Zoology of Atlas-Groups: Dessins D’enfants, Finite Geometries and Quantum Commutation

Planat, Michel; Zainuddin, Hishamuddin

doi:10.3390/math5010006

Cited by 16 publications

(26 citation statements)

References 41 publications

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“…Why not use this knot group for quantum computing? In [3,4,22,23], M. Planat et al studied the representation of knot groups and the usage for quantum computing. Here, we discussed a direct relation between the knot complement and quantum computing via the Berry phase.…”

Section: Discussionmentioning

confidence: 99%

Topological Quantum Computing and 3-Manifolds

Asselmeyer-Maluga

2021

Quantum Reports

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In this paper, we will present some ideas to use 3D topology for quantum computing. Topological quantum computing in the usual sense works with an encoding of information as knotted quantum states of topological phases of matter, thus being locked into topology to prevent decay. Today, the basic structure is a 2D system to realize anyons with braiding operations. From the topological point of view, we have to deal with surface topology. However, usual materials are 3D objects. Possible topologies for these objects can be more complex than surfaces. From the topological point of view, Thurston’s geometrization theorem gives the main description of 3-dimensional manifolds. Here, complements of knots do play a prominent role and are in principle the main parts to understand 3-manifold topology. For that purpose, we will construct a quantum system on the complements of a knot in the 3-sphere. The whole system depends strongly on the topology of this complement, which is determined by non-contractible, closed curves. Every curve gives a contribution to the quantum states by a phase (Berry phase). Therefore, the quantum states can be manipulated by using the knot group (fundamental group of the knot complement). The universality of these operations was already showed by M. Planat et al.

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Section: Discussionmentioning

confidence: 99%

Topological Quantum Computing and 3-Manifolds

Asselmeyer-Maluga

2021

Quantum Reports

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“…Why not use this knot group for quantum computing? In [24,25,26,27] M. Planat et.al. studied the representation of knot groups and the usage for quantum computing.…”

Section: Discussionmentioning

confidence: 99%

3D Topological Quantum Computing

Asselmeyer-Maluga

2021

Preprint

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In this paper we will present some ideas to use 3D topology for quantum computing extending ideas from a previous paper. Topological quantum computing used "knotted" quantum states of topological phases of matter, called anyons. But anyons are connected with surface topology. But surfaces have (usually) abelian fundamental groups and therefore one needs non-abelian anyons to use it for quantum computing. But usual materials are 3D objects which can admit more complicated topologies. Here, complements of knots do play a prominent role and are in principle the main parts to understand 3-manifold topology. For that purpose, we will construct a quantum system on the complements of a knot in the 3-sphere (see arXiv:2102.04452 for previous work). The whole system is designed as knotted superconductor where every crossing is a Josephson junction and the qubit is realized as flux qubit. We discuss the properties of this systems in particular the fluxion quantization by using the A-polynomial of the knot. Furthermore we showed that 2-qubit operations can be realized by linked (knotted) superconductors again coupled via a Josephson junction.

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“…A so-called Veldkamp space of the doily V(W(3, 2))-which is a parabolic quadric and isomorphic to PG(4, 2) with 31 points (and 155 lines), of which 15 are generated by single-point perp-sets, 10 by grids, and 6 by ovoids-has also been discussed [90]. Finally, a Mermin pentagram has been discussed in [92].…”

Section: Geometric Viewmentioning

confidence: 99%

Symmetries and Geometries of Qubits, and Their Uses

Rau

2021

Symmetry

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The symmetry SU(2) and its geometric Bloch Sphere rendering have been successfully applied to the study of a single qubit (spin-1/2); however, the extension of such symmetries and geometries to multiple qubits—even just two—has been investigated far less, despite the centrality of such systems for quantum information processes. In the last two decades, two different approaches, with independent starting points and motivations, have been combined for this purpose. One approach has been to develop the unitary time evolution of two or more qubits in order to study quantum correlations; by exploiting the relevant Lie algebras and, especially, sub-algebras of the Hamiltonians involved, researchers have arrived at connections to finite projective geometries and combinatorial designs. Independently, geometers, by studying projective ring lines and associated finite geometries, have come to parallel conclusions. This review brings together the Lie-algebraic/group-representation perspective of quantum physics and the geometric–algebraic one, as well as their connections to complex quaternions. Altogether, this may be seen as further development of Felix Klein’s Erlangen Program for symmetries and geometries. In particular, the fifteen generators of the continuous SU(4) Lie group for two qubits can be placed in one-to-one correspondence with finite projective geometries, combinatorial Steiner designs, and finite quaternionic groups. The very different perspectives that we consider may provide further insight into quantum information problems. Extensions are considered for multiple qubits, as well as higher-spin or higher-dimensional qudits.

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Zoology of Atlas-Groups: Dessins D’enfants, Finite Geometries and Quantum Commutation

Cited by 16 publications

References 41 publications

Topological Quantum Computing and 3-Manifolds

Topological Quantum Computing and 3-Manifolds

3D Topological Quantum Computing

Symmetries and Geometries of Qubits, and Their Uses

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