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

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

Logics for reasoning about quantum states have been given in the literature. In this paper, we extend one such logic with temporal constructs mimicking the standard computational tree logic used to reason about classical transition systems. We investigate the model-checking problem for this temporal quantum logic and illustrate its use by reasoning about the BB84 key distribution protocol.

We introduce the language QML, a functional language for quantum computations on finite types. QML introduces quantum data and control structures, and integrates reversible and irreversible quantum computation. QML is based on strict linear logic, hence weakenings, which may lead to decoherence, have to be explicit. We present an operational semantics of QML programs using quantum circuits, and a denotational semantics using superoperators.

We show how to model the semantics of quantum programs that give classical output during their execution. That is, in our model even non-terminating programs may have output. The modelling interprets a program as a measurement process on the machines state, with the classical output as measurement result. The semantics presented here are fully abstract in the sense that two programs are equal if and only if they give the same outputs in any composition.

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

We provide a type system inspired by affine intuitionistic logic for the calculus of Higher-Order Mobile Embedded Resources (Homer), resulting in the first process calculus combining affine linear (non-copyable) and non-linear (copyable) higher-order mobile processes, nested locations, and local names. The type system guarantees that linear resources are neither copied nor embedded in non-linear resources during computation. We exemplify the use of the calculus by modelling a simplistic e-cash Smart Card system, the security of which depends on the interplay between (linear) mobile hardware, embedded (non-linear) mobile processes, and local names. A purely linear calculus would not be able to express that embedded software processes may be copied. Conversely, a purely non-linear calculus would not be able to express that mobile hardware processes cannot be copied.

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

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We provide a computational definition of the notions of vector space and bilinear functions. We use this result to introduce a minimal language combining higher-order computation and linear algebra. This language extends the λ-calculus with the possibility to make arbitrary linear combinations of terms α.t + β.u. We describe how to “execute” this language in terms of a few rewrite rules, and justify them through the two fundamental requirements that the language be a language of linear operators, and that it be higher-order. We mention the perspectives of this work in the field of quantum computation, whose circuits we show can be easily encoded in the calculus. Finally, we prove the confluence of the entire calculus.