Biunitary constructions in quantum information

Reutter, David; Vicary, Jamie

doi:10.48550/arxiv.1609.07775

Cited by 13 publications

(34 citation statements)

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“…The current authors have previously shown that CQM methods can be extended to a highercategorical setting [54,63,64], developing the work of Baez on a categorified notion of Hilbert space [7], and this paper develops these ideas further.…”

Section: Related Workmentioning

confidence: 91%

“…The graphical notation we employ can be described formally as a shaded planar algebra, although we do not use that terminology in this paper, preferring a more elementary presentation. The relationship between shaded planar algebras and Hadamard matrices was first suggested by Jones [34], and developed by the present authors [54,63,64].…”

Section: Related Workmentioning

confidence: 95%

“…To help the reader understand these new 5 Here and throughout, we use the term '2-category' to refer to the weak structure, which is sometimes called 'bicategory'. concepts, we also give a direct account of the formalism in elementary terms, that can be used without reference to the higher categorical technology (see also [54]). In this direct perspective, regions are labelled by finite sets.…”

Section: Related Workmentioning

confidence: 99%

“…On a finite-dimensional Hilbert space H, a unitary error basis is a basis of unitary operators U i : H − → H such that Tr(U † i U j ) = δ i j dim(H). UEBs have an elegant presentation in terms of our graphical calculus [54,64]. Theorem 5.4 (See [54]).…”

Section: Unitary Error Basis Codesmentioning

confidence: 99%

See 3 more Smart Citations

Shaded Tangles for the Design and Verification of Quantum Programs (Extended Abstract)

Reutter

Vicary

2018

Electron. Proc. Theor. Comput. Sci.

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We give a scheme for interpreting shaded tangles as quantum programs, with the property that isotopic tangles yield equivalent programs. We analyze many known quantum programs in this way-including entanglement manipulation and error correction-and in each case present a fullytopological formal verification, yielding in several cases substantial new insight into how the program works. We also use our methods to identify several new or generalized procedures.

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Section: Related Workmentioning

confidence: 91%

Section: Related Workmentioning

confidence: 95%

Section: Related Workmentioning

confidence: 99%

Section: Unitary Error Basis Codesmentioning

confidence: 99%

See 2 more Smart Citations

Shaded Tangles for the Design and Verification of Quantum Programs (Extended Abstract)

Reutter

Vicary

2018

Electron. Proc. Theor. Comput. Sci.

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“…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. Examples include the theory of biunitaries, which underlies several constructions in quantum information theory [RV16], but also classical cryptography with "one time pads" [SV13]; the various theories called "ZX calculi", axiomatisations of fragments of the theory of qubits, including stabiliser quantum mechanics [Bac14a]; and our own ZW calculus [Had15], the subject of Chapter 5, towards which we will be working in this chapter.…”

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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The Morita Theory of Quantum Graph Isomorphisms

Musto

Reutter

Verdon

2018

Commun. Math. Phys.

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We classify instances of quantum pseudo-telepathy in the graph isomorphism game, exploiting the recently discovered connection between quantum information and the theory of quantum automorphism groups. Specifically, we show that graphs quantum isomorphic to a given graph are in bijective correspondence with Morita equivalence classes of certain Frobenius algebras in the category of finite-dimensional representations of the quantum automorphism algebra of that graph. We show that such a Frobenius algebra may be constructed from a central type subgroup of the classical automorphism group, whose action on the graph has coisotropic vertex stabilisers. In particular, if the original graph has no quantum symmetries, quantum isomorphic graphs are classified by such subgroups. We show that all quantum isomorphic graph pairs corresponding to a well-known family of binary constraint systems arise from this group-theoretical construction. We use our classification to show that, of the small order vertextransitive graphs with no quantum symmetry, none is quantum isomorphic to a non-isomorphic graph. We show that this is in fact asymptotically almost surely true of all graphs.

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Biunitary constructions in quantum information

Cited by 13 publications

References 47 publications

Shaded Tangles for the Design and Verification of Quantum Programs (Extended Abstract)

Shaded Tangles for the Design and Verification of Quantum Programs (Extended Abstract)

The algebra of entanglement and the geometry of composition

The Morita Theory of Quantum Graph Isomorphisms

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