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

量子低密度奇偶校验码（LDPC）是已知的唯一一类具有有限容错纠错阈值和渐近有限速率共存的量子纠错码。简而言之，使用这些代码进行相干保护，原则上可以构建大型量子计算机，与其他现有方案相比，它需要更少的冗余量子位。好的有限速率量子LDPC码非常少：第一个解析例子几年前才被构建出来。这些代码将被用作构建具有不同寻常性质的新型非局部统计力学模型的框架。研究这些模型将提高我们对量子计算相关的量子理论问题的理解。这些研究也将提供对非局部模型的一般性质的见解。这类模型的许多研究集中在粒子之间的相互作用是随机选择的情况下；这些倾向于产生一般的平均场行为，这是很容易理解的。相比之下，在这项工作中，模型将使用以前从未见过的特征来构建，甚至在局部环境中被证明是不可能的。同时，由于与原始量子编码的连接，这些模型可以保证具有一些非常受欢迎的资格，例如，几个不同的热力学相和它们之间的非平凡的“对偶”映射，“拓扑”相，其中不同的基态不能通过局部测量来区分，等等。在更雄心勃勃的潜在应用中，有一个是一致的量子引力理论，在这个理论中，宇宙本身将通过一些量子密码从混沌中出现。该奖项支持与量子纠错码相关的非局部离散和连续统计力学模型的物理理论研究。这种码的一个重要特征是存在解码阈值，其中足够大的码可以有效地处理低于阈值的任何噪声水平，但不能高于它。本文将构建和研究与译码跃迁相关的无序自旋模型（这些模型具有精确的Wegner自对偶性）、与容错译码相关的大规范群相关模型以及具有广泛基态熵的模型，包括推广描述Kitaev环向码基态的Wen互chen - simons理论的U(1)规范理论。与量子LDPC码相关的模型预计会特别有趣，因为它们的相互作用项涉及有限数量的参与粒子。这些模型的低能部分预计将由非平凡扩展缺陷主导，这些扩展缺陷推广了拓扑缺陷的概念，如畴壁，漩涡等。新的物理学包括由缺陷类的广泛熵驱动的相变，来自描述原始量子代码的指数级大量维度。结果将与传统上处理类似模型的几个已建立的物理领域相关：自旋玻璃的统计力学，相变理论等，并具有扩展到许多其他领域的潜在应用。

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.