The aim of this paper is to introduce a general model of quantum computation, the quantum calculus: both unitary transformations and projective measurements are allowed; furthermore a complete classical control, including conditional structures and loops, is available. Complementary to its operational semantics, we introduce a pure denotational semantics for the quantum calculus. Based on probabilistic power domains [Jones, C. and G. D. Plotkin, A probabilistic powerdomain of evaluations, in: LICS, 1989, pp. 186–195. URL http://homepages.inf.ed.ac.uk/gdp/publications/Prob_Powerdomain.pdf], this pure denotational semantics associates with any description of a computation in the quantum calculus its action in a mathematical setting. Adequacy between operational and pure denotational semantics is established. Additionally to this pure denotational semantics, an observable denotational semantics is introduced. Following the work by Selinger, this observable denotational semantics is based on density matrices and super-operators. Finally, we establish an exact abstraction connection between these two semantics.
The aim of this paper is to introduce a general model of quantum computation, the quantum calculus: both unitary transformations and projective measurements are allowed; furthermore a complete classical control, including conditional structures and loops, is available. Complementary to its operational semantics, we introduce a pure denotational semantics for the quantum calculus. Based on probabilistic power domains [Jones, C. and G. D. Plotkin, A probabilistic powerdomain of evaluations, in: LICS, 1989, pp. 186–195. URL http://homepages.inf.ed.ac.uk/gdp/publications/Prob_Powerdomain.pdf], this pure denotational semantics associates with any description of a computation in the quantum calculus its action in a mathematical setting. Adequacy between operational and pure denotational semantics is established. Additionally to this pure denotational semantics, an observable denotational semantics is introduced. Following the work by Selinger, this observable denotational semantics is based on density matrices and super-operators. Finally, we establish an exact abstraction connection between these two semantics.
This paper studies the linear fragment of the programing language for quantum computation with classical control described in [Selinger, P. and B. Valiron, A lambda calculus for quantum computation with classical control, in: P. Urzyczyn, editor, Proceedings of the Seventh International Conference on Typed Lambda Calculi and Applications (TLCA 2005), Lecture Notes in Computer Science 3461 (2005), pp. 354–368]. We sketch the language, and discuss equivalence of terms. We also describe a fully abstract denotational semantics based on completely positive maps.
In this thesis, we study quantum programming languages, and focus on the quantum lambda-calculus, a paradigmatic language defined by Selinger and Valiron, which mixes a rich classical control flow (higher order functions, etc.) and quantum data (qubit creation, measure, and unitary operations). In this thesis, we search for a denotational semantics of this language, in other words, we extract the meaning and behaviour of programs by interpreting them in an adequately chosen mathematical universe. In order to model the quantum lambda-calculus, one must adapt the traditional denotational models to the mathematics of quantum computation (Hilbert spaces, density matrices, etc.). We have three main contributions:Firstly, we extend game semantics to the quantum case. Game semantics is a dynamic approach to denotational semantics, which focuses on the sequence of interactions between a program and its environment. We obtain the first denotational model of the full quantum lambda-calculus which is both compositional and interactive, meaning that it represents the dynamic of the execution. Our model generalise the recently developed model for the probabilistic case.Secondly, we formally link our model to the only pre-existing denotational model of the full lambda-calculus, the quantum relational model of Pagani, Selinger and Valiron.Lastly, we prove that our game model is fully abstract, the gold standard of denotational semantics, which ensures a total correspondence between the programming language and the model. We deduce that the quantum relational model was also fully abstract, which was an open problem.This thesis is constituted of three parts. The first part details some preliminaries on category theory, used for defining a formal framework including both the quantum game model and the quantum relational model, some preliminaries on the mathematics of quantum computation, and then some preliminaries on quantum programming and its models. The second part starts with some preliminaries on game semantics and defines our quantum game model in the simplified case of the linear quantum lambda-calculus. The third part builds on the second part to define the full quantum game model, prove its full abstraction, and deduce that the quantum relational model is fully abstract too.
Several domains [S. Abramsky. A Cook's tour of a simple quantum programming language. 3rd International Symposium on Domain Theory, Xi'an, China, May 2004; B. Coecke and K. Martin. A partial order on classical and quantum states. Technical report, PRG-RR-02-07, 2002; Ph. Jorrand and S. Perdrix. Towards a quantum calculus. In To appear in Proceedings of the 4th International Workshop on Quantum Programming Languages, ENTCS, 2006; P. Selinger. Towards a quantum programming language. Mathematical Structures in Computer Science, 14(4):527–586, 2004] can be used to define the semantics of quantum programs. Among them Abramsky [S. Abramsky. A Cook's tour of a simple quantum programming language. 3rd International Symposium on Domain Theory, Xi'an, China, May 2004] has introduced a semantics based on probabilistic power domains, whereas the one by Selinger [P. Selinger. Towards a quantum programming language. Mathematical Structures in Computer Science, 14(4):527–586, 2004] 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 [S. Abramsky. A Cook's tour of a simple quantum programming language. 3rd International Symposium on Domain Theory, Xi'an, China, May 2004]. The observable semantics, à la Selinger [P. Selinger. Towards a quantum programming language. Mathematical Structures in Computer Science, 14(4):527–586, 2004], 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 [P. Cousot. Types as abstract interpretations. In POPL, pages 316–331, 1997] or interpretation relations are established between them, leading to a hierarchy of quantum semantics.
Measurement-based quantum computation has emerged from the physics community as a new approach to quantum computation where the notion of measurement is the main driving force of computation. This is in contrast with the more traditional circuit model which is based on unitary operations. Among measurement-based quantum computation methods, the recently introduced one-way quantum computer stands out as fundamental. We develop a rigorous mathematical model underlying the one-way quantum computer and present a concrete syntax and operational semantics for programs, which we call patterns, and an algebra of these patterns derived from a denotational semantics. More importantly, we present a calculus for reasoning locally and compositionally about these patterns. We present a rewrite theory and prove a general standardization theorem which allows all patterns to be put in a semantically equivalent standard form. Standardization has far-reaching consequences: a new physical architecture based on performing all the entanglement in the beginning, parallelization by exposing the dependency structure of measurements and expressiveness theorems. Furthermore we formalize several other measurement-based models: Teleportation, Phase and Pauli models and present compositional embeddings of them into and from the one-way model. This allows us to transfer all the theory we develop for the one-way model to these models. This shows that the framework we have developed has a general impact on measurement-based computation and is not just particular to the one-way quantum computer.
Measurement-based quantum computation has emerged from the physics community as a new approach to quantum computation where the notion of measurement is the main driving force of computation. This is in contrast with the more traditional circuit model that is based on unitary operations. Among measurement-based quantum computation methods, the recently introduced one-way quantum computer [Raussendorf and Briegel 2001] stands out as fundamental. We develop a rigorous mathematical model underlying the one-way quantum computer and present a concrete syntax and operational semantics for programs, which we call patterns , and an algebra of these patterns derived from a denotational semantics. More importantly, we present a calculus for reasoning locally and compositionally about these patterns. We present a rewrite theory and prove a general standardization theorem which allows all patterns to be put in a semantically equivalent standard form. Standardization has far-reaching consequences: a new physical architecture based on performing all the entanglement in the beginning, parallelization by exposing the dependency structure of measurements and expressiveness theorems. Furthermore we formalize several other measurement-based models, for example, Teleportation, Phase and Pauli models and present compositional embeddings of them into and from the one-way model. This allows us to transfer all the theory we develop for the one-way model to these models. This shows that the framework we have developed has a general impact on measurement-based computation and is not just particular to the one-way quantum computer.