An Algebra of Pure Quantum Programming

We develop a sound and complete equational theory for the functional quantum programming language QML. The soundness and completeness of the theory are with respect to the previously-developed denotational semantics of QML. The completeness proof also gives rise to a normalisation algorithm following the normalisation by evaluation approach. The current work focuses on the pure fragment of QML omitting measurements.Comment: To appear in ENTCS, 3rd International Workshop on Quantum Programming Languages, 2005. 21 Page

Introduction

This chapter will motivate why it is useful to consider the topic of derivations and filtering in more detail. We will argue against the popular belief that the minimalist program and optimality theory are incompatible theories in that the former places the explanatory burden on the generative device (the computational system) whereas the latter places it on the fi ltering device (the OT evaluator). Although this belief may be correct in as far as it describes existing tendencies, we will argue that minimalist and optimality theoretic approaches normally adopt more or less the same global architecture of grammar: both assume that a generator defines a set S of potentially well-formed expressions that can be generated on the basis of a given input and that there is an evaluator that selects the expressions from S that are actually grammatical in a given language L. For this reason, we believe that it has a high priority to investigate the role of the two components in more detail in the hope that this will provide a better understanding of the differences and similarities between the two approaches. We will conclude this introduction with a brief review of the studies collected in this book.

Applying quantitative semantics to higher-order quantum computing

Finding a denotational semantics for higher order quantum computation is a long-standing problem in the semantics of quantum programming languages. Most past approaches to this problem fell short in one way or another, either limiting the language to an unusably small finitary fragment, or giving up important features of quantum physics such as entanglement. In this paper, we propose a denotational semantics for a quantum lambda calculus with recursion and an infinite data type, using constructions from quantitative semantics of linear logic

Probabilistic call by push value

We introduce a probabilistic extension of Levy's Call-By-Push-Value. This extension consists simply in adding a " flipping coin " boolean closed atomic expression. This language can be understood as a major generalization of Scott's PCF encompassing both call-by-name and call-by-value and featuring recursive (possibly lazy) data types. We interpret the language in the previously introduced denotational model of probabilistic coherence spaces, a categorical model of full classical Linear Logic, interpreting data types as coalgebras for the resource comonad. We prove adequacy and full abstraction, generalizing earlier results to a much more realistic and powerful programming language