machines and recognizing moments of joint friction and inertia. Moreover, another problem, such as identifying the reason for the inconsistency between expected and observed behaviour is a challenging task, which could be solved using Quantum algorithms.
Quantum computing can simulate complex problems of chemistry, physics and biology using small-scale (50-100 qubit) 'quantum simulators', which could be available in coming years [32]. The expertise of an extensive range of researchers and fundamental aspects of classical computing can work together to understand and harness the capabilities of quantum technology. Further, quantum simulators can realize the real system while solving the complex problems (which is difficult to solve on the classical system or supercomputer) in a controlled manner to measure the influence of various parameters on each other. Quantum simulators can take advantage of essential properties of quantum computing such as entanglement and superposition while designing it.
To improve the security for implanted medical devices, cares and online communication, there is a need for cryptography. Nevertheless, the various generally used cryptosystems will be damaged once large quantum computers come into existence. Post-quantum cryptography denotes the cryptographic algorithms (generally public-key algorithms). It is assumed that the attacker used a large quantum computer to attack in post-quantum cryptography, and these systems attempt to stay secure in this situation [33]. Post-quantum cryptography has to maintain integrity and confidentiality while preventing from different kinds of attacks. Presently, post-quantum cryptography research is typically concentrated on six techniques such as symmetric key quantum resistance, supersingular elliptic curve isogeny cryptography, code-based cryptography, hash-based cryptography, multivariate cryptography and lattice-based cryptography [34]. Another challenge within post-quantum cryptography is “Agility”; there is a need to find out the right areas to incorporate agility. Therefore, future systems should build in such a way, which must be able to predict the possible security problems. Further, there is a need for testing and validation design by developing new automated tools to identify and fix the fault at runtime dynamically. Moreover, the reconfiguration of legacy devices with cryptosystems is still an open problem, which needs to be solved by incorporating agility in the legacy applications.
The development of classical computers was accompanied by the advancements in the skills of numerical weather prediction in the 1950s. Since then the predictions of weather forecasts have greatly enhanced in the last few decades. This development has been catapulted by the improved hardware and software but has been limited by the fundamental principle on which these traditional computers are built, i.e. bits or 0s and 1s. For the purpose of colossal calculations required, the classical computers are stacked to build what are known as supercomputers. These supercomputers perform computations day and night to generate forecasts of the atmosphere, ocean, land and other components of the Earth system. The state-of-the-art predictions, although have improved with time, but still need lot of upgradations for societal applications such as flood forecasting, urban modelling, sub-surface flow modelling and allied complex tasks. These developments have been limited by the computational power available today. With the hope of industrial quantum computers becoming a reality, the next generation Earth System Models would be able to run at a much higher spatial and temporal resolutions. There is a need to diligently study the applicability of quantum computing to numerical weather predictions [249].
9. SUMMARY AND CONCLUSIONS
In this paper, a systematic review of the existing literature of quantum computing is presented. It has been identified that quantum-mechanical phenomena such as entanglement and superposition are expected to play an important role while solving computational problems. We proposed a taxonomy of quantum computing and mapped to various related studies based on proposed taxonomy to identify the research gaps. Various quantum software tools and technologies are discussed. Further, post-quantum cryptography and industrial quantum computers are discussed. Various open challenges are identified, and promising future directions are proposed. The fusion of all the performance attributes in a single quantum computing technique is still ambiguous until now. In future, to make a quantum computer which can perform concurrent operations; it is required to have a quantum computing technique that can allow quantum I/O with all the necessary classified features. The suggested taxonomies framework can be used to contrast various existing quantum computing techniques for determining the optimal strategy that can be applied on classical computing infrastructure. However, the scaling of qubits, trade-off between speed and the decoherence time is the topic of research in the field of quantum computing. Quantum computers are developed to increase the security rate in communication and computations via decreasing the computational time. To secure the classical cryptography primitives and protocols with the usage of quantum computer’s ability in solving the mathematical problems in few milliseconds, post -quantum cryptography mechanisms are designed. Post-quantum cryptography strengthens the symmetric cryptography primitives and protocols against well-known quantum attacks. Further, it has taken three hard mathematical problems (integer factorization, discrete logarithmic, and elliptic-curve discrete logarithm) in asymmetric key cryptography to secure the cryptography primitives and protocols. In conclusion, the characteristics of post-quantum cryptography increase the computational efficiency and security of many futuristic applications. Furthermore, the present-day industrial quantum computers are not yet there to replace classical supercomputers owing to the challenges in scaling up on the number of qubits that can be practically realized hitherto. When that might happen is an open question. Though the next decade is going to be highly exciting for industrial quantum computers, there is still uncertainty on when the quantum computers will start to replace their classical counterparts in complex tasks. However, digital supercomputers are here to stay, even if quantum becomes a reality, as an addendum to the quantum computers of the future. There is one crucial design challenge; to run a quantum algorithm efficiently. A large number of physical qubits are required, which need a close and continuous connection between the classical platform and quantum chip, forming a huge control overhead. It is very challenging to attain fault-tolerant and reliable quantum computations because of quantum error correction, which is still an open problem. Due to the fragile nature of quantum states, there is a need to operate bits at very low temperatures and fabrication should be accurate. Further, to improve the scalability and efficiency of machine learning algorithms, quantum technology can be used in handling a large dataset with a large number of devices (100 – implementing hybrid computing practically to solve today’s hardest business problems. Quantum simulators can be designed for simulations for complex quantum experiments, which can take advantage of important properties of quantum computing such as entanglement and superposition while designing it. Presently, Artificial Intelligence based robotics are dealing with different kinds of problems using graph search to deduct new information, but complexity is increasing with the increase in data. Quantum computing can reduce the complexity of robotic mechanism by using quantum random walks instead of graph search. Other various fields such as computer security, biomedicine, the development of new materials and the economy, might be reorganized by the advancement in quantum computing. Appendix A: List of Abbreviations
Notation Description
ABC Simple Matrix Scheme or ABC in short AKCN Asymmetric Key Consensus with Noise BB84 Quantum key distribution scheme Bi-GISIS Bilateral Generalization Inhomogeneous Short Integer Solution CHP CNOT-Hadamard-Phase - Scott Aaronson CK+ Extended Cohn-Kanade (CK+) database Cirq Software library for writing, manipulating, and optimizing quantum circuits DWDM Dense Wavelength Division Multiplexing EFC Extension Field Cancellation EMBLEM Error-blocked Multi-Bit LWE-based Encapsulation EQCS Egyptian Quantum Computing Society GeMSS Great Multivariate Short Signature HFE Hidden Field Equation ILWE Integer Module Learning With Errors KCL Key Consensus from Lattice LanQ A quantum imperative programming language LDPC Low-density parity-check LUOV UOV + PRNG + Field Lifting + Simplified Secret Key MDPC Moderate Density Parity Check MLWE Module Learning With Errors MPKC Multivariate Public Key Cryptosystems MPLWE Middle Product Learning With Errors MQDSS MQ (multivariate quadratic) Digital Signature Scheme NC-Rainbow Non-Commutative Rainbow OKC Optimal Key Consensus OKCN Optimally-Balanced Key Consensus with Noise PRNG Pseudorandom Number Generator R. EMBLEM Ring Error-blocked Multi-Bit LWE-based Encapsulation RLWE Ring Learning With Errors RDSS Rainbow Digital Signature Schemes R-LRS2 Rainbow Low Resolution Spectrograph 2 SRP MPKC encryption scheme called SRP SIS Short Integer Solution staq Full-stack quantum processing toolkit written in standard C++ TRMS Tractable Rational Map Signature TTS Tame Transformation Signature QCAD Quantum Computer Aided Design QCGPU Quantum Computing GPU qchas A library for implementing Quantum Algorithms QC-LDPC Quasi-Cyclic Low-Density Parity Codes QC-LRPC Quasi-Cyclic Low-Rank Parity-Check QIO Quantum Input Output QMDD Quantum Multiple-valued Decision Diagram QOCS Qualified One-way Costs Shifting QuEST Quantum Exact Simulation Toolkit UOV Unbalanced Oil and Vinegar
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