This thesis provides an introduction to the concepts of soundness and completeness in the context of quantum logic. As quantum logic is based on a lattice and not Boolean logic as is the case of classical logic, its meta analysis is very different. This thesis is devoted to studying the differences that this structure provides. In terms of quatum computation, the thesis provides an introduction to decidability as it is used in classical computation theory using Turing machines and then presents a comparison to quantum computation structure. The aim of the thesis is to present the existing basic structures of classical and quantum logic, as well as classical and quantum computation that will enable the development of the technical connections between the two in terms of specific models and finally to their relationship to classical and quantum complexity.
This thesis provides an introduction to the concepts of soundness and completeness in the context of quantum logic. As quantum logic is based on a lattice and not Boolean logic as is the case of classical logic, its meta analysis is very different. This thesis is devoted to studying the differences that this structure provides. In terms of quatum computation, the thesis provides an introduction to decidability as it is used in classical computation theory using Turing machines and then presents a comparison to quantum computation structure. The aim of the thesis is to present the existing basic structures of classical and quantum logic, as well as classical and quantum computation that will enable the development of the technical connections between the two in terms of specific models and finally to their relationship to classical and quantum complexity.
This thesis covers the topic of Quantum Computation, in both its theoretical and experimental aspects. The Theory of Quantum Computing is an extension of the standard Theory of Computation inspired by the principles of Quantum Mechanics. The discovery of efficient quantum algorithms for problems for which no efficient classical algorithms are known challenges the Strong Church-Turing Thesis, which states that all computational models are essentially equivalent in terms of what they can compute and how efficiently they can do so.
In Part I of this thesis, we study the nature of this difference. We first introduce a general and simplified framework for characterising the essence of a quantum theory, in contrast with a deterministic or a probabilistic theory. We put forth the thesis that the most fundamental axioms are those related to measurement, from which these differences emanate. We cover some variations on these theories, and show some interesting structural relationships between them. For example, the only models which can be simultaneously quantum and probabilistic are the deterministic ones.
We then use this meta-model to re-introduce in a succinct and uniform manner the deterministic, probabilistic and quantum versions of the Turing Machine and circuit computational models. Moreover, we generalise the circuit model on arbitrary semirings. We thus succeed in providing a new classification of existing complexity classes by varying the semirings and vector norms with which their vector space of states is defined. In particular, the standard models of deterministic, probabilistic, and quantum computing can be characterised (with the same norm) by defining circuits on the boolean algebra, the positive rational or reals, and the rational or reals, respectively. We further explore this model with other non-standard semirings. We show that quaternion based models are equivalent to quantum computation. Finally, we strengthen this “algebraic hierarchy” by using the tensor formula formalism to construct a family of complete (or promise complete) problems for the corresponding complexity classes.
Part II concerns experimental Quantum Computing by Nuclear Magnetic Resonance (NMR). An important obstacle in this approach is the inability to properly initialise the quantum register, which is related to the signal-to-noise problem in NMR spectroscopy. One of the most promising techniques to solve it is Algorithmic Cooling, a generalisation of the polarisation transfer techniques already used by NMR spectroscopists. We discuss the traditional adiabatic approaches and their limitations. We describe a variation on this technique, the non-adiabatic approach, which uses the environment in order to cool beyond these limits. In particular, we provide a new, efficient non-adiabatic cooling algorithm which could achieve near-zero spin temperatures in liquid-state NMR already with more reasonably sized registers (30–60 spins). Finally, we report on the successful laboratory realisation of the first ever non-adiabatic cooling experiment. This constitutes, we hope, the first step towards the fully fledged development of this promising technique, with applications far beyond NMR-based Quantum Computing.
§1. Introduction. Our understanding of Nature comes in layers, so should the development of logic. Classic logic is an indispensable part of our knowledge, and its interactions with computer science have recently dramatically changed our life. A new layer of logic has been developing ever since the discovery of quantum mechanics. G. D. Birkhoff and von Neumann introduced quantum logic in a seminal paper in 1936 [1]. But the definition of quantum logic varies among authors (see [2]). How to capture the logic structure inherent in quantum mechanics is very interesting and challenging. Given the close connection between classical logic and theoretical computer science as exemplified by the coincidence of computable functions through Turing machines, recursive function theory, and λ-calculus, we are interested in how to gain some insights about quantum logic from quantum computing. In this note we make some observations about quantum logic as motivated by quantum computing (see [5]) and hope more people will explore this connection. The quantum logic as envisioned by Birkhoff and von Neumann is based on the lattice of closed subspaces of a Hilbert space, usually an infinite dimensional one. The quantum logic of a fixed Hilbert space ℍ in this note is the variety of all the true equations with finitely many variables using the connectives meet, join and negation. Quantum computing is theoretically based on quantum systems with finite dimensional Hilbert spaces, especially the states space of a qubit ℂ2. (Actually the qubit is merely a convenience.
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.
In this paper we study quantum computation from a complexity theoretic viewpoint. Our first result is the existence of an efficient universal quantum Turing machine in Deutsch's model of a quantum Turing machine (QTM) [Proc. Roy. Soc. London Ser. A, 400 (1985), pp. 97--117]. This construction is substantially more complicated than the corresponding construction for classical Turing machines (TMs); in fact, even simple primitives such as looping, branching, and composition are not straightforward in the context of quantum Turing machines. We establish how these familiar primitives can be implemented and introduce some new, purely quantum mechanical primitives, such as changing the computational basis and carrying out an arbitrary unitary transformation of polynomially bounded dimension. We also consider the precision to which the transition amplitudes of a quantum Turing machine need to be specified. We prove that O(logT) bits of precision suffice to support a T step computation. This justifies the claim that the quantum Turing machine model should be regarded as a discrete model of computation and not an analog one. We give the first formal evidence that quantum Turing machines violate the modern (complexity theoretic) formulation of the Church--Turing thesis. We show the existence of a problem, relative to an oracle, that can be solved in polynomial time on a quantum Turing machine, but requires superpolynomial time on a bounded-error probabilistic Turing machine, and thus not in the class $\BPP$. The class $\BQP$ of languages that are efficiently decidable (with small error-probability) on a quantum Turing machine satisfies $\BPP \subseteq \BQP \subseteq \Ptime^{\SP}$. Therefore, there is no possibility of giving a mathematical proof that quantum Turing machines are more powerful than classical probabilistic Turing machines (in the unrelativized setting) unless there is a major breakthrough in complexity theory.
This paper deals with the foundations of quantum mechanics. We start by outlining the characterisation, due to Birkhoff and Von Neumann, of the logical structures of the theories of classical physics and quantum mechanics, as boolean and modular lattices respectively. We then derive these descriptions from what we claim are basic properties of any physical theory - i.e. the notion that a quantity in such a theory may be analysed into parts and that the results of this analysis may be treated in languages with an underlying boolean structure. We shall see that in the course of constructing a model of a theory with these properties different indistinguishable possibilities will arise for how the elements of the model may be named, that is to say different possibilities arise for how they can be associated with points from Set. Taking a particular collection of possibilities gives the usual boolean lattice of the propositions of classical physics. Taking all possibilities - in a sense, the set of all things that may be described by physical theories - gives the lattice of quantum mechanical propositions. This gives an interpretation of quantum mechanics as the complete set of such possible descriptions, the complete physical description of the world.
In the last few years, theoretical study of quantum systems serving as computational devices has achieved tremendous progress. We now have strong theoretical evidence that quantum computers, if built, might be used as a dramatically powerful computational tool. This review is about to tell the story of theoretical quantum computation. I left out the developing topic of experimental realizations of the model, and neglected other closely related topics which are quantum information and quantum communication. As a result of narrowing the scope of this paper, I hope it has gained the benefit of being an almost self contained introduction to the exciting field of quantum computation. The review begins with background on theoretical computer science, Turing machines and Boolean circuits. In light of these models, I define quantum computers, and discuss the issue of universal quantum gates. Quantum algorithms, including Shor's factorization algorithm and Grover's algorithm for searching databases, are explained. I will devote much attention to understanding what the origins of the quantum computational power are, and what the limits of this power are. Finally, I describe the recent theoretical results which show that quantum computers maintain their complexity power even in the presence of noise, inaccuracies and finite precision. I tried to put all results in their context, asking what the implications to other issues in computer science and physics are. In the end of this review I make these connections explicit, discussing the possible implications of quantum computation on fundamental physical questions, such as the transition from quantum to classical physics.