Abstract structure of unitary oracles for quantum algorithms

Zeng, William J.; Vicary, Jamie

doi:10.4204/eptcs.172.19

Cited by 7 publications

(10 citation statements)

References 75 publications

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“…Diagrammatic algebra has helped revealing structural similarities between quantum algorithms [Vic13,Zen15], and the equivalence of certain protocols in terms of the abstract operations needed to implement them [Vic12,RV17]. Moreover, it has naturally led to the problem of axiomatising fragments of quantum theory as independent higher algebraic theories: once we have packaged such a fragment into a theory, we can study it and tweak it on its own, forgetting the context from which it came, and gaining new perspective from its interpretation in different contexts.…”

Section: From Abelian Groups To Hilbert Spacesmentioning

confidence: 99%

The algebra of entanglement and the geometry of composition

Hadzihasanovic

2017

Preprint

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String diagrams turn algebraic equations into topological moves. These moves have often-recurring shapes, involving the sliding of one diagram past another. In the past, this fact has mostly been considered in terms of its computational convenience; this thesis investigates its deeper reasons.In the first part of the thesis, we individuate, at its root, the dual nature of polygraphs -freely generated higher categories -as presentations of higher algebraic theories, and as combinatorial descriptions of "directed spaces": CW complexes whose cells have their boundary subdivided into an input and an output region. Operations of polygraphs modelled on operations of topological spaces, including an asymmetric tensor product refining the topological product of spaces, can then be used as the foundation of a compositional universal algebra, where sliding moves of string diagrams for an algebraic theory arise from tensor products of sub-theories. Such compositions come automatically with higher-dimensional coherence cells. We provide several examples of compositional reconstructions of higher algebraic theories, including theories of braids, homomorphisms of algebras, and distributive laws of monads, from operations on polygraphs.In this regard, the standard formalism of polygraphs, based on strict ω-categories, suffers from some technical problems and inadequacies, including the difficulty of computing tensor products. As a solution, we propose a notion of regular polygraph, barring cell boundaries that are degenerate, that is, not homeomorphic to a disk of the appropriate dimension. We develop the theory of globular posets, based on ideas of poset topology, in order to specify a category of shapes for cells of regular polygraphs. We prove that these shapes satisfy our non-degeneracy requirement, and show how to calculate their tensor products. Then, we introduce a notion of weak unit for regular polygraphs, allowing us to recover weakly degenerate boundaries in low dimensions. We prove that the existence of weak units is equivalent to the existence of cells satisfying certain divisibility properties -an elementary notion of equivalence cell -which prompts new questions on the relation between units and equivalences in higher categories.In the second part of the thesis, we turn to specific applications of diagrammatic algebra to quantum theory. First, we re-evaluate certain aspects of quantum theory from the point of view of categorical universal algebra, which leads us to define a 2-dimensional refinement of the category of Hilbert spaces. Then, we focus on the problem of axiomatising fragments of quantum theory, and present the ZW calculus, the first complete diagrammatic axiomatisation of the theory of qubits.The ZW calculus has specific advantages over its predecessors, the ZX calculi, including the existence of a computationally meaningful normal form, and of a fragment whose diagrams can be interpreted physically as setups of fermionic oscillators. Moreover, the choice of its generators reflects an operational clas...

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Section: From Abelian Groups To Hilbert Spacesmentioning

confidence: 99%

The algebra of entanglement and the geometry of composition

Hadzihasanovic

2017

Preprint

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“…In quantum foundations, the CQM framework has been applied to study features of causality [CPV13, CK15,Coe16] and non-locality [CDKW12,Gog15a], both operationally and in connection with the sheaf-theoretic framework for contextuality of [AB14, AMB12, ABK + 15] (which is also applicable to OPTs [CY16]). In quantum information and computation, the framework has been applied to the study of quantum algorithms and protocols [CD11, Vic12b, ZV14, Vic12a, VV16, CK17, Zam12], measurement-based and clusterstate quantum computing [Dun15,Hor11], complementarity [MV15,DD16,ZV14], and the information theoretic characterisation of quantum theory [HK16]. Relational models for non-deterministic classical computation have also been explored using tools from CQM, both as toy models for quantum theory [Pav09, Abr13, Gog15c, Mar, Coe16, CE12, BD15] and in their own right as models of computation [Pav09,BV14].…”

Section: Some Application Of Cqmmentioning

confidence: 99%

“…A number of applications of CQM rely on strong complementarity as their active ingredient: most relevant are its appearance as an abstract version of the quantum Fourier transform in group-theoretic quantum algorithms [Vic12b,ZV14,Zen15,GZ15a], and the role it plays in connecting Mermin-type non-locality scenarios to the structure of phase groups in abstract process theories [CDKW12,CES10,GZ15b].…”

Section: Applications Of Strong Complementaritymentioning

confidence: 99%

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Categorical Quantum Dynamics

Gogioso

2017

Preprint

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“…A section of the "Dodo book" [CK17] is dedicated to the description of quantum algorithms in ZX-calculus (in particular Deutsch-Jozsa and Grover), and a few articles [ZV14,Vic13] address the diagrammatic description of quantum oracles, but we show that the recent developments in the formalism, mainly the scalable construction [CHP19,CP20] and the discard construction [CJPV19] allow for more self-contained, accurate and compact ZX-based proofs of quantum algorithms. Notice that depending on the algorithm we also use generators of the ZH-calculus [BK19] a variant of the ZX-calculus.…”

Section: Introductionmentioning

confidence: 99%

Quantum Algorithms and Oracles with the Scalable ZX-calculus

Carette,

D'Anello,

Perdrix

2021

Preprint

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The ZX-calculus was introduced as a graphical language able to represent specific quantum primitives in an intuitive way. The recent completeness results have shown the theoretical possibility of a purely graphical description of quantum processes. However, in practice, such approaches are limited by the intrinsic low level nature of ZX calculus. The scalable notations have been proposed as an attempt to recover an higher level point of view while maintaining the topological rewriting rules of a graphical language. We demonstrate that the scalable ZX-calculus provides a formal, intuitive, and compact framework to describe and prove quantum algorithms. As a proof of concept, we consider the standard oracle-based quantum algorithms: Deutsch-Jozsa, Bernstein-Vazirani, Simon, and Grover algorithms, and we show they can be described and proved graphically.

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Abstract structure of unitary oracles for quantum algorithms

Cited by 7 publications

References 75 publications

The algebra of entanglement and the geometry of composition

The algebra of entanglement and the geometry of composition

Categorical Quantum Dynamics

Quantum Algorithms and Oracles with the Scalable ZX-calculus

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