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12 pages, 1404 KiB

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S L (2, C)

Scheme Processing of Singularities in Quantum Computing and Genetics

by Michel Planat, Marcelo M. Amaral, David Chester and Klee Irwin

Axioms 2023, 12(3), 233; https://doi.org/10.3390/axioms12030233 - 23 Feb 2023

Cited by 4 | Viewed by 2682

Abstract

Revealing the time structure of physical or biological objects is usually performed thanks to the tools of signal processing such as the fast Fourier transform, Ramanujan sum signal processing, and many other techniques. For space-time topological objects in physics and biology, we propose [...] Read more.

S L (2, C)

. Such topological objects possess an homotopy structure encoded in their fundamental group, and the related

S L (2, C)

multivariate polynomial character variety contains a plethora of singularities somehow analogous to the frequency spectrum in time structures. Our approach is applied to a model of quantum computing based on an Akbulut cork in exotic

R_{4}

, to an hyperbolic model of topological quantum computing based on magic states and to microRNAs in genetics. Such diverse topics reveal the manifold of possibilities of using the concept of a scheme spectrum. Full article

(This article belongs to the Special Issue Advances in Algebraic Geometry)

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13 pages, 956 KiB

Open AccessArticle

Quantum Computation and Measurements from an Exotic Space-Time R⁴

by Michel Planat, Raymond Aschheim, Marcelo M. Amaral and Klee Irwin

Symmetry 2020, 12(5), 736; https://doi.org/10.3390/sym12050736 - 5 May 2020

Cited by 7 | Viewed by 3769

Abstract

The authors previously found a model of universal quantum computation by making use of the coset structure of subgroups of a free group G with relations. A valid subgroup H of index d in G leads to a ‘magic’ state

|ψ⟩

in d [...] Read more.

|ψ⟩

in d-dimensional Hilbert space that encodes a minimal informationally complete quantum measurement (or MIC), possibly carrying a finite ‘contextual’ geometry. In the present work, we choose G as the fundamental group

π_{1} (V)

of an exotic 4-manifold V, more precisely a ‘small exotic’ (space-time)

R^{4}

(that is homeomorphic and isometric, but not diffeomorphic to the Euclidean

R^{4}

). Our selected example, due to S. Akbulut and R. E. Gompf, has two remarkable properties: (a) it shows the occurrence of standard contextual geometries such as the Fano plane (at index 7), Mermin’s pentagram (at index 10), the two-qubit commutation picture

G Q (2, 2)

(at index 15), and the combinatorial Grassmannian Gr

(2, 8)

(at index 28); and (b) it allows the interpretation of MICs measurements as arising from such exotic (space-time)

R^{4}

s. Our new picture relating a topological quantum computing and exotic space-time is also intended to become an approach of ‘quantum gravity’. Full article

(This article belongs to the Special Issue Symmetry in Quantum Systems)

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