Universal Quantum Computing and Three-Manifolds

Planat, Michel; Aschheim, Raymond; Amaral, Marcelo M.; Irwin, Klee

doi:10.3390/sym10120773

Cited by 16 publications

(22 citation statements)

References 44 publications

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“…Finally, the group theoretical approach may be related to the theory of 3-manifolds. According to the Poincaré conjecture (now a theorem) every simply connected closed 3-manifold is homeomorphic to the 3-sphere S 3 , alias the house of qubits [16]. However, one can dress S 3 as a 3-manifold M that looses the homeomorphism to S 3 following the work of W. Thurston [17].…”

Section: Algebraic Geometrical Models Of Secondary Structuresmentioning

confidence: 99%

Quantum Information in the Protein Codes, 3-Manifolds and the Kummer Surface

et al. 2021

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Every protein consists of a linear sequence over an alphabet of 20 letters/amino acids. The sequence unfolds in the 3-dimensional space through secondary (local foldings), tertiary (bonds) and quaternary (disjoint multiple) structures. The mere existence of the genetic code for the 20 letters of the linear chain could be predicted with the (informationally complete) irreducible characters of the finite group Gn:=Zn⋊2O (with n=5 or 7 and 2O the binary octahedral group) in our previous two papers. It turns out that some quaternary structures of protein complexes display n-fold symmetries. We propose an approach of secondary structures based on free group theory. Our results are compared to other approaches of predicting secondary structures of proteins in terms of α helices, β sheets and coils, or more refined techniques. It is shown that the secondary structure of proteins shows similarities to the structure of some hyperbolic 3-manifolds. The hyperbolic 3-manifold of smallest volume—Gieseking manifold—some other 3 manifolds and the oriented hypercartographic group are singled out as tentative models of such secondary structures. For the quaternary structure, there are links to the Kummer surface.

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Section: Algebraic Geometrical Models Of Secondary Structuresmentioning

confidence: 99%

Quantum Information in the Protein Codes, 3-Manifolds and the Kummer Surface

et al. 2021

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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%

“…Therefore, finally we get a complete set of operations to realize any quantum circuit: a 1-qubit operation by the knot group of the trefoil knot and a 2-qubit operation by the complement of the link (Hopf link for instance). For the universality of these operations we refer to the work of M. Planat et al [3,4], which was the main inspiration of this work. This paper followed the idea to use knots directly for quantum computing.…”

Section: Introductionmentioning

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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“…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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Universal Quantum Computing and Three-Manifolds

Cited by 16 publications

References 44 publications

Quantum Information in the Protein Codes, 3-Manifolds and the Kummer Surface

Quantum Information in the Protein Codes, 3-Manifolds and the Kummer Surface

Topological Quantum Computing and 3-Manifolds

3D Topological Quantum Computing

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