This thesis introduces the language QML, a functional language for quantum computations on finite types. QML exhibits quantum data and control structures, and integrates reversible and irreversible quantum computations. The design of QML is guided by the categorical semantics: QML programs are in-terpreted by morphisms in the category FQC of finite quantum computations, which provides a constructive operational semantics of irreversible quantum computations, realisable as quantum circuits. The quantum circuit model is also given a formal categorical definition via the category FQC. QML integrates reversible and irreversible quantum computations in one language, using first order strict linear logic to make weakenings, which may lead to the collapse of the quantum wavefunction, explicit. Strict programs are free from measurement, and hence preserve superpositions and entanglement. A denotational semantics of QML programs is presented, which maps QML terms

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We define a language CQP (Communicating Quantum Processes) for modelling systems which combine quantum and classical communication and computation. CQP combines the communication primitives of the picalculus with primitives for measurement and transformation of quantum state; in particular, quantum bits (qubits) can be transmitted from process to process along communication channels. CQP has a static type system which classifies channels, distinguishes between quantum and classical data, and controls the use of quantum state. We formally define the syntax, operational semantics and type system of CQP, prove that the semantics preserves typing, and prove that typing guarantees that each qubit is owned by a unique process within a system. We illustrate CQP by defining models of several quantum communication systems, and outline our plans for using CQP as the foundation for formal analysis and verification of combined quantum and classical systems. 1

We present a quantum interpretation of the perfect part of linear logic, by means of quantum coherent spaces. In particular this yields a novel interpretation of the reduction of the wave packet as the expression of η-conversion, a.k.a, extensionality.

We show that quantum computation can be decomposed in a pure classical (functional) part and an effectful part modeling probabilities and measurement. The effectful part can be modeled using a generalization of monads called arrows. Both the functional and effectful parts can be elegantly expressed in the Haskell programming language. 1

Quantum pseudo-telepathy is an intriguing phenomenon which results from the application of quantum information theory to communication complexity. To demonstrate this phenomenon researchers in the field of quantum communication complexity devised a number of quantum non-locality games. The setting of these games is as follows: the players are separated so that no communication between them is possible and are given a certain computational task. When the players have access to a quantum resource called entanglement, they can accomplish the task: something that is impossible in a classical setting. To an observer who is unfamiliar with the laws of quantum mechanics it seems that the players employ some sort of telepathy; that is, they somehow exchange information without sharing a communication channel. This paper provides a formal framework for specifying, implementing, and analysing quantum non-locality games.

Category theory is a very general formalism, but there is a certain special way that physicists use categories which turns out to have close analogues in topology, logic and computation. A category has objects and morphisms, which represent things and ways to go between things. In physics, the objects are often physical systems, and the morphisms are processes turning a state of one physical system into a state of another system — perhaps

The article discusses causal models, such as Petri nets and event structures, how they have been rediscovered in a wide variety of recent applications, and why they are fundamental to computer science. A discussion of their present limitations leads to their extension with symmetry. The consequences, actual and potential, are discussed.

We apply the notion of quantum predicate proposed by D’Hondt and Panangaden to analyze a purely quantum language fragment which describes the quantum part of a future quantum computer in Knill’s architecture. The denotational semantics, weakest precondition semantics, and weakest liberal precondition semantics of this language fragment are introduced. To help reasoning about quantum programs involving quantum loops, we extend proof rules for classical probabilistic programs to our purely quantum programs. 1

Abstract. We propose an extension of the traditional λ-calculus in which terms are used to control an outside computing device (quantum computer, DNA computer...). We introduce two new binders: ν and ρ. In νx.M, x denotes an abstract resource of the outside computing device, whereas in ρx.M, x denotes a concrete resource. These two binders have different properties (in terms of α-conversion, scope extrusion, convertibility) than the ones of standard λ-binder. We illustrate the potential benefits of our approach with a study of a quantum computing language in which these new binders prove meaningful. We introduce a typing system for this quantum computing framework in which linearity is only required for concrete quantum bits offering a greater expressiveness than previous propositions. 1