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Quantum and classical structures in nondeterministic computation
 Proceedings of Quanum Interaction 2009, Lecture
"... Abstract. In categorical quantum mechanics, classical structures characterize the classical interfaces of quantum resources on one hand, while on the other hand giving rise to some quantum phenomena. In the standard Hilbert space model of quantum theories, classical structures over a space correspon ..."
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Abstract. In categorical quantum mechanics, classical structures characterize the classical interfaces of quantum resources on one hand, while on the other hand giving rise to some quantum phenomena. In the standard Hilbert space model of quantum theories, classical structures over a space correspond to its orthonormal bases. In the present paper, we show that classical structures in the category of relations correspond to direct sums of abelian groups. Although relations are, of course, not an interesting model of quantum computation, this result has some interesting computational interpretations. If relations are viewed as denotations of nondeterministic programs, it uncovers a wide variety of nonstandard quantum structures in this familiar area of classical computation. Ironically, it also opens up a version of what in philosophy of quantum mechanics would be called an onticepistemic gap, as it provides no interface to these nonstandard quantum structures. 1
2008) Classical and quantum structures
"... In recent work, symmetric daggermonoidal (SDM) categories have emerged as a convenient categorical formalization of quantum mechanics. The objects represent physical systems, the morphisms physical operations, whereas the tensors describe composite systems. Classical data turn out to correspond to ..."
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In recent work, symmetric daggermonoidal (SDM) categories have emerged as a convenient categorical formalization of quantum mechanics. The objects represent physical systems, the morphisms physical operations, whereas the tensors describe composite systems. Classical data turn out to correspond to Frobenius algebras with some additional properties. They express the distinguishing capabilities of classical data: in contrast with quantum data, classical data can be copied and deleted. The algebraic approach thus shifts the paradigm of ”quantization ” of a classical theory to ”classicization ” of a quantum theory. Remarkably, the simple SDM framework suffices not only for this conceptual shift, but even allows us to distinguish the deterministic classical operations (i.e. functions) from the nondeterministic classical operations (i.e. relations), and the probabilistic classical operations (stochastic maps). Moreover, a combination of some basic categorical constructions (due to Kleisli, resp. Grothendieck) with the categorical presentations of quantum states, provides a resource sensitive account of various quantumclassical interactions: of classical control of quantum data, of classical data arising from quantum measurements, as well as of the classical data processing inbetween controls and measurements. A salient feature here is the graphical calculus for categorical quantum mechanics, which allows a purely diagrammatic representation of classicalquantum interaction. 1
2009) Graph states and the necessity of Euler decomposition
 In: Proceedings of Computability in Europe: Mathematical Theory and Computational Practice (CiE’09
"... Abstract. Coecke and Duncan recently introduced a categorical formalisation of the interaction of complementary quantum observables. In this paper we use their diagrammatic language to study graph states, a computationally interesting class of quantum states. We give a graphical proof of the fixpoin ..."
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Abstract. Coecke and Duncan recently introduced a categorical formalisation of the interaction of complementary quantum observables. In this paper we use their diagrammatic language to study graph states, a computationally interesting class of quantum states. We give a graphical proof of the fixpoint property of graph states. We then introduce a new equation, for the Euler decomposition of the Hadamard gate, and demonstrate that Van den Nest’s theorem—locally equivalent graphs represent the same entanglement—is equivalent to this new axiom. Finally we prove that the Euler decomposition equation is not derivable from the existing axioms.
A unified sentence space for categorical distributionalcompositional semantics: Theory and experiments
 In Proceedings of COLING: Posters
, 2012
"... This short paper summarizes a faithful implementation of the categorical framework of Coecke et al. (2010), the aim of which is to provide compositionality in distributional models of lexical semantics. Based on Frobenius Algebras, our method enable us to (1) have a unifying meaning space for phrase ..."
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This short paper summarizes a faithful implementation of the categorical framework of Coecke et al. (2010), the aim of which is to provide compositionality in distributional models of lexical semantics. Based on Frobenius Algebras, our method enable us to (1) have a unifying meaning space for phrases and sentences of different structure and word vectors, (2) stay faithful to the linguistic types suggested by the underlying typelogic, and (3) perform the concrete computations in lower dimensions by reducing the space complexity. We experiment with two different parameters of the model and apply the setting to a verb disambiguation and a term/definition classification task with promising results.
Geometry of abstraction in quantum computation
"... Quantum algorithms are sequences of abstract operations, performed on nonexistent computers. They are in obvious need of categorical semantics. We present some steps in this direction, following earlier contributions of Abramsky, Coecke and Selinger. In particular, we analyze function abstraction i ..."
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Quantum algorithms are sequences of abstract operations, performed on nonexistent computers. They are in obvious need of categorical semantics. We present some steps in this direction, following earlier contributions of Abramsky, Coecke and Selinger. In particular, we analyze function abstraction in quantum computation, which turns out to characterize its classical interfaces. Some quantum algorithms provide feasible solutions of important hard problems, such as factoring and discrete log (which are the building blocks of modern cryptography). It is of a great practical interest to precisely characterize the computational resources needed to execute such quantum algorithms. There are many ideas how to build a quantum computer. Can we prove some necessary conditions? Categorical semantics help with such questions. We show how to implement an important family of quantum algorithms using just abelian groups and relations.
Quantum picturalism
, 2009
"... Why did it take us 50 years since the birth of the quantum mechanical formalism to discover that unknown quantum states cannot be cloned? Yet, the proof of the ‘nocloning theorem’ is easy, and its consequences and potential for applications are immense. Similarly, why did it take us 60 years to dis ..."
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Why did it take us 50 years since the birth of the quantum mechanical formalism to discover that unknown quantum states cannot be cloned? Yet, the proof of the ‘nocloning theorem’ is easy, and its consequences and potential for applications are immense. Similarly, why did it take us 60 years to discover the conceptually intriguing and easily derivable physical phenomenon of ‘quantum teleportation’? We claim that the quantum mechanical formalism doesn’t support our intuition, nor does it elucidate the key concepts that govern the behaviour of the entities that are subject to the laws of quantum physics. The arrays of complex numbers are kin to the arrays of 0s and 1s of the early days of computer programming practice. Using a technical term from computer science, the quantum mechanical formalism is ‘lowlevel’. In this review we present steps towards a diagrammatic ‘highlevel ’ alternative for the Hilbert space formalism, one which appeals to our intuition. The diagrammatic language as it currently stands allows for intuitive reasoning about interacting quantum systems, and trivialises many otherwise involved and tedious computations. It clearly exposes limitations such as the nocloning theorem, and phenomena such as quantum teleportation. As a logic, it supports ‘automation’: it enables a (classical) computer to reason about interacting quantum systems, prove theorems, and design protocols. It allows for a wider variety of underlying theories, and can be easily modified, having the potential to provide the required stepstone towards a deeper conceptual understanding of quantum theory, as well as its
Generalised ProofNets for Compact Categories with Biproducts
, 2009
"... Just as conventional functional programs may be understood as proofs in an intuitionistic logic, so quantum processes can also be viewed as proofs in a suitable logic. We describe such a logic, the logic of compact closed categories and biproducts, presented both as a sequent calculus and as a syste ..."
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Just as conventional functional programs may be understood as proofs in an intuitionistic logic, so quantum processes can also be viewed as proofs in a suitable logic. We describe such a logic, the logic of compact closed categories and biproducts, presented both as a sequent calculus and as a system of proofnets. This logic captures much of the necessary structure needed to represent quantum processes under classical control, while remaining agnostic to the fine details. We demonstrate how to represent quantum processes as proofnets, and show that the dynamic behaviour of a quantum process is captured by the cutelimination procedure for the logic. We show that the cut elimination procedure is strongly normalising: that is, that every legal way of simplifying a proofnet leads to the same, unique, normal form. Finally, taking some initial set of operations
The Frobenius Anatomy of Relative Pronouns
"... This paper develops a compositional vectorbased semantics of relative pronouns within a categorical framework. Frobenius algebras are used to formalise the operations required to model the semantics of relative pronouns, including passing information between the relative clause and the modified nou ..."
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This paper develops a compositional vectorbased semantics of relative pronouns within a categorical framework. Frobenius algebras are used to formalise the operations required to model the semantics of relative pronouns, including passing information between the relative clause and the modified noun phrase, as well as copying, combining, and discarding parts of the relative clause. We develop two instantiations of the abstract semantics, one based on a truththeoretic approach and one based on corpus statistics. 1