Wave-Style Token Machines and Quantum Lambda Calculi

Particle-style token machines are a way to interpret proofs and programs, when the latter are written following the principles of linear logic. In this paper, we show that token machines also make sense when the programs at hand are those of a simple quantum lambda-calculus with implicit qubits. This, however, requires generalising the concept of a token machine to one in which more than one particle travel around the term at the same time. The presence of multiple tokens is intimately related to entanglement and allows us to give a simple operational semantics to the calculus, coherently with the principles of quantum computation.Comment: In Proceedings LINEARITY 2014, arXiv:1502.0441

Game semantics for quantum programming

Quantum programming languages permit a hardware independent, high-level description of quantum algo rithms. In particular, the quantum lambda-calculus is a higher-order programming language with quantum primitives, mixing quantum data and classical control. Giving satisfactory denotational semantics to the quantum lambda-calculus is a challenging problem that has attracted significant interest in the past few years. Several models have been proposed but for those that address the whole quantum λ-calculus, they either do not represent the dynamics of computation, or they lack the compositionality one often expects from denotational models. In this paper, we give the first compositional and interactive model of the full quantum lambda-calculus, based on game semantics. To achieve this we introduce a model of quantum games and strategies, combining quantum data with a representation of the dynamics of computation inspired from causal models of concurrent systems. In this model we first give a computationally adequate interpretation of the affine fragment. Then, we extend the model with a notion of symmetry, allowing us to deal with replication. In this refined setting, we interpret and prove adequacy for the full quantum lambda-calculus. We do this both from a sequential and a parallel interpretation, the latter representing faithfully the causal independence between sub-computations

Coinductive Techniques on a Linear Quantum λ-Calculus

In this thesis, it is examined the issue of equivalence between linear terms in higher order languages, that is, in languages which allow to use functions as variables, and where variables which appear in the terms must be used exactly once. The work is developed focusing on the bisimulation method, with the purpose to compare this technique with that which has become the standard for the comparison between the terms of a language, i.e. the context equivalence. The thesis is divided into three parts: in the first one, the introduction of the bisimulation and context equivalence techniques takes place within a deterministic linear and typed language. In the second part, the same techniques are reformulated for a language that, while preserving the linearity, loses the deterministic connotation, allowing the terms to evaluate to a set of values each one having a certain probability to appear in the end of calculation. In the last part, a quantum language is examined, discussing the advantages of quantum computation, which allows to speed-up many of the algorithms of computation. Here one gives the concept of quantum program, which is inextricably linked to the (quantum) register where the qubits used in the computation are stored, entailing a more complex notion of equivalence between terms. The techniques to demonstrate that bisimulation is a congruence are not standard and have been used for the first time by Howe for untyped languages: within the thesis, one shows that bisimulation is a congruence in all considered languages but it coincides with the context equivalence relation only for the deterministic one. Indeed, extending the techniques already used by Howe to the probabilistic and quantum environment, it is shown, as non trivial result, that in probabilistic and quantum linear languages the bisimulation is contained in context equivalence relation