Abstract

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.

Highlights

  • Quantum computation, a paradigm that exploits the quantum physical aspects of reality, promises to have a huge impact in computing

  • In order to define the operational semantics of the quantum λ-calculus, we provide a reminder on some basic notions regarding the mathematical representation of quantum states

  • We review some of the basic structures behind the denotational semantics of [Pagani et al 2014] ś these structures will play a role in our game semantics

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Summary

INTRODUCTION

A paradigm that exploits the quantum physical aspects of reality, promises to have a huge impact in computing. It is static (it is similar to the weighted relational models [Laird et al 2013]): it collects all completed executions of the classical layer of a quantum program, and annotates each with a quantum weight formalized as a morphism in CPM As a consequence, it carries no sequential information (nor does it claim to) and, fundamentally, cannot represent the evaluation order. Static semantics can be opposed to dynamic or interactive semantics, that display information about the sequentiality and dynamics of execution ś such semantics include game semantics and the geometry of interaction mentioned above In this family, and besides the models of fragments mentioned above, a recent breakthrough was achieved in [Dal Lago et al 2017]; yielding an adequate model of the full quantum λ-calculus based on an extension of the geometry of interaction.

THE QUANTUM λ-CALCULUS
Syntax and Typing
Pure Quantum States and Their Operations
Operational Semantics
Mixed Quantum States and Completely Positive Maps
LINEAR PROBABILISTIC GAMES
Games and Non-Deterministic Strategies
Probabilistic Strategies
LINEAR QUANTUM GAMES
Quantum Games and Strategies
A Compact Closed Category of Linear Quantum Strategies
ADEQUACY FOR THE AFFINE FRAGMENT
Call-By-Value Primitives
Soundness and Adequacy for the Affine Quantum λ-Calculus
REPLICATION AND THE FULL QUANTUM λ-CALCULUS
Quantum Games With Symmetry
Interpretation of the Full Quantum λ-Calculus
CONCLUSION
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