Holographic software for quantum networks

Jaffe, Arthur; Liu, Zhengwei; Wozniakowski, Alex

doi:10.1007/s11425-017-9207-3

Cited by 33 publications

(48 citation statements)

References 81 publications

(112 reference statements)

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“…Greenberg, Horne and Zeilinger introduced a multipartite resource state for quantum information, called the GHZ state, denoted by |GHZ [GHZ89]. In [JLW16a], Jaffe, Wozniakowski and the authur find another resource state following topological intuition, called |M ax . They both generalize the Bell state.…”

Section: 2mentioning

confidence: 99%

Quon Language: Surface Algebras and Fourier Duality

Liu

2019

Commun. Math. Phys.

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Quon language is a 3D picture language that we can apply to simulate mathematical concepts. We introduce the surface algebras as an extension of the notion of planar algebras to higher genus surface. We prove that there is a unique one-parameter extension. The 2D defects on the surfaces are quons, and surface tangles are transformations. We use quon language to simulate graphic states that appear in quantum information, and to simulate interesting quantities in modular tensor categories. This simulation relates the pictorial Fourier duality of surface tangles and the algebraic Fourier duality induced by the S matrix of the modular tensor category. The pictorial Fourier duality also coincides with the graphic duality on the sphere. For each pair of dual graphs, we obtain an algebraic identity related to the S matrix. These identities include well-known ones, such as the Verlinde formula; partially known ones, such as the 6j-symbol self-duality; and completely new ones.

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Section: 2mentioning

confidence: 99%

Quon Language: Surface Algebras and Fourier Duality

Liu

2019

Commun. Math. Phys.

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“…In a categorical setting, the Reidemeister move (R2) follows from the invertibility of a braiding, while (R3) follows from the coherence axioms of a braided monoidal category and the naturality of a braiding. Therefore, 3D-isotopy is modelled by a braided autonomous category augmented with the loop axiom (R1) [Sel10], which appears so useful that it is exploited by several graphical languages for quantum information and computation [RV19,JLW18].…”

Section: Braided Autonomous Category / 3d Isotopymentioning

confidence: 99%

A Simplified Stabilizer ZX-calculus

Backens¹,

Perdrix²,

Wang³

2017

Electron. Proc. Theor. Comput. Sci.

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The stabilizer ZX-calculus is a rigorous graphical language for reasoning about quantum mechanics. The language is sound and complete: a stabilizer ZX-diagram can be transformed into another one if and only if these two diagrams represent the same quantum evolution or quantum state. We show that the stabilizer ZX-calculus can be simplified, removing unnecessary equations while keeping only the essential axioms which potentially capture fundamental structures of quantum mechanics. We thus give a significantly smaller set of axioms and prove that meta-rules like 'colour symmetry' and 'upside-down symmetry', which were considered as axioms in previous versions of the language, can in fact be derived. In particular, we show that the additional symbol and one of the rules which had been recently introduced to keep track of scalars (diagrams with no inputs or outputs) are not necessary.

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“…We obtain fundamental diagrams for resource states, measurements, the Pauli matrices X Y Z , , , and the string Fourier transform F S . Our strings are charged, with a label indicating the charge  Î k ; d this means that we consider charges modulo d. The reader may wish to read complete details about the two-string language that we present in [15,21]. However, in order to make this paper self-contained, we explain in this section those aspects of the language that we require in this paper-without repeating the detailed proofs.…”

Section: Fundamental Diagrams In the Two-string Languagementioning

confidence: 99%

“…Experimental work on ñ |GHZ was achieved in [24][25][26]. We introduced our n-qudit resource state ñ |Max in [15]. This state generalizes the Bell state (16) and has the diagrammatic representation…”

Section: Multipartite Resource Statesmentioning

confidence: 99%