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

建造量子计算机的主要挑战是量子组件容易出错。纠错可以用来克服这一挑战，但它对未来的量子计算机硬件提出了严格的要求。量子纠错的一种很有前途的方法是所谓的量子低密度奇偶校验码(LDPC)。如果成功，原则上可以使用这些代码建造一台大型量子计算机。与其他现有的方案相比，它的效率要高得多，需要更少的冗余量子比特，即量子比特。研究这些密码将加深我们对与量子计算有关的量子理论问题的理解。这个项目将为研究生提供极好的机会。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译码阈值，其中足够大的码可以有效地处理低于阈值但不高于阈值的任何噪声水平。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)，与容错译码相关的大规范群模型，以及具有广泛的基态熵的模型，包括推广了温氏互陈-西蒙斯理论的描述Kitaev环码基态的U(1)规范理论，将被构建和研究。与量子LDPC码相关的模型预计将特别有趣，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的扩展缺陷主导，这些缺陷概括了拓扑学缺陷的概念，如域壁、涡旋等。新物理包括由缺陷类别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，潜在的应用范围扩展到许多其他领域。

The main challenge for building a quantum computer is that quantum components are prone to error. Error correction can be used to overcome this challenge but it places stringent requirements on future quantum computer hardware. One promising method of quantum error correction is the so-called Quantum Low-Density-Parity-Check (LDPC) codes. If successful, using these codes a large quantum computer could in principle be built. Compared to other existing schemes, it would be much more efficient, requiring fewer redundant quantum bits, called qubits. Studying these codes will improve our understanding of the quantum theoretical problems related to quantum computation. This project will provide excellent opportunities for graduate students. 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 large code 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 that generalize the notion of topological defects like domain walls, vortices, etc. New physics includes a phase transition driven by an extensive entropy of defect classes, coming from the exponentially large number of dimensions describing the original quantum code. Results will be relevant to several 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.