N ov 2 01 8 Logical structures underlying quantum computing

Entropy

In this work we advance a generalization of quantum computational logics capable of dealing with some important examples of quantum algorithms. We outline an algebraic axiomatization of these structures.

This monograph gives an overview of main results obtained within the solution of the project "Algebraic methods in quantum logic", being realized in the cooperation of the Faculty of Science of Masaryk University in Brno and the Faculty of Science of Palacký University in Olomouc. An innovated approach of the studied branch of mathematics consists of several chapters. The first chapter comprises of summarization of the results. The further chapters deal with more detailed and illustrated view of the studied topic.

Foundations of Physics

We show that in quantum logic of closed subspaces of Hilbert space one cannot substitute quantum operations for classical (standard Hilbert space) ones and treat them as primitive operations. We consider two possible ways of such a substitution and arrive at operation algebras that are not lattices what proves the claim. We devise algorithms and programs which write down any two-variable expression in an orthomodular lattice by means of classical and quantum operations in an identical form. Our results show that lattice structure and classical operations uniquely determine quantum logic underlying Hilbert space. As a consequence of our result, recent proposals for a deduction theorem with quantum operations in an orthomodular lattice as well as a, substitution of quantum operations for the usual standard Hilbert space ones in quantum logic prove to be misleading. Quantum computer quantum logic is also discussed.

Trends in logic, 2018

VOLUME 48 The book series Trends in Logic covers essentially the same areas as the journal Studia Logica, that is, contemporary formal logic and its applications and relations to other disciplines. The series aims at publishing monographs and thematically coherent volumes dealing with important developments in logic and presenting significant contributions to logical research. The series is open to contributions devoted to topics ranging from algebraic logic, model theory, proof theory, philosophical logic, non-classical logic, and logic in computer science to mathematical linguistics and formal epistemology. However, this list is not exhaustive, moreover, the range of applications, comparisons and sources of inspiration is open and evolves over time.

2005

The (meta)logic underlying classical theory of computation is Boolean (twovalued) logic. Quantum logic was proposed by Birkhoff and von Neumann as a logic of quantum mechanics more than sixty years ago. It is currently understood as a logic whose truth values are taken from an orthomodular lattice. 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. We introduce the notion of orthomodular lattice-valued (quantum) automaton. Various properties of automata are carefully reexamined in the framework of quantum logic by employing an approach of semantic analysis. We define the class of regular languages accepted by orthomodular lattice-valued automata. The acceptance abilities of orthomodular lattice-valued nondeterministic automata and their various modifications (such as deterministic automata and automata with ε−moves) are compared. The closure properties of orthomodular lattice-valued regular languages are derived. The Kleene theorem about equivalence of regular expressions and finite automata is generalized into quantum logic. We also present a pumping lemma for orthomodular lattice-valued regular languages. It is found that the universal validity of many properties (for example, the Kleene theorem, the equivalence of deterministic and nondeterministic automata) 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.

Trends in Logic, 2003

Quantum computation has suggested new forms of quantum logic, called quantum computational logics ([CDCGL02]). The basic semantic idea is the following: the meaning of a sentence is identified with a quregister, a system of qubits, representing a possible pure state of a compound quantum system. The generalization to mixed states, which might be useful to analyse entanglement-phenomena, is due to Gudder ([Gu02]). Quantum computational logics represent non standard examples of unsharp quantum logic, where the non-contradiction principle is violated, while conjunctions and disjunctions are strongly non-idempotent. In this framework, any sentence α of the language gives rise to a quantum tree: a kind of quantum circuit that transforms the quregister associated to the atomic subformulas of α into the quregister associated to α.

International Journal of Theoretical Physics, 2010

Your article is protected by copyright and all rights are held exclusively by Springer Science+Business Media, LLC. This e-offprint is for personal use only and shall not be selfarchived in electronic repositories. If you wish to self-archive your work, please use the accepted author's version for posting to your own website or your institution's repository. You may further deposit the accepted author's version on a funder's repository at a funder's request, provided it is not made publicly available until 12 months after publication.

Foundations of Physics, 1974

The logic of quantum mechanical propositions-called quantum logic-is constructed on the basis of the operational foundation of logic. Some obvious modifications of the operational method, which come from the incommensurability of the quantum mechanical propositions, lead to the effective quantum logic. It is shown in this paper that in the framework of a calculization of this effective quantum logic the negation of apropositionisuniquelydefined (Theorem I), and that a weak form of the quasimodular law can be derived (Theorem II). Taking account of the definiteness of truth values for quantum mechanical propositions, the calculus of full quantum logic can be derived (Theorem 111). This calculus represents an orthocomplemented quasimodular lattice which has as a model the lattice of subspaces of Hilbert space.

Soft Computing, 2015

The standard theory of quantum computation relies on the idea that the basic information quantity is represented by a superposition of elements of the canonical basis and the notion of probability naturally follows from the Born rule. In this work we consider three valued quantum computational logics. More specifically, we will focus on the Hilbert space C 3 , we discuss extensions of several gates to this space and, using the notion of effect probability, we provide a characterization of its states.

International Journal of Quantum Information, 2005

The theory of logical gates in quantum computation has suggested new forms of quantum logic, called quantum computational logics. The basic semantic idea is the following: the meaning of a sentence is identified with a quregister (a system of qubits) or, more generally, with a mixture of quregisters (called qumix ). In this framework, any sentence α of the language gives rise to a quantum tree: a kind of quantum circuit that transforms the quregister (qumix) associated to the atomic subformulas of α into the quregister (qumix) associated to α. A variant of the quantum computational semantics is represented by the quantum holistic semantics, which permits us to represent entangled meanings. Physical models of quantum computational logics can be built by means of Mach-Zehnder interferometers.