Giuseppe Sergioli

Representing continuous t-norms in quantum computation with mixed states

A model of quantum computation is discussed in (Aharanov et al 1997 Proc. 13th Annual ACM Symp. on Theory of Computation, STOC pp 20–30) and (Tarasov 2002 J. Phys. A: Math. Gen. 35 5207–35) in which quantum gates are represented by quantum operations acting on mixed states. It allows one to use a quantum computational model in which connectives of a four-valued logic can be realized as quantum gates. In this model, we give a representation of certain functions, known as t-norms (Menger 1942 Proc. Natl Acad. Sci. USA 37 57–60), that generalize the triangle inequality for the probability distribution-valued metrics. As a consequence an interpretation of the standard operations associated with the basic fuzzy logic (Hajek 1998 Metamathematics of Fuzzy Logic (Trends in Logic vol 4) (Dordrecht: Kluwer)) is provided in the frame of quantum computation.

Epistemic Quantum Computational Structures in a Hilbert-space Environment

Quantum computation and quantum computational logics are intrinsically connected with some puzzling epistemic problems. In the framework of a quantum computational approach to epistemic logic we investigate the following question: is it possible to interpret the basic epistemic operations (having information, knowing) as special kinds of Hilbert-space operations? We show that non-trivial knowledge operations cannot be represented by unitary operators. We introduce the notions of strong epistemic quantum computational structure and of epistemic quantum computational structure, where knowledge operations are identified with special examples of quantum operations. This represents the basic tool for developing an epistemic quantum computational semantics, where epistemic sentences (like “Alice knows that the spin-value in the x-direction is up”) are interpreted as quantum pieces of information that may be stored by quantum objects.

Some generalizations of fuzzy structures in quantum computational logic

Quantum computational logics provide a fertile common ground for a unified treatment of vagueness and uncertainty. In this paper, we describe an approach to the logic of quantum computation that has been recently taken up and developed. After reporting on the state of the art, we explore some future research perspectives in the light of some recent limitative results whose general significance will be duly assessed.

Holistic logical arguments in quantum computation

Abstract Quantum computational logics represent a logical abstraction from the circuit-theory in quantum computation. In these logics formulas are supposed to denote pieces of quantum information (qubits, quregisters or mixtures of quregisters), while logical connectives correspond to (quantum logical) gates that transform quantum information in a reversible way. The characteristic holistic features of the quantum theoretic formalism (which play an essential role in entanglement-phenomena) can be used in order to develop a holistic version of the quantum computational semantics. In contrast with the compositional character of most standard semantic approaches, meanings of formulas are here dealt with as global abstract objects that determine the contextual meanings of the formulas’ components (from the whole to the parts). We present a survey of the most significant logical arguments that are valid or that are possibly violated in the framework of this semantics. Some logical features that may appear prima facie strange seem to reflect pretty well informal arguments that are currently used in our rational activity.

Quantum teleportation and quantum epistemic semantics

AbstractQuantum information gives rise to some puzzling epistemic problems that can be interestingly investigated from a logical point of view. A characteristic example is represented by teleportation phenomena, where knowledge and actions of observers (epistemic agents) play a relevant role. By abstracting from teleportation, we propose a simplified semantics for a language that consists of two parts: 1)the quantum computational sub-language, whose sentences α represent pieces of quantum information (which are supposed to be stored by some quantum systems)2)the classical epistemic sub-language, whose atomic sentences have the following forms: agentahas a probabilistic information about the sentence α; agentaknows the sentence α. Interestingly enough, some conceptual difficulties of standard epistemic logics can be avoided in this framework.

Quantum information, cognition, and music.

Parallelism represents an essential aspect of human mind/brain activities. One can recognize some common features between psychological parallelism and the characteristic parallel structures that arise in quantum theory and in quantum computation. The article is devoted to a discussion of the following questions: a comparison between classical probabilistic Turing machines and quantum Turing machines. possible applications of the quantum computational semantics to cognitive problems. parallelism in music.

A quantum-inspired version of the nearest mean classifier

We introduce a framework suitable for describing standard classification problems using the mathematical language of quantum states. In particular, we provide a one-to-one correspondence between real objects and pure density operators. This correspondence enables us: (1) to represent the nearest mean classifier (NMC) in terms of quantum objects, (2) to introduce a quantum-inspired version of the NMC called quantum classifier (QC). By comparing the QC with the NMC on different datasets, we show how the first classifier is able to provide additional information that can be beneficial on a classical computer with respect to the second classifier.

Fuzzy approach for Toffoli gate in quantum computation with mixed states

In the framework of quantum computation with mixed states, a fuzzy representation based on continuous t -norms for Toffoli gate is introduced. In this representation, the incidence of nonseparability is specially investigated.

Two cooperative versions of the Guessing Secrets problem

We investigate two cooperative variants (with and without lies) of the Guessing Secrets problem, introduced in [L. Chung, R. Graham, F.T. Leighton, Guessing secrets, Electronic Journal of Combinatorics 8 (2001)] in the attempt to model an interactive situation arising in the World Wide Web, in relation to the efficient delivery of Internet content. After placing bounds on the cardinality of the smallest set of questions needed to win the game, we establish that the algebra of all the states of knowledge induced by any designated game is a pseudocomplemented lattice. In particular, its join semilattice reduct is embeddable into the corresponding reduct of the Boolean algebra 2^N^-^1, where N is the cardinality of the search space.

A many-valued approach to quantum computational logics

Quantum computational logics are special examples of quantum logic where formulas are supposed to denote pieces of quantum information (qubit-systems or mixtures of qubit-systems), while logical connectives are interpreted as reversible quantum logical gates. Hence, any formula of the quantum computational language represents a synthetic logical description of a quantum circuit. We investigate a many-valued approach to quantum information, where the basic notion of qubit has been replaced by the more general notion of qudit. The qudit-semantics allows us to represent as reversible gates some basic logical operations of Łukasiewicz many-valued logics. In the final part of the article we discuss some problems that concern possible implementations of gates by means of optical devices.