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Interacting quantum observables
 of Lecture Notes in Computer Science
, 2008
"... Abstract. We formalise the constructive content of an essential feature of quantum mechanics: the interaction of complementary quantum observables, and information flow mediated by them. Using a general categorical formulation, we show that pairs of mutually unbiased quantum account on the quantum d ..."
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Cited by 56 (27 self)
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Abstract. We formalise the constructive content of an essential feature of quantum mechanics: the interaction of complementary quantum observables, and information flow mediated by them. Using a general categorical formulation, we show that pairs of mutually unbiased quantum account on the quantum data encoded in complex phases, and prove a normal form theorem for it. Together these enable us to describe all observables of finite dimensional Hilbert space quantum mechanics. The resulting equations suffice to perform computations with elementary quantum gates, translate between distinct quantum computational models, establish the equivalence of entangled quantum states, and simulate quantum algorithms such as the quantum Fourier transform. All these computations moreover happen within an intuitive diagrammatic calculus. 1
Simulations in Coalgebra
 THEOR. COMP. SCI
, 2003
"... A new approach to simulations is proposed within the theory of coalgebras by taking a notion of order on a functor as primitive. Such an order forms a basic building block for a "lax relation lifting", or "relator" as used by other authors. Simulations appear as coalgebras of thi ..."
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Cited by 28 (2 self)
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A new approach to simulations is proposed within the theory of coalgebras by taking a notion of order on a functor as primitive. Such an order forms a basic building block for a "lax relation lifting", or "relator" as used by other authors. Simulations appear as coalgebras of this lifted functor, and similarity as greatest simulation. Twoway similarity is then similarity in both directions. In general, it is different from bisimilarity (in the usual coalgebraic sense), but a su#cient condition is formulated (and illustrated) to ensure that bisimilarity and twoway similarity coincide. Also, suitable conditions are identified which ensures that similarity on a final coalgebra forms an (algebraic) dcpo structure. This involves a close investigation of the iterated applications F (#) and F (1) of a functor F with an order to the initial and final sets.
New Directions in Categorical Logic, for Classical, Probabilistic and Quantum Logic
, 2014
"... ..."
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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Cited by 16 (4 self)
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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
Graphical reasoning in compact closed categories for quantum computation
 AMAI
, 2009
"... Compact closed categories provide a foundational formalism for a variety of important domains, including quantum computation. These categories have a natural visualisation as a form of graphs. We present a formalism for equational reasoning about such graphs and develop this into a generic proof sys ..."
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Cited by 15 (4 self)
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Compact closed categories provide a foundational formalism for a variety of important domains, including quantum computation. These categories have a natural visualisation as a form of graphs. We present a formalism for equational reasoning about such graphs and develop this into a generic proof system with a fixed logical kernel for reasoning about compact closed categories. A salient feature of our system is that it provides a formal and declarative account of derived results that can include ‘ellipses’style notation. We illustrate the framework by instantiating it for a graphical language of quantum computation and show how this can be used to perform symbolic computation.
Quantum logic in dagger kernel categories
 Order
"... This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial inject ..."
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Cited by 14 (12 self)
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This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial injections, Hilbert spaces (also modulo phase), and Boolean algebras, and (2) have interesting categorical/logical/ordertheoretic properties, in terms of kernel fibrations, such as existence of pullbacks, factorisation, orthomodularity, atomicity and completeness. For instance, the Sasaki hook and andthen connectives are obtained, as adjoints, via the existentialpullback adjunction between fibres. 1
Rewriting measurementbased quantum computations with generalised flow
 in Proc. of ICALP’10
, 2010
"... Abstract. We present a method for verifying measurementbased quantum computations, by producing a quantum circuit equivalent to a given deterministic measurement pattern. We define a diagrammatic presentation of the pattern, and produce a circuit via a rewriting strategy based on the generalised ..."
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Cited by 12 (2 self)
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Abstract. We present a method for verifying measurementbased quantum computations, by producing a quantum circuit equivalent to a given deterministic measurement pattern. We define a diagrammatic presentation of the pattern, and produce a circuit via a rewriting strategy based on the generalised flow of the pattern. Unlike other methods for translating measurement patterns with generalised flow to circuits, this method uses neither ancilla qubits nor acausal loops. In addition this method can detect errors in the correction strategy used to make the pattern deterministic. 1
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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Cited by 10 (7 self)
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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.
Categories for the practising physicist
"... in a somewhat unconventional manner. Our main focus will be on monoidal categories, mainly symmetric ones, for which we propose a physical interpretation. Special attention is given to the category of sets and relations, posetal categories, diagrammatic calculi, strictification, compact categories, ..."
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Cited by 6 (3 self)
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in a somewhat unconventional manner. Our main focus will be on monoidal categories, mainly symmetric ones, for which we propose a physical interpretation. Special attention is given to the category of sets and relations, posetal categories, diagrammatic calculi, strictification, compact categories, biproduct categories and abstract matrix calculi, internal structures, and topological quantum field theories. In our attempt to complement the existing literature we (on purpose) omitted some very basic topics for which we point to other available sources. 0 Prologue: cooking with vegetables Consider a raw potato. Conveniently, we refer to it as A. Raw potato A admits several states e.g. ‘dirty’, ‘clean’, ‘skinned’,... We usually don’t eat raw potatoes so we need to process A such that it becomes eatable. We refer to this cooked version of A as B. Also B admits several states e.g. ‘boiled’, ‘fried’, ‘baked with skin’, ‘baked without skin’,... Correspondingly, there are several ways to turn raw potato A into cooked potato B e.g. ‘boiling’, ‘frying’, ‘baking’, respectively referred to as f, f ′ and f ′ ′. We make the fact that these cooking processes apply to raw potato A and produce cooked potato B explicit by labelled arrows: A f ✲ B A f ′