Compositional and holistic quantum computational semantics

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.

Mathematica Slovaca, 2016

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 m...

2016

By using the abstract structures investigated in the first Part of this article, we develop a semantics for an epistemic language, which expresses sentences like "Alice knows that Bob does not understand that PI is irrational". One is dealing with a holistic form of quantum computational semantics, where entanglement plays a fundamental role, thus, the meaning of a global expression determines the contextual meanings of its parts, but generally not the other way around. The epistemic situations represented in this semantics seem to reflect some characteristic limitations of the real processes of acquiring information. Since knowledge is not generally closed under logical consequence, the unpleasant phenomenon of logical omniscience is here avoided.

International Journal of Theoretical Physics, 2013

By using the abstract structures investigated in the first Part of this article, we develop a semantics for an epistemic language, which expresses sentences like "Alice knows that Bob does not understand that π is irrational". One is dealing with a holistic form of quantum computational semantics, where entanglement plays a fundamental role; thus, the meaning of a global expression determines the contextual meanings of its parts, but generally not the other way around. The epistemic situations represented in this semantics seem to reflect some characteristic limitations of the real processes of acquiring information. Since knowledge is not generally closed under logical consequence, the unpleasant phenomenon of logical omniscience is here avoided.

Electronic Notes in Theoretical Computer Science, 2008

Several domains can be used to define the semantics of quantum programs. Among them Abramsky [1] has introduced a semantics based on probabilistic power domains, whereas the one by Selinger associates with every program a completely positive map. In this paper, we mainly introduce a semantical domain based on admissible transformations, i.e. multisets of linear operators. In order to establish a comparison with existing domains, we introduce a simple quantum imperative language (QIL), equipped with three different denotational semantics, called pure, observable, and admissible respectively. The pure semantics is a natural extension of probabilistic (classical) semantics and is similar to the semantics proposed by Abramsky [1]. The observable semantics,à la Selinger [16], associates with any program a superoperator over density matrices. Finally, we introduce an admissible semantics which associates with any program an admissible transformation. These semantics are not equivalent, but exact abstraction or interpretation relations are established between them, leading to a hierarchy of quantum semantics. 2 Trace decreasing, completely positive maps. 3 Multisets of linear operators satisfying a completeness condition, see section 2.

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.

Soft Computing, 2015

Quantum information has suggested new forms of quantum logic, called quantum computational logics, where meanings of sentences are represented by pieces of quantum information (generally, density operators of some Hilbert spaces), which can be stored and transmitted by means of quantum particles. This approach can be applied to a semantic characterization of epistemic logical operations, which may occur in sentences like "At time t Bob knows that at time t Alice knows that the spin-value is up". Each epistemic agent (say, Alice, Bob,...) has a characteristic truth-perspective, corresponding to a particular orthonormal basis of the Hilbert space C 2 . From a physical point of view, a truth-perspective can be associated to an apparatus that allows one to measure a given observable. An important feature that characterizes the knowledge of any agent is the amount of information that is accessible to him/her (technically, a special set of density operators, which also represents the internal memory of the agent in question). One can prove that interesting epistemic operations are special examples of quantum channels, which generally are not unitary. The act of knowing may involve some intrinsic irreversibility due to possible measurement-procedures or to a loss of information about the environment. We also illustrate some relativistic-like effects that arise in the behavior of epistemic agents.

We view the vectors of a distributional semantic model as vectors of a semi-module over the semi-ring of the real interval I = [0, 1]. We show that the quantum logic of projectors is distributive and includes a Boolean sublattice. The compositional functional semantics of pregroup grammars interprets words and sentences as vectors such that the first order formula, the pregroup vector and the semantic vector interpreting a sentence are equivalent.

Fuzzy Sets and Systems, 2016

Quantum computation has suggested new forms of quantum logic, called quantum computational logics. In these logics wellformed formulas are supposed to denote pieces of quantum information: possible pure states of quantum systems that can store the information in question. At the same time, the logical connectives are interpreted as quantum logical gates: unitary operators that process quantum information in a reversible way, giving rise to quantum circuits. Quantum computational logics have been mainly studied as sentential logics (whose alphabet consists of atomic sentences and of logical connectives). In this article we propose a semantic characterization for a first-order epistemic quantum computational logic, whose language can express sentences like "Alice knows that everybody knows that she is pretty". One can prove that (unlike the case of logical connectives) both quantifiers and epistemic operators cannot be generally represented as (reversible) quantum logical gates. The "act of knowing" and the use of universal (or existential) assertions seem to involve some irreversible "theoretic jumps", which are similar to quantum measurements. Since all epistemic agents are characterized by specific epistemic domains (which contain all pieces of information accessible to them), the unrealistic phenomenon of logical omniscience is here avoided: knowing a given sentence does not imply knowing all its logical consequences.

Journal of Artificial Intelligence Research

Quantum Natural Language Processing (QNLP) deals with the design and implementation of NLP models intended to be run on quantum hardware. In this paper, we present results on the first NLP experiments conducted on Noisy Intermediate-Scale Quantum (NISQ) computers for datasets of size greater than 100 sentences. Exploiting the formal similarity of the compositional model of meaning by Coecke, Sadrzadeh, and Clark (2010) with quantum theory, we create representations for sentences that have a natural mapping to quantum circuits. We use these representations to implement and successfully train NLP models that solve simple sentence classification tasks on quantum hardware. We conduct quantum simulations that compare the syntax-sensitive model of Coecke et al. with two baselines that use less or no syntax; specifically, we implement the quantum analogues of a “bag-of-words” model, where syntax is not taken into account at all, and of a word-sequence model, where only word order is respec...