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Mathematical Structures in Computer Science

Probability in quantum computation and quantum computational logics: a survey

María Luisa Dalla Chiara, Roberto Giuntini, Giuseppe Sergioli

Published Mar 28, 2014• 6 citations
Publisher

Abstract

Quantum computation and quantum computational logics give rise to some non-standard probability spaces that are interesting from a formal point of view. In this framework, events represent quantum pieces of information (qubits, quregisters, mixtures of quregisters), while operations on events are identified with quantum logic gates (which correspond to dynamic reversible quantum processes). We investigate the notion of Shi–Aharonov quantum computational algebra. This structure plays the role for quantum computation that is played by σ-complete Boolean algebras in classical probability theory.

DOI
10.1017/s0960129512000734

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Quantum computation and quantum computational logics give rise to some non-standard probability spaces that are interesting from a formal point of view. In this framework, events represent quantum pieces of information (qubits, quregisters, mixtures of quregisters), while operations on events are identified with quantum logic gates (which correspond to dynamic reversible quantum processes). We investigate the notion of Shi–Aharonov quantum computational algebra. This structure plays the role for quantum computation that is played by σ-complete Boolean algebras in classical probability theory.
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 in a pure state) or, more generally, with a mixture of quregisters (called qumix). Following an approach proposed by Domenech and Freytes, we apply residuated structures associated with fuzzy logic to develop certain aspects of information processing in quantum computing from a logical perspective. For this purpose, we introduce an axiomatic system whose natural interpretation is the irreversible quantum Poincaré algebra. Such a system allows to establish a completeness theorem for the treatment of quantum information.
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 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.
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.
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 quantum information quantity, represented by a quregister (a system of qudits) or, more generally, by a mixture of quregisters (called qumix), whose dimension depends on the logical complexity of the sentence. At the same time, the logical connectives are interpreted as logical operations defined in terms of quantum logical gates. Physical models of quantum computational logics can be built by means of Mach-Zehnder interferometers.
Quantum computation theory has inspired new forms of quantum logic, called quantum computational logics, where formulas are supposed to denote pieces of quantum information, while logical connectives are interpreted as special examples of quantum logical gates. The most natural semantics for these logics is a form of holistic semantics, where meanings behave in a contextual way. In this framework, the concept of quantum probability can assume different forms. We distinguish an absolute concept of probability, based on the idea of quantum truth, from a relative concept of probability (a form of transition-probability, connected with the notion of fidelity between quantum states). Quantum information has brought about some intriguing epistemic situations. A typical example is represented by teleportation-experiments. In some previous works we have studied a quantum version of the epistemic operations “to know”, “to believe”, “to understand”. In this article, we investigate another epistemic operation (which is informally used in a number of interesting quantum situations): the operation “being probabilistically informed”.