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Topological Quantum Error-Correcting Codes beyond dimension 2

Abstract : Error correction is the set of techniques used in order to store, process and transmit information reliably in a noisy context. The classical theory of error correction is based on encoding classical information redundantly. A major endeavor of the theory is to find optimal trade-offs between redundancy, which we try to minimize, and noise tolerance, which we try to maximize. The quantum theory of error correction cannot directly imitate the redundant schemes of the classical theory because it has to cope with the no-cloning theorem: quantum information cannot be copied. Quantum error correction is nonetheless possible by spreading the information on more quantum memory elements than would be necessary. In quantum information theory, dilution of the information replaces redundancy since copying is forbidden by the laws of quantum mechanics. Besides this conceptual difference, quantum error correction inherits a lot from its classical counterpart. In this PhD thesis we are concerned with a class of quantum error correcting codes whose classical counterpart was defined in 1961 by Gallager. At that time quantum information was not even a research domain yet. This class is the family of low density parity check (LDPC) codes. Informally a code is said to be LDPC if the constraints imposed to ensure redundancy in the classical setting or dilution in the quantum setting are local. More precisely this PhD thesis focuses on a subset of the LDPC quantum error correcting codes: the homological quantum error correcting codes. These codes take their name from the mathematical field of homology, whose objects of study are sequences of linear maps such that the kernel of a map contains the image of its left neighbour. Originally introduced to study the topology of geometric shapes, homology theory now encompasses more algebraic branches as well, where the focus is more abstract and combinatorial. The same is true of homological codes: they were introduced in 1996 by Kitaev with a quantum code that has the shape of a torus. They now form a vast family of quantum LDPC codes, some more inspired from geometry than others. Homological quantum codes have been designed from spherical, Euclidean and hyperbolic geometries, from 2-dimensional and 4-dimensional objects, from objects with increasing and unbounded dimension and from hypergraph or homological products. After we introduce some general quantum information concepts in the first chapter of this manuscript, we focus in the two following ones on families of quantum codes based on 4-dimensional hyperbolic objects. We highlight the interplay between their geometric side, given by manifolds, and their combinatorial sides, given by abstract polytopes. We use both sides to analyze the corresponding quantum codes. In the fourth and last chapter we analyze a family of quantum codes based on spherical objects of arbitrary dimension. To have more flexibility in the design of quantum codes, we use combinatorial objects that realize this spherical geometry: hypercube complexes. This setting allows us to introduce a new link between classical and quantum error correction where classical codes are used to introduce homology in hypercube complexes.
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Vivien Londe. Topological Quantum Error-Correcting Codes beyond dimension 2. Mathematics [math]. Inria Paris; Université de bordeaux, 2019. English. ⟨tel-02429868⟩

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