The search for a semantics for higher-order quantum computation leads naturallyto the study of categories of normed cones. In the first part of this paper, we develop the theory of continuous normed cones, and prove some of their basic properties, includinga Hahn-Banach style theorem. We then describe two different concrete *-autonomous categories of normed cones. The first of these categories is built from completelypositive maps as in the author's semantics of first-order quantum computation. The second category is a reformulation of Girard's quantum coherent spaces. We also pointout why ultimately, neither of these categories is a satisfactory model of higher-order quantum computation.
The search for a semantics for higher-order quantum computation leads naturallyto the study of categories of normed cones. In the first part of this paper, we develop the theory of continuous normed cones, and prove some of their basic properties, includinga Hahn-Banach style theorem. We then describe two different concrete *-autonomous categories of normed cones. The first of these categories is built from completelypositive maps as in the author's semantics of first-order quantum computation. The second category is a reformulation of Girard's quantum coherent spaces. We also pointout why ultimately, neither of these categories is a satisfactory model of higher-order quantum computation.
The objective of this thesis is to develop a semantics for higher-order quantum information. Following the work done in the author's M.Sc. thesis, we study a lambda calculus for quantum computation with classical control. The language features two important properties. The first one, arising from the so-called no-cloning theorem of quantum computation, is the need for a distinction between duplicable and non-duplicable elements. For keeping track of duplicability at higher order, we use a type system inspired by the resource-sensitive linear logic. The second important aspect is the probability inherent to measurement, the only operation for retrieving classical data from quantum data. This forces us into choosing a reduction strategy for being able to define an operational semantics. We address the question of a denotational semantics in two respects. First, we restrict the study to the strictly linear aspect of the language. Doing so, we suppress the need for distinguishing between duplicable and non-duplicable elements and we can focus on the description of quantum features at higher order. Using the category of completely positive maps as a framework, we build a fully abstract denotational model of the strictly linear fragment of the language. The study of the full language is more demanding. For dealing with the probabilistic aspect of the language, we use a method inspired by Moggi and build a computational model with a distinction between values and computations. For the distinction between duplicability and non-duplicability in the calculus, we adapt Bierman's linear category, where the duplication is considered as a comonad with specific properties. The resulting model is what we call a linear category for duplication. Finally, we only focus on the fragment of the language that contains the aforementioned elements, and remove the classical Boolean and quantum Boolean features to get a generic computational linear lambda-calculus. In this idealized setting, we show that such a language have a full and complete interpretation in a linear category for duplication.
While much of the current study on quantum computation employs low-level formalisms such as quantum circuits, several high-level languages/calculi have been recently proposed aiming at structured quantum programming. The current work contributes to the semantical study of such languages by providing interaction-based semantics of a functional quantum programming language; the latter is, much like Selinger and Valiron's, based on linear lambda calculus and equipped with features like the ! modality and recursion. The proposed denotational model is the first one that supports the full features of a quantum functional programming language; we prove adequacy of our semantics. The construction of our model is by a series of existing techniques taken from the semantics of classical computation as well as from process theory. The most notable among them is Girard's Geometry of Interaction (GoI), categorically formulated by Abramsky, Haghverdi and Scott. The mathematical genericity of these techniques---largely due to their categorical formulation---is exploited for our move from classical to quantum.
Quantum Coherent Spaces were introduced by Girard as a quantum framework where to interpret the exponential-free fragment of Linear Logic. Aim of this paper is to extend Girard's interpretation to a subsystem of linear logic with bounded exponentials. We provide deduction rules for the bounded exponentials and, correspondingly, we introduce the novel notion of bounded exponentials of Quantum Coherent Spaces. We show that the latter provide a categorical model of our system. In order to do that, we first study properties of the category of Quantum Coherent Spaces.
While much of the current study on quantum computation employs low-level formalisms such as quantum circuits, several high-level languages/calculi have been recently proposed aiming at structured quantum programming. The current work contributes to the semantical study of such languages, by providing interaction-based semantics of a functional quantum programming language, the latter is based on linear lambda calculus and is equipped with features like the! modality and recursion. The proposed denotational model is the first one that supports the full features of a quantum functional programming language, we also prove adequacy of our semantics. The construction of our model is by a series of existing techniques taken from the semantics of classical computation as well as from process theory. The most notable among them is Girard's Geometry of Interaction (GoI), categorically formulated by Abramsky, Haghverdi and Scott. The mathematical genericity of these techniques - largely dueto their categorical formulation - is exploited for our move from classical to quantum.
Processes with indefinite causal orders (ICOs), such as the quantum switch, are higher-order quantum processes that superpose the order in which quantum operations are performed. Such coherent control yields computational advantages but is not faithfully captured by existing quantum programming languages: either they are restricted to the unitary case, and thus cannot combine ICOs with measurement, or they treat coherent control nonlinearly. In both cases, they do not realize the full computational power of ICOs. We introduce a higher-order quantum functional language that supports general quantum computation, not merely the permutation of channels, and whose linear type system allows quantum control to be well-defined beyond the unitary case, on arbitrary quantum channels. We equip this language with a small-step operational semantics that synchronizes measurement outcomes across superposed branches, using device references and a memory function. We also give a denotational semantics by means of completely positive maps. With linearity as the only constraint, some well-typed terms would denote unphysical maps. We therefore impose a typing discipline that goes beyond linearity, and interpret programs in the causal category Caus[CPM], under which every well-typed program is physically meaningful, a property that can be checked statically and efficiently. We prove soundness, and study the language's expressive power: it can express every quantum channel at first order, and at second order a large subclass of the so-called quantum circuits with quantum control (QC-QCs), containing the quantum switch. Last but not least, we show that this language is well-designed enough to be extended to the nonlinear setting with recursion.
We propose a new 'quantum domain theory' in which Scott-continuous functions are replaced by Scott-continuous natural transformations. Completely positive maps are widely accepted as a model of first-order quantum computation. We begin by establishing a categorical characterization of completely positive maps as natural families of positive maps. We explore this categorical characterization by building various representations of quantum computation based on different structures: affine maps between cones of positive elements, morphisms of algebras of effects, and affine maps of convex sets of states. By focusing on convex dcpos, we develop a quantum domain theory and show that it supports some important constructions such as tensor products by quantum data, and lifting.