This paper investigates the core principles of quantum computation, providing an in-depth understanding of quantum phenomena and illustrating how these principles form the scientific foundation of the field. The pivotal physical concepts, such as properties of subatomic particles, including electrons and photons, as well as their mathematical description through linear algebra are examined. It focuses on the qubit, the quantum analogue of a classical bit, featuring properties like superposition, entanglement, and wave function collapse, which redefine the traditional concept of information processing. The mathematical structures that underlie quantum system modelling—vector spaces, tensor products, and Dirac notation—are also systematically explained. Essential components of quantum computation involve logic gates such as Pauli matrices and Hadamard gates, which demonstrate the novel application of qubits in unique ways. This sets up an important theme of the interdependence and potential for transformation of quantum states through the illustration of advanced ideas of quantum entanglement and teleportation. Quantum circuits serve as an outstanding example of how reversibility and superposition are exploited to solve complex problems at unprecedented levels of efficiency. The paper also highlights the Deutsch-Jozsa algorithm as a compelling example of quantum superiority over traditional computational approaches. Together, these findings emphasize the transformative potential of quantum computing for science and industry.
The purpose of this dissertation is to construct a radically new type of mathematics whose underlying logic differs from the ordinary classical logic used in standard mathematics, and which we feel may be more natural for applications in quantum mechanics. Specifically, we begin by constructing a first order quantum logic, the development of which closely parallels that of ordinary (classical) first order logic --- the essential differences are in the nature of the logical axioms, which, in our construction, are motivated by quantum theory. After showing that the axiomatic first order logic we develop is sound and complete (with respect to a particular class of models), this logic is then used as a foundation on which to build (axiomatic) mathematical systems --- and we refer to the resulting new mathematics as "quantum mathematics." As noted above, the hope is that this form of mathematics is more natural than classical mathematics for the description of quantum systems, and will enable us to address some foundational aspects of quantum theory which are still troublesome --- e.g. the measurement problem --- as well as possibly even inform our thinking about quantum gravity.\n\nAfter constructing the underlying logic, we investigate properties of several mathematical systems --- e.g. axiom systems for abstract algebras, group theory, linear algebra, etc. --- in the presence of this quantum logic. In the process, we demonstrate that the resulting quantum mathematical systems have some strange, but very interesting features, which indicates a richness in the structure of mathematics that is classically inaccessible. Moreover, some of these features do indeed suggest possible applications to foundational questions in quantum theory. \n\nWe continue our investigation of quantum mathematics by constructing an axiomatic quantum set theory, which we show satisfies certain desirable criteria. Ultimately, we hope that such a set theory will lead to a foundation for quantum mathematics in a sense which parallels the foundational role of classical set theory in classical mathematics. One immediate application of the quantum set theory we develop is to provide a foundation on which to construct quantum natural numbers, which are the quantum analog of the classical counting numbers. It turns out that in a special class of models, there exists a 1-1 correspondence between the quantum natural numbers and bounded observables in quantum theory whose eigenvalues are (ordinary) natural numbers. This 1-1 correspondence is remarkably satisfying, and not only gives us great confidence in our quantum set theory, but indicates the naturalness of such models for quantum theory itself. We go on to develop a Peano-like arithmetic for these new "numbers," as well as consider some of its consequences. Finally, we conclude by summarizing our results, and discussing directions for future work.
. Quantum computers take advantage of interfering quantum alternatives in order to handle problems that might be too time consuming with algorithms based on classical logic. Developing quantum computers requires new ways of thinking beyond those in the familiar classical world. To help in this thinking, we give a description of the foundational ideas that hold in all of our successful physical models, including quantum theory. Our emphasis will be on the proper interpretation of our theories, and not just their statements. Our tact will be to build on the concept of information, which lies central to the operation of not just computers, but the Universe. For application to quantum computing, the essence of quantum theory is given, together with special precautions and limitations. Introduction. Having a grasp on the ideas behind a theory helps to apply it correctly, to understand its limitations, and to generate new ideas. Getting a firm hold on quantum theory is not an easy task, because our experiences and even our genetic predispositions have been developed in a world in which quantum effects are largely washed out. Remarkably, our predilection for finding logic behind the behavior of what we observe, including that of electrons and atoms, has led us to quantum theory, a description of nature that is hard for us to conceptualize, but is logical, accurate, and explains a wide variety of phenomena with only a few statements and input. As background to quantum theory and quantum computing, an attempt is made here to give the primitive notions and essential observations that underlie current physical theories, so that foundational ideas are explicit, and a common language is established. In our description, information storage and transfer is made central. A short description of quantum theory follows, and then applied to quantum computing, focusing on what the theory says, and particularly does not say, in areas where conceptual difficulties have arisen. Physical theory and reality. A physical theory is a logical model capable of making predictions of what we observe. It is judged by its accuracy in matching measurements, and by its economy, i.e. whether the proposed theory has only a few relationships and input data needed for its ability to explain observations over a wide realm.
The (meta)logic underlying classical theory of computation is Boolean (two-valued) logic. Quantum logic was proposed by Birkhoff and von Neumann as a logic of quantum mechanics more than sixty years ago. The major difference between Boolean logic and quantum logic is that the latter does not enjoy distributivity in general. The rapid development of quantum computation in recent years stimulates us to establish a theory of computation based on quantum logic. The present paper is the first step toward such a new theory and it focuses on the simplest models of computation, namely finite automata. It is found that the universal validity of many properties of automata depend heavily upon the distributivity of the underlying logic. This indicates that these properties does not universally hold in the realm of quantum logic. On the other hand, we show that a local validity of them can be recovered by imposing a certain commutativity to the (atomic) statements about the automata under consideration. This reveals an essential difference between the classical theory of computation and the computation theory based on quantum logic.
We are currently in the midst of a second quantum revolution: transition from quantum theory to quantum engineering (e.g. quantum computing, communication, sensing). The main purpose of quantum theory is to find fundamental rules governing the existing physical systems. In contrast, quantum engineering aims at designing and implementing new systems (machines, devices, etc.) to achieve some desirable tasks, based on quantum theory. From experience in today's engineering, it is not always easy for a human designer to completely understand the behaviours of the system that she/he is designing, and an error in her/his design may cause serious problems and even disasters. Consequently, theories and methodologies for verification of correctness, safety and reliability of complex engineering systems have been systematically studied in various engineering fields. In particular, computer scientists have developed techniques to verify the correctness of both hardware and software as well as the security of communication protocols. Human intuition is poorly adapted to the quantum world compared to the classical world, which implies that human engineers will make many more mistakes in designing and implementing complex quantum systems such as quantum computer hardware and software and communication protocols. Even worse, because of the essential differences between the classical and quantum worlds, verification techniques for classical systems cannot be directly used for quantum systems. Novel verification techniques will be indispensable for the coming era of quantum engineering and technology. Model-checking is an effective technique to check whether a system satisfies a desired property. The properties that are checked are usually specified in a temporal logic; typical properties are deadlock freedom, invariants, safety, and request–response properties. The systems being checked are mathematically modelled as e.g. (finite-state) automata, transition systems, Markov chains and Markov decision processes [1]. In the last three decades, model-checking has become one of the dominant techniques for verification of computer hardware and software, and has proved mature as witnessed by a large number of successful industrial applications. Techniques of model-checking were even applied in systems biology recently. With the emergence of quantum engineering and quantum technology, a question then naturally arises: is it possible to apply model-checking techniques in verifying the correctness and safety of quantum engineering systems, and how can this be done? Unfortunately, due to some essential differences between classical and quantum systems, it seems unlikely that the classical model-checking techniques can be directly applied to quantum systems. Basically, to make model-checking techniques effective for quantum systems, the following three problems must be systematically addressed: System modelling and property specification: Behaviours of quantum systems cannot be described using classical modelling methods and, consequently, the properties of the quantum systems to be checked cannot be formalized by classical specification languages. As a result, novel conceptual frameworks must be proposed to properly model and consider quantum systems, including formal models and formal descriptions of temporal properties of quantum systems. Quantum measurements: Model-checking is usually applied to check the long-term behaviours of systems. However, to check whether a quantum system satisfies a certain property at a time point, one has to perform a quantum measurement on the system, which can change the state of the system. This makes studying the long-term behaviours of quantum systems much harder than with classical systems. Algorithms: Classical model-checking algorithms normally assume the state spaces to be finite or countably infinite. However, state spaces of quantum systems are inherently continuous. To develop algorithms for model-checking quantum systems, deep mathematical properties of the systems have to be exploited, so that a finite (or countably infinite) number of representative elements in the state spaces will suffice. Note that the state space of any quantum system has a natural linear algebraic structure. A well developed algorithm for verifying quantum systems should make clever use of this structure. Despite the difficulties discussed above, a few model-checking techniques for quantum systems have been developed in the last 10 years. The earliest work mainly targeted checking quantum communication protocols: Taking the probabilism arising from quantum measurements into account, the probabilistic model-checker PRISM was used in [2] to verify the correctness of quantum protocols, including superdense coding, quantum teleportation and quantum error correction. A branching-time temporal extension of exogenous quantum propositional logic was introduced and then the model-checking problem for this logic was studied in [3], with verification of the correctness of quantum key distribution BB84 as an application. A linear temporal extension of exogenous quantum propositional logic was then defined and the corresponding model-checking problem was investigated in [4]. Model-checking techniques were developed in [5] for quantum communication protocols modelled in process algebra CQP (communicating quantum processes) [6]. A model-checker for quantum communication protocols was also developed in [7], where only the protocols that can be modelled as quantum circuits expressible in the stabilizer formalism were considered. This technique was further extended beyond stabilizer states and used to check equivalence of quantum protocols. A research line pursued by the authors and their collaborators is to develop model-checking techniques that can be used not only for quantum communication protocols but also for general quantum systems, including physical systems and quantum programs. Quantum automata were adopted in [8,9] as the model of the systems: Definition 1(Quantum automata [10]). A quantum automaton is a 4-tuple |
A=(H,Act,{Uα:α∈Act},H0)|, where: 1) |
H| is a finite-dimensional Hilbert space, called the state space; 2) |
Act| is a finite set of action names; 3) for each action name |
α∈Act|, Uα is a unitary operator on |
H|; 4) |
H0⊆H| is the subspace of initial states. A quantum automaton behaves as follows: it starts from some initial state in |
H0|, and at each step it performs a unitary transformation Uα for some |
α∈Act|. An algorithm for checking certain linear-time properties (e.g. invariants and safety properties) was proposed in [8], where, following Birkhoff–von Neumann quantum logic, closed subspaces of the state Hilbert space are used as the atomic propositions about the state of the system, and the checked linear-time properties are defined as infinite sequences of sets of atomic propositions. Furthermore, the decidability or undecidability of several reachability problems for quantum automata were established in [9]. The model-checking problem for a larger class of quantum systems than quantum automata, namely quantum Markov chains, was studied in [11]. Note that continuous-time quantum Markov processes have been studied intensively in mathematical physics. Discrete-time quantum Markov chains were recently introduced as a semantic model for quantum programs. Definition 2 (Quantum Markov chains [11]). A quantum Markov chain is a triple |
(H,E,H0)|, where |
H| and |
H0| are the same as in Definition 1, and |
E| is a super-operator on |
H|. A quantum Markov chain starts in an initial state in |
H0|, and at each step it performs the (same) quantum operation modelled by the super-operator |
E|. Note that the (discrete-time) dynamics of closed quantum systems is usually depicted by unitary operators, and the behaviours of open quantum systems are described by super-operators. Obviously, the notion of quantum automata can be generalized by replacing unitary operators Uα in Definition 1 by super-operators |
Eα|. Furthermore, quantum Markov decision processes [12] can be defined by introducing decision strategies into such generalized quantum automata. Several algorithms for checking the reachability of quantum Markov chains and quantum Markov decision processes have been developed. As in checking classical Markov chains and Markov decision processes, graph reachability is a key to these algorithms. However, classical graph theory is not suited to our purpose; instead, a new theory of quantum graphs (i.e. graphs in a Hilbert space with an adjacency relation induced by a super-operator) was developed and, in particular, an algorithm for the BSCC (bottom strongly connected components) decomposition of the state Hilbert spaces was found in [11]. Another decomposition technique, namely periodic decomposition, for quantum Markov chains has recently been proposed. The notion of the super-operator-valued Markov chain is introduced in [13] as a higher-level model of quantum programs and quantum cryptographic protocols. Definition 3 (Super-operator-valued Markov chains [13]). A labelled super-operator-valued Markov chain over a set AP of predefined atomic propositions is a 5-tuple |
(S,s0,H,Q,L)|, where: 1) S is a finite set of classical states with s0 ∈ S being the initial state; 2) |
H| is a finite-dimensional Hilbert space, called the quantum state space; 3) |
Q:S×S→SOH| is a transition super-operator function, where |
SOH| denotes the set of trace-nonincreasing super-operators on |
H| and, for each s ∈ S, ∑t ∈ SQ(s, t) is trace-preserving; and 4) L: S → 2AP is a labelling function. A super-operator-valued Markov chain has two state spaces, a classical one and a quantum one, which are connected through the transition super-operator function. It behaves in a similar manner to classical Markov chains. It starts from the classical initial state s0 but with the quantum initial state unspecified (it can be taken arbitrarily). Then at each step, given the current classical state s and quantum state ρ, it proceeds to a classical state t with probability tr[Q(s, t)(ρ)], and the accompanying quantum state evolves into Q(s, t)(ρ)/tr[Q(s, t)(ρ)] provided that tr[Q(s, t)(ρ)] ≠ 0. The normalization requirement that ∑t ∈ SQ(s, t) is trace-preserving guarantees that the probabilities of going from s to some classical state sum up to 1. As the atomic propositions are taken to be classical (they apply only to classical states), this Markov chain model is suitable for verification of quantum systems against classical properties, such as running time, termination, reachability, etc. One distinct feature of this model, however, is that it allows us to check the properties of the system once and for all; i.e. the verified results apply to all initial quantum states. For example, the model-checking algorithm for the reachability problem essentially calculates a positive operator Π, accounting for all (classical) paths satisfying the concerned property. Then the reachability probability when the Markov chain starts in the initial quantum state ρ is simply tr(Πρ). A corresponding computation tree logic (CTL) for super-operator-valued Markov chains was defined, and algorithms for checking such properties were developed in [13]. A tool implementation of these algorithms has been provided [14] based on a probabilistic model-checker. Algorithms for model-checking ω-regular properties, a general class of properties subsuming linear temporal logic (LTL) formulas, against super-operator-valued Markov chains have been proposed [15], thus allowing analysis of a wide range of properties such as repeated reachability, reachability in a restricted order, and nested Until properties. Furthermore, the reachability problem of a recursive extension of super-operator-valued Markov chains was studied in [16], with the application of analysing quantum programs with procedure calls. As reviewed in previous sections, several theoretical frameworks and algorithms of quantum model-checking have been developed. Certainly, however, quantum model-checking is still at a very early stage of its development; in particular, its applications are only at the level of toy examples. We envisage that, in the future, quantum model-checking techniques can be applied to the following areas: Checking physical systems: Physicists have already considered the algorithmic checking problem of certain properties of quantum systems, e.g. quantum measurement occurrence [17] and reachability of quantum states [18]. Quantum model-checking can offer a systematic view of this line of research. Verification of quantum circuits: Verification of circuits has been one of the major application areas of classical model-checking. But model-checking applied to verification of quantum circuits is an area to be systematically exploited. Analysis and verification of quantum programs: Another important application area of classical model-checking is analysis and verification of programs. Several techniques for analysis and verification of quantum programs have been reported in the last few years [19,20]. However, model-checking techniques specifically designed for quantum programs are still missing. Verification of security of quantum communication protocols: Applications of model-checking mentioned in the section entitled ‘Early research on model-checking of quantum systems' focus on verification of the correctness of quantum communication protocols. However, verification of the security of quantum protocols is much more difficult, and model-checking applied to it is an interesting topic for future research.
The physics and the mathematics of computation are examined to provide a foundation and perspective for the investigation of the quantum mechanics of computation. Our purpose is to explore the fundamental limits and constraints imposed on computation by Nature through the laws of physics and the mathematics of computational complexity. Inasmuch as information storage and transmission are an integral part of computation, their physical bounds are considered. The computer is viewed both physically and mathematically as a dynamical system, and is depicted in terms of the basic Turing machine paradigm. Three fundamental classes of the Turing machine are defined; the deterministic, stochastic and quantum Turing machines. Hamiltonian models and physical realizations of quantum computing are described. Quantum computers can perform some tasks which have no classical analogue, but they cannot compute functions that are non-computable by classical means. Some classically intractable problems can be solved with quantum computers.