Collaborative Research: Statistical Mechanics of Non-local Disordered Models Associated with Quantum LDPC Codes

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Quantum low-density-parity-check (LDPC) codes is the only class of quantum error correcting codes where an asymptotically finite rate isknown to coexist with a finite fault-tolerant error-correction threshold. In simple terms, using these codes for coherence protection, a large quantum computer can in principle be built, and compared to other existing schemes, it would require fewer redundant qubits. Good finite-rate quantum LDPC codes are preciously few: the firstanalytical examples have only been constructed a few years ago. These codes will be used as a scaffold to construct novel non-local statistical mechanical models with unusual properties. Studying these models will improve our understanding of the quantum theoretical problems related to quantum computation.These studies will also offer an insight in general properties of non-local models. Many studies of such models concentrated on caseswhere interactions between particles are chosen randomly; these tend to produce generic mean-field behavior which is well understood. In contrast, in this work models will be constructed with features never seen before and even proved to be impossible in a local setting. At the same time, because of the connection to the original quantum codes, these models can be guaranteed to have some highlysought-after qualifications, e.g., several distinct thermodynamical phases and non-trivial "duality" mappings between them, "topological"phases where different ground states cannot be distinguished by a local measurement, etc. Among the more ambitious potential applications is a consistent theory of quantum gravity where the universe itself would emerge from chaos via some quantum code.The award supports theoretical research on physics of non-local discrete and continuous statistical-mechanical models associated with quantum error correcting codes. An important feature of such codes is the existence of the decoding threshold, where a sufficiently largecode can deal effectively with any noise level below the threshold, but not above it.Disordered spin models associated with decoding transition (these models have exact Wegner's self-duality), related models with large gauge groups associated with fault-tolerant decoding, as well as models with extensive ground state entropy, including U(1) gauge theories which generalize Wen's mutual Chern-Simons theory describing the ground state of Kitaev's toric code will be constructed and studied. Models associated with quantum LDPC codes are expected to be particularly interesting since their interaction terms involve a limited number of participating particles. The low-energy sectors of these models are expected to be dominated by non-trivial extended defects which generalize the notion of topological defects like domain walls, vortices, etc. New physics includes a phase transition driven by an extensive entropy of defectclasses, coming from the exponentially large number of dimensions describing the original quantum code. Results will be relevant toseveral established fields of physics traditionally dealing with similar models: statistical mechanics of spin glasses, phase transition theory, etc., with potential applications extending to many other fields.