Hyperbolic delaunay complexes and voronoi diagrams made practical

Богданов, М. Б.; Devillers, Olivier; Teillaud, Monique

doi:10.1145/2462356.2462365

Cited by 10 publications

(23 citation statements)

References 28 publications

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“…Remark 3.6. The hyperbolic Delaunay complex of [5], defined on p. 7 there, satisfies the empty circumspheres condition with only metric spheres. Thus it is a subcomplex of our Delaunay tessellation (by Theorem 5.9, the geometric dual to the Voronoi tessellation).…”

Section: The Delaunay Tessellation and The Finite Casementioning

confidence: 99%

The Delaunay tessellation in hyperbolic space

DeBlois

2016

Math. Proc. Camb. Phil. Soc.

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Abstract. The Delaunay tessellation of a locally finite subset of the hyperbolic space H n is constructed via convex hulls in R n+1 . For finite and lattice-invariant sets it is proven to be a polyhedral decomposition, and versions (necessarily modified from the Euclidean setting) of the empty circumspheres condition and geometric duality with the Voronoi tessellation are proved. Some pathological examples of infinite, non lattice-invariant sets are exhibited.The main theorem of this paper describes a "convex hull construction" of canonical polyhedral decompositions with prescribed vertex set for arbitrary complete, finite-volume hyperbolic manifolds and locally finite subsets. Various versions of this construction have been useful in the study of hyperbolic manifolds, beginning with work of Epstein-Penner in which the "vertex set" is essentially a collection of horospherical cusp neighborhoods [13]. Charney-Davis-Moussong gave a version for finite subsets of closed hyperbolic manifolds [7], generalizing earlier work of Näätänen-Penner [15]. More recently, Cooper-Long translated the Epstein-Penner construction to the setting of convex projective manifolds [8].Our results are complementary to [13] and generalize the main case of [7]. The main new case here, which is an important tool in our subsequent works [11] and [10], covers finite subsets of finite-volume non-compact hyperbolic manifolds. Compared to previous work this case exhibits substantial differences in the nature of the cells produced and of the decomposition's "geometric duality" relationship with the Voronoi tessellation. As we will describe below the statement, the source of these differences also significantly complicates the proof. We call our decomposition the "Delaunay tessellation" because it is characterized by an empty circumspheres condition, property (2) below. It is constructed in Definition 3.1.Theorem 6.23. Let Γ < SO + (1, n) be a torsion-free lattice and S a non-empty, locally finite, Γ-invariant set in H n . The Delaunay tessellation of S is a locally finite, Γ-invariant collection of convex polyhedra (the cells) whose union is H n , satisfying:(1) Each face of each cell is a cell, and distinct cells that intersect do so in a face of each; i.e. it is a polyhedral complex in the sense of eg. [9, Dfn. 2.1.5], with vertex set S. (2) For each metric ball or horoball B of H n that intersects S but only on its boundary, ie. such that S = ∂B satisfies B ∩ S = S ∩ S, the closed convex hull of S ∩ S in H n is a Delaunay cell contained in B. Each Delaunay cell has this form. (3) For each parabolic fixed point U of Γ such that there is a horoball centered at U and disjoint from S, there is a unique horosphere S centered at U such that the closed convex hull of S ∩ S in H n is a Γ U -invariant n-cell, where Γ U is the stabilizer of U in Γ. Each other cell is compact and has a metric circumsphere. The Delaunay tessellation is uniquely determined by condition (2) above.Here and in the remainder of the paper, we use the hyperboloid model of hyperbo...

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Section: The Delaunay Tessellation and The Finite Casementioning

confidence: 99%

The Delaunay tessellation in hyperbolic space

DeBlois

2016

Math. Proc. Camb. Phil. Soc.

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“…log[dim H Λ (Σ)]) for a topological quantum code on a closed genus g surface is proportional to g. On the other hand, for a fine triangulation with bounded geometry, where by bounded geometry we mean that the edge lengths and angles are bounded from above and below, the number of physical qubits n is proportional to the surface area A Σ and the code distance d scales as log(n) [26]. Therefore, according to (6), by using hyperbolic surfaces of increasing genus we can construct a family of hyperbolic Turaev-Viro codes with constant encoding rate and increasing code distance. Note that the encoding rate for a quantum code defined on a Euclidean surface is O(1/d 2 ) and goes to 0 as one goes to large distances [13].…”

Section: Hyperbolic Turaev-viro Codementioning

confidence: 99%

Universal logical gates with constant overhead: instantaneous Dehn twists for hyperbolic quantum codes

Lavasani

Zhu

Barkeshli

2019

Quantum

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A basic question in the theory of fault-tolerant quantum computation is to understand the fundamental resource costs for performing a universal logical set of gates on encoded qubits to arbitrary accuracy. Here we consider qubits encoded with constant space overhead (i.e. finite encoding rate) in the limit of arbitrarily large code distance d through the use of topological codes associated to triangulations of hyperbolic surfaces. We introduce explicit protocols to demonstrate how Dehn twists of the hyperbolic surface can be implemented on the code through constant depth unitary circuits, without increasing the space overhead. The circuit for a given Dehn twist consists of a permutation of physical qubits, followed by a constant depth local unitary circuit, where locality here is defined with respect to a hyperbolic metric that defines the code. Applying our results to the hyperbolic Fibonacci Turaev-Viro code implies the possibility of applying universal logical gate sets on encoded qubits through constant depth unitary circuits and with constant space overhead. Our circuits are inherently protected from errors as they map local operators to local operators while changing the size of their support by at most a constant factor; in the presence of noisy syndrome measurements, our results suggest the possibility of universal fault tolerant quantum computation with constant space overhead and time overhead of O(d/ log d). For quantum circuits that allow parallel gate operations, this yields the optimal scaling of spacetime overhead known to date.

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“…Both are static, i.e., they first compute the whole Euclidean triangulation before performing the extraction. The third one (omitted in this abstract, see [6]) is dynamic: it allows to add a point and to update the hyperbolic Delaunay complex while the Euclidean Delaunay triangulation is updated.…”

Section: Extracting Dt H (P) From Dt E (P): Algorithmsmentioning

confidence: 99%

“…The construction of the Delaunay complex and the Voronoi diagram in H 2 was implemented using CGAL [14,40,25] (see [6] for a description). The implementation will soon be submitted for future integration in CGAL.…”

Section: Implementation In Hmentioning

confidence: 99%