Aspects combinatoires de modèles quantique

par Victor Nador

Projet de thèse en Informatique

Sous la direction de Adrian Tanasa.

Thèses en préparation à Bordeaux , dans le cadre de École doctorale de mathématiques et informatique , en partenariat avec LaBRI - Laboratoire Bordelais de Recherche en Informatique (laboratoire) et de Combinatoire et algorithmiques (equipe de recherche) depuis le 22-09-2020 .

  • Résumé

    Divers modèles quantiques apparaissent dans un large éventail de domaines de la recherche fondamentale, tel que le domaine de l'information quantique. Comprendre pleinement les aspects théoriques de l'informations quantiques est aujourd'hui une question ouverte particulièrement importante. L'étude théorique de ces modèles exige des compétences interdisciplinaires allant de la combinatoire à l'informatique mathématique à la physique quantique.

  • Titre traduit

    Combinatorial aspects of quantum models

  • Résumé

    Various quantum models appear in a wide range of domains of fundamental research, such as the celebrated domain of quantum information. Understanding the formal aspects of these quantum models is nowadays a particularly important open question. The theoretical study of these models requires inter-disciplinary skills ranging from combinatorics and mathematical computer science to quantum physics. Quantum models which are good candidates to define and implement concretely, in various ways, quantum computers, see the general references [1], [2] or [3], are also closely related to condensed matter and quantum gravity models. Thus, a celebrated model for topological quantum computing is the Kitaev model [4]. This model assigns to an oriented surface a certain finite-dimensional vector space, called the protected space. Indeed, defining the qubits - the quantum analogues of the usual bits - into topological degrees of freedom makes them more robust to destructive interactions with the environment (decoherence, heat etc.). This makes the qubits scalable so that a large quantum computer based on many qubits could be built in the future. Kitaev model for topological quantum computing has been extensively studied since its proposal, and in [5] it was proved that it is equivalent to the combinatorial quantization of a certain topological quantum field theory, which is further related to various quantum gravity models. In 2015, using a certain condensed matter model, the Sachdev-Ye model [6] , Kitaev proposed in a series of talks [7] in Santa Barbara (California, USA) a toy model for holography known today as the Sachdev-Ye-Kitaev (SYK) model. Let us mention here that the holographic principle states that the description of a volume of space can be encoded on a lower-dimensional boundary and it was first proposed by Nobel prize winner G. 't Hooft. This model is now kwon as the Sachdev- Ye-Kitaev (SYK) model. Since 2015, the SYK model has attracted a huge amount of interest. In 2016, Nobel prize winner Witten proposed in [8] a non-trivial relation of the SYK model to Gurau's colored random tensor model. This is known today as the Gurau-Witten model. Klebanov and Tarnopolsky then proposed several SYK-like tensor models [9] based on the tensor model [10] initially defined and studied by S. Carrozza and A. Tanasa. In [9], a variation of this model is also proposed, variation which is expressed in terms of the multi-orientable tensor model initially defined in [11]. From a combinatorial point of view, these random tensor models are particularly rich. Thus, the dominant graphs of each of the terms of the large N expansion have been studied in detail by Gurau and Schaeffer [12] for the colored tensor model, and then by Fusy and Tanasa [13] for the multi-orientable tensor model. Various other preliminary combinatorial results have been obtained since then by Fusy et. al. [14], and also by Bonzom et. al. [15], [16]. Another domain where quantum information theory and quantum gravity are prone to have fertile interactions is the domain of quantum reference frames. Indeed, information is the most natural language to make sense of the notions of space and time at the fundamental level. Information can be distinguished as fungible or non-fungible. Explicitly, fungible information deals with abstract information for which encoding is not relevant. For example, one can encode a qubit in a spin etc. Non-fungible information, on the contrary, will depend on the encoding. General questions such as how precise can be a quantum reference frame, what are its symmetry transformations, how non-fungible information coarse-grain,s etc. are then fundamental aspects to explore from a quantum information perspective [5]. Quantum reference frames can also be seen as an alternative resource for performing quantum computing [5], which could be more robust than the standard entanglement resource - used for example by Google for its recent quantum computer built through the use of a superconducting processor [17]. The PhD thus proposes to study the various combinatorial properties of the quantum models above. More concretely, if both the Kitaev quantum information model and respectively the SYK models are low-dimensional models (three-dimensional for the Kitaev model and respectively two-dimensional (respectively one-dimensional, via the holographic principle)), we propose to study here the combinatorial properties of higher dimensional generalizations of these models. This objective is particularly natural, since quantum model which are pertinent for physics are known to be four-dimensional models. The main method planned to be used is the Gurau-Schaeffer combinatorial approach developped in [12]. Our objectives is highly non-trivial, because the classes of graphs associated to these models is particularly larger in higher dimensions.