Quantum Computation and Quantum Information by Michael E. Nielson and Isaac L. Chuang

Quantum computing is a multidisciplinary field at the intersectionof quantum physics, computer science, and mathematics, aiming to har-ness quantum mechanical phenomena for computational advantage. Thisarticle provides a comprehensive review of both the foundational princi-ples and advanced theoretical models of quantum computing. We beginwith the basic framework of quantum computation, introducing the con-cept of qubits as two-level quantum systems, the linear algebra of Hilbertspaces, Dirac’s bra–ket notation, and the postulates of quantum mechan-ics (state superposition, unitary evolution, and projective measurement).We then discuss quantum logic gates and circuits, highlighting how clas-sical reversible computation principles extend to universal quantum gatesets. Next, we survey key quantum algorithms (from Shor’s factoring toGrover’s searching and beyond) and the complexity-theoretic implicationsof quantum computers, contrasting the class BQP with classical complex-ity classes. We cover the theoretical foundations of quantum error cor-rection and fault tolerance, explaining how quantum information can beprotected from decoherence using redundancy and how fault-tolerant pro-tocols can, in principle, allow scalable quantum computation despite noisyhardware. We then explore advanced models and paradigms of quantumcomputation, including measurement-based quantum computing (one-wayquantum computing), topological quantum computation with anyons, adi-abatic quantum computing and quantum annealing, continuous-variablequantum computing, and related approaches. We also provide an overviewof quantum information theory as it relates to computation, discussingentanglement, quantum entropy, and information-theoretic limits like theHolevo bound. Finally, we review the leading physical implementations(quantum hardware models) of quantum computers—trapped ions, super-conducting circuits, photonic systems, solid-state spin qubits, and oth-ers—outlining the challenges and achievements of each approach.

Quantum information theory has provided the formal framework for describing how information is stored, transmitted, and transformed in physical quantum systems [Renes, 2022; Tomamichel, 2015; Wilde, 2013]. Its entropic formulations underpin our understanding of quantum computation, communication, and cryptography. Yet this theory traditionally treats all quantum operations as freely available, ignoring computational restrictions. In practice, however, any manipulation of quantum information must be performed by devices of bounded complexity and runtime. Capturing such realistic constraints requires extending quantum information theory to include computational efficiency as a fundamental component. This work takes a first step toward building a computational version of quantum information theory, one that treats efficiency as part of the theory itself. The goal is to understand how the behavior of quantum information changes when the parties involved can only perform computationally efficient operations. This approach bridges the abstract, ideal setting of quantum information theory with the practical limitations of real quantum devices, offering a means to study information processing under realistic resource constraints. At the center of this work are two new quantities: the quantum computational min-entropy and the quantum computational max-entropy. These entropies extend standard quantum entropies by explicitly limiting the computational power of the observer or adversary. The quantum computational min-entropy captures how unpredictable a quantum system A remains to an observer holding system B, when that observer is restricted to quantum circuits of bounded size. Formally, for a bipartite state ρ_{AB}, we define {H^c}^s_{min}(A|B)_{ρ} ≔ -log d_A max_{ℰ^s_{B→A'}} F((𝕀_A ⊗ ℰ^s)(ρ_{AB}),|Φ_{AA'}⟩⟨Φ_{AA'}|) , where the maximization is over quantum channels that can be implemented by circuits of size at most s, and F denotes fidelity with a maximally entangled state. In the classical setting, the min-entropy can be expressed through the maximal probability of correctly guessing a random variable given some side-information. In the fully quantum setting, this idea extends to uncertainty about quantum information [König et al., 2009], quantifying how well one system can be inferred from another using local quantum operations. Our definition generalizes this operational viewpoint by restricting the computational power of the observer to efficient quantum circuits. This definition extends the operational meaning of the information-theoretic quantum min-entropy [König et al., 2009] by incorporating computational constraints, and it provides the fully quantum counterpart of the classical unpredictability entropy [Hsiao et al., 2007]. We establish fundamental properties for the computational min-entropy, including monotonicity in the circuit size and smoothing parameters, efficient data-processing inequalities, and fully quantum leakage and purification chain rules, which were left as open questions in earlier definitions of quantum computational entropies [Yi-Hsiu Chen et al., 2017; Munson et al., 2025]. For classical–quantum states, it coincides with the previously defined quantum computational unpredictability entropy [Noam Avidan and Rotem Arnon, 2025], showing that the new definition correctly generalizes known results. We also introduce the quantum computational max-entropy through a duality relation [Tomamichel et al., 2010] with the min-entropy using a fixed purification. Finally, we prove unconditional separations between the computational and information-theoretic entropies, demonstrating that computational restrictions can fundamentally alter entropic behavior even for simple states. These results establish the fundamental mathematical framework for studying quantum information within realistic computational constraints. By integrating efficiency directly into entropic quantities, they open the door to a fully developed computational quantum information theory that parallels its information-theoretic counterpart. Such a framework provides the foundation for analyzing cryptographic security against computationally bounded quantum adversaries [Noam Avidan and Rotem Arnon, 2025] and the limits of efficient quantum state manipulation. More broadly, it suggests that many core notions in quantum information theory may have refined computational analogues yet to be explored.

Quantum teleportation is one of the most basic quantum protocols, which transfers an unknown quantum state from one location to another through local operation and classical communication by using shared quantum entanglement without physical transfer of the information carrier. And it has been widely used in various quantum information protocols such as entanglement swapping, quantum repeaters, quantum gate teleportation, quantum computation based on measurement, and quantum teleportation networks, which have important application value in quantum computation and quantum information. Quantum teleportation is a naturally bipartite process, in which an unknown quantum state can only be transmitted from one node to another. With the further development of quantum information research, it is necessary to transfer quantum states or quantum information among more and more nodes. Multipartite quantum protocols are expected to form fundamental components for larger-scale quantum communication and computation. A bipartite quantum teleportation should be extended to a multipartite protocol known as a quantum teleportation network. In this paper, a multifunctional quantum teleportation network is proposed theoretically. We first propose a special method of constructing four-partite quantum resources in continuous variables (CVs), and based on this, construct two different types of CV quantum teleportation networks. One type of network contains just one quantum teleportation process consisting of a sender, a receiver and two controllers. In this type of network, the unknown quantum state can be recovered at any other node according to the requirement after the measurement in the input node, which enriches the transfer direction and transfer mode of the unknown quantum state. And meanwhile, the two controllers can control the transfer of a quantum state from the sender to the receiver by restricting the sender and receiver’s access to their information, which makes the quantum teleportation network controllable. The other type of network has two quantum teleportation processes, each containing only a sender, a receiver and no controllers, which increases the number of quantum states that can be transmitted. Then we analyze the dependence of the fidelity of each quantum teleportation network on different physical parameters, and compare the characteristics, advantages and disadvantages among different types of quantum teleportation networks. The scheme for constructing a multifunctional quantum teleportation network in this paper shows some advantages, such as the greater number of quantum nodes, diversity of types, simple operation procedure. And all these advantages provide a broader application prospect for establishing larger and more complex quantum information networks in the future and quicken the pace of the application of quantum information.

Quantum mechanics is the most ground-breaking and fascinating theoretical concept developed in physics during the past century. Much of our present understanding of the microscopic world and its extension into the macroscopic world, including modern technical applications, is based upon quantum mechanics. We have experienced a remarkable development of information and communication technology during the past two decades, to a large extent depending upon successful fabrication of smaller and smaller components and circuits. However, we are finally approaching the physical limits of component miniaturization as we enter a microscopic world ruled by quantum mechanics.

Quantum Computer: Quantum Model and Reality

The ability to transfer information securely is essential in the modern world we live in. While the field of cryptography has long researched ways to do this, events over the last couple of years have brought the issues of secure information transfer squarely into the public eye. As a consequence, security and cryptography have quickly become issues of social and political importance, sparking debates about the values of privacy and the role of government. An essential ingredient of modern cryptography is the generation of randomness: cryptographic techniques are built on the premise that one has access to random bits. It is well known, however, that computers cannot produce algorithmically random sequences – that is, sequences with maximal algorithmic information content – but are doomed to produce 'pseudorandomness'. This lack of randomness can be, and has been, exploited to compromise security, see [4]. The active field of quantum information theory has proposed approaches to provide supposedly 'unbreakable' security by exploiting various quantum phenomena. This security unfortunately relies on assumptions about the nature of quantum measurements and their ability to generate random bits. Anton Zeilinger summarises this by postulating that the simplest quantum systems, qubit, can hold only one bit of classical information [6]. This foundational principle is in line with a wider paradigm shift to view quantum information as an extension of classical information, but it is nonetheless unsatisfying to simply postulate this principle, given its importance in determining the practical advantages of quantum information theory. In order to understand better how quantum mechanics can help generate meaningful information, we instead look to relate the outcomes of quantum measurements to formal properties of the system based on more fundamental assumptions. Indeed, we have shown that a) some of the postulated properties of quantum information follow from the formal structure of the theory and b) a purely formal notion of information within a quantum world can generate, via measurement, meaningful and useful information in the macroscopic world (for example, in cryptography). The indeterminism of quantum measurements can be formalised via the notion of value indefiniteness. To explain this concept, let us consider an arbitrary quantum system and ask whether the outcome of a measurement of any observable quantity A (such as the energy of the system, its angular momentum (spin), etc.) is determined prior to the measurement. If this is the case, then we say the observable is value definite with value v(A); otherwise, the observable is value indefinite and v(A) is undefined. Mathematically, one can reduce all observable quantities to so-called projection observables, which can only take the values 0 or 1. Thus, the question of whether the outcome of several measurements can be simultaneously determined in advance can be rephrased in terms of the information 'carried' by a particular system in a definite quantum state. Classically, one expects that all quantities are determined in advance, and hence all observables are value definite. Quantum mechanically, however, the belief is that this is not the case, and the information content of quantum systems is limited. Formulating carefully the notion of value indefiniteness allows us to formalise the notion of (quantum) indeterminism; however, this doesn't help clarify whether quantum systems are indeed value indefinite or not. Staying in this formal framework, the Kochen-Specker theorem [5] provides a first positive result, showing that at least some observables must be value indefinite if one makes the assumption known as non-contextuality, which states that any definite values that exist must be independent of other compatible measurements that may or may not be performed on the system. Under the same assumption, this theorem can be strengthened to show that only one single observable can have the definite value 1 (see [2]). Furthermore, only observables that can be measured simultaneously with this single one can have the definite value 0; the rest must all be value indefinite. Since the preparation of a system involves precisely ensuring that, usually via measurement, the system is in a definite state with respect to some desired observable, this result shows that no other incompatible observable can be value definite. That is, preparing a system in a definite state by making the 'preparation' observable value definite specifies completely the information content of the quantum system. This is an example of syntactical quantum information acquiring meaning at the level of the quantum itself. Can the syntactical information at the level of the quantum generate meaning outside the quantum, that is, at the macroscopic level? The results cited above hold in the Hilbert-space framework of quantum mechanics and are formulated only in terms of a syntactical notion of information. Their real importance becomes evident when one interprets them in the context of quantum measurements. Specifically, if we prepare a quantum system in a known state, they allow us to 'locate' observables which we can measure, but which are value indefinite; that is, observables whose measurement results are not specified by any pre-existing property of the quantum system. Furthermore, with respect to a mathematical model of unpredictability which we developed in [3], the results of these measurements can be shown to be absolutely unpredictable. In this way we get a mathematical explanation and justification of the largely accepted intuition that quantum mechanics is inherently unpredictable, and that this unpredictability arises from the phenomenon of indefiniteness within the quantum world. Furthermore, if one considers a hypothetical infinite sequence generated by the repeated measurement of a quantum value indefinite observable, one can prove that these sequences must be strongly incomputable, technically 'bi-immune' [1]. Such sequences cannot be generated by any Turing machine or classical computer, showing that value indefiniteness leads to a clear classical/quantum split in a purely algorithmic context. Importantly, bi-immunity is a property of observed, macroscopic quantities, quite separate from the quantum framework in which the value indefiniteness is formalised. This form of macroscopic meaning created from the lack of syntactic information is precisely the scenario that quantum random number generators try to create, and which is essential for the development and certification of quantum cryptographic systems. The fact that the macroscopic information created goes beyond anything classically obtainable serves as a valuable practical resource, outside of and removed from the quantum formalism. To conclude, quantum information creates meaning within the quantum and via measure- ment, the lack of information within the quantum, creates meaning and valuable information at the macroscopic level. Acknowledgments The second author thanks Prof. S. Marcus for useful conversations on quantum information theory. This work was supported in part by Marie Curie FP7-PEOPLE-2010-IRSES Grant RANPHYS. References and Notes Abbott, A.A.; Calude, C.S.; Svozil, K. Strong Kochen-Specker theorem and incomputability of quantum randomness. Physical Review A 2012, 86, 062109. Abbott, A.A.; Calude, C.S.; Svozil, K. Value-indefinite observables are almost everywhere. Physical Review A 2013, 89, 032109. Abbott, A.A.; Calude, C.S.; Svozil, K. On the unpredictability of individual quantum measurement outcomes. CDMTCS Research Report 2014, 458. Calude, C.S. Quantum randomness & cryptology. CyberTalk 2013, 3, 42–43. Kochen, S.; Specker, E. The problem of hidden variables in quantum mechanics. Journal of Mathematics and Mechanics 1967, 17, 59–87. Zeilinger, A. A foundational principle for quantum mechanics. Foundations of Physics 1999, 29(4),631–643.

Quantum computers will make it possible to solve problems faster and more effectively than conventional computers currently can. Commercial quantum computers might be available in the next several years. Performing quantum operations on quantum registers is crucial to the building blocks of a quantum computing system. When these events occur in a system, they result in something fundamentally different from what you expect from a computer system. Using quantum physics, it is feasible that some computer jobs may be done more rapidly than they would be if the tasks were conducted using conventional Boolean logic and classical computers. This work discussed the recent developments and in-depth state of the art literature in Quantum Computing (QC) and related areas. This work primarily focuses on core concepts of Quantum computing such as Quantum mechanics, quantum gates, and quantum circuits, which serve as pillars for Quantum Computing and Quantum Information. Further, the QC-based concepts are implemented using a case study. The case study explores quantum computing techniques to address a real-life financial problem QAOA on IBM's Quantum Computer using Qiskit.

Chapter 9 - Quantum Information Science from the Perspective of a Device and Materials Engineer

By the birth of Quantum Physics, many new areas were opened for research and development in the world of science and technology. One such field is Quantum Computing and Communication where there is room for all that was once a dream in the field of computing and communication. Ultra high speed, immunity to eavesdropping, security, and name it and it may be possible. This paper gives a brief review of what is happening in the field of quantum computing and communication. We try to give an idea on polarization, quantum entanglement and their use in quantum communication and also briefly explain quantum communication networks, teleportation over optic fiber cables, quantum cryptography, quantum based satellite communication and quantum memory.

The novel field of quantum computation and quantum information has gathered significant momentum in the last few years. It has the potential to radically impact the future of information technology and influence the development of modern society. The construction of practical, general purpose quantum computers has been challenging, but quantum cryptographic and communication devices have been available in the commercial marketplace for several years. Quantum networks have been built in various cities around the world and a dedicated satellite has been launched by China to provide secure quantum communication. Such new technologies demand rigorous analysis and verification before they can be trusted in safety- and security-critical applications. Experience with classical hardware and software systems has shown the difficulty of achieving robust and reliable implementations. We present CCS q , a concurrent language for describing quantum systems, and develop verification techniques for checking equivalence between CCS q processes. CCS q has well-defined operational and superoperator semantics for protocols that are functional , in the sense of computing a deterministic input-output relation for all interleavings arising from concurrency in the system. We have implemented QEC (Quantum Equivalence Checker), a tool that takes the specification and implementation of quantum protocols, described in CCS q , and automatically checks their equivalence. QEC is the first fully automatic equivalence checking tool for concurrent quantum systems. For efficiency purposes, we restrict ourselves to Clifford operators in the stabilizer formalism, but we are able to verify protocols over all input states. We have specified and verified a collection of interesting and practical quantum protocols, ranging from quantum communication and quantum cryptography to quantum error correction.

Background: Passwords are still the predominant way of authentication in information systems, and are mostly at user's responsibility. They conceive, use, re-use, abuse and forget passwords. In absence of strict password policies and at minimum required user training, passwords tend to be short, easy to remember, connected to the user's personal or professional life and consequently easy to break. The additional problem with passwords is their aging: Moore's law is affecting the available computing power to crack passwords and those deemed secure today may easily be broken in the near future. Objective: The aim of this paper is to study various scenarios of the effect the Moore's law is having on passwords and their security. In addition, advancements in other fields, e.g. quantum computing and Internet of Things, are taken into the account. Method: We analyzed various password types and the lengths required to withstand an off-line brute-force attack. The analysis was performed under various scenarios and combinations thereof: the Moore's law will continue to be in the effect for years to come with varying parameters, quantum computing will become feasible, improvements in hash tables computations will speed up the cracking process, and others. Results: The paper shows the minimum password length in characters for each password type under various scenarios. Even the most optimistic scenario shows that the minimum required password length today should be of 11 randomly drawn characters, rendering most of the passwords inappropriate due to their poor memorability. Conclusion: The current textual passwords are cursed by the Moore's law and other advancements in the field. Soon, classical textual passwords will need to be replaced by other mechanisms, which are, fortunately, already emerging.

Quantum information science is one of the frontier areas of science and technology. It is also an interdisciplinary area of research where scientists from physics, mathematics, and computer science can contribute alike. One of the main goals is how well one can process information using laws of quantum theory. Quantum information theory aims to process information that is not amenable with classical devices. At the same time this also provides new insights into the nature of the quantum world. Quantum information processing includes quantum computation, quantum communication, quantum cryptography and various information processing tasks. Not only does this provide a fundamental arena to investigate quantum phenomena but also gives new technological benefits like quantum teleportation, remote state preparation and secure communications. In India there is a growing interest in this emerging field. In view of this we had organized an International Program on Quantum Information (IPQI-2010) at the Institute of Physics, Bhubaneswar during January 4–January 30, 2010. In this programme, there were experts from all over the world and topics of focus were on various issues related to quantum information science and general quantum theory, characterization and quantification of entanglement, quantum channels and quantum operations, role of entanglement in quantum algorithms and quantum computation, developing new quantum algorithms, understanding of capacities of quantum channels, and new communication tasks in the multiparty case.

The rapid growth in Artificial Intelligence demands the progress of advanced computation techniques to support new complex algorithms. Traditional computational methods, mainly relying on classical architectures, become inadequate to satisfy the needs of state -of -the -art AI applications which demand greater computing power and efficiency. The paper deals with advanced computation techniques and their applicability and effectiveness for optimization in AI algorithms. It focuses on quantum computing, distributed systems, and neuromorphic computing as computational paradigms. Quantum computing uses principles of superposition and entanglement to give exponential speedups for specific problems of search and factorization. Distributed systems run enormous datasets and complex computations across a large number of computing resources and provide a scalable solution with efficiency. Neuromorphic computing works like the neural structure of the human brain and performs real -time processing in an energy -efficient manner. In this paper, we present a set of experiments that reveal how these cutting -edge computing technologies greatly improve the performance of AI algorithms. Quantum algorithms run on a simulated quantum processor exhibited marked computational time decreases for search and factorization problems. Distributed neural networks trained on a Hadoop cluster showed linear scalability with the addition of nodes, thus decreasing training time. That is to say, utilizing spiking neural networks allowed neuromorphic hardware to realize real -time processing while consuming very minimal energy, hence outperforming traditional architectures on tasks such as image recognition. The research builds on a number of unique data sets, with graphs showing how computational performance may be improved. In summary, our findings suggest that the marriage of these advanced computation techniques can empower the creation of more efficient and scalable AI systems and thus outline the future course of developments in the domain. This paper discusses the insights from the findings and future research directions. This paper explores advanced computation techniques crucial for optimizing complex AI algorithms, focusing on quantum computing, distributed systems, and neuromorphic computing. Through simulated experiments on quantum processors, distributed neural networks, and neuromorphic hardware, the research demonstrates significant improvements in processing speed, scalability, and energy efficiency. The findings suggest that integrating these techniques can lead to the development of more efficient and scalable AI systems, with significant implications for future AI advancements. This research is significant as it highlights the potential of cuttingedge computation techniques to revolutionize AI by improving processing efficiency, scalability, and energy consumption, paving the way for more robust and capable AI systems.

Traditional electronic and communication devices operate by processing binary information encoded as bits. Such digital devices have led to the most advanced technologies that we encounter in our everyday lives and they influence virtually every aspect of our society. Nonetheless, there exists a much richer way to encode and process information. By encoding information in quantum mechanical states as qubits, phenomena such as coherence and entanglement can be harnessed to execute tasks that are intractable to digital devices. Under this paradigm, it should be possible to realize quantum computers, quantum communication networks and quantum sensors that outperform their classical counterparts. The electronic spin states of color-center defects in the semiconductor silicon carbide have recently emerged as promising qubit candidates. They have long-lived quantum coherence up to room temperature, they can be controlled with mature magnetic resonance techniques, and they have a built-in optical interface operating near the telecommunication bands. In this thesis I will present two of our contributions to this field. The first is the electric-field control of electron spin qubits. This development lays foundation for quantum electronics that operate via electrical gating, much like traditional electronics. The second is the universal control and entanglement of electron and nuclear spin qubits in an ensemble under ambient conditions. This development lays foundation for quantum devices that have a built-in redundancy and can operate in real-world conditions. Both developments represent important steps towards practical quantum devices in an electronic grade material.

The Symmetries of Quantum and Classical Information. The Ressurected “Ether" of Quantum Information