格量子码与物质的奇异拓扑相
Lattice quantum codes and exotic topological phases of matter
摘要：
本论文探讨是否有可能建立一个稳固的量子信息记忆装置。我们发现了一个三维带隙晶格自旋模型，它第一次证明了在有限温度下可靠的量子存储是可能的，至少在某种程度上是可能的。当量子信息被编码到该模型的高度纠缠的基态中并受到热误差时，由于宏观能垒使误差保持良好的局部化，因此在很长一段时间内，这些误差仍然可以很容易地校正而无需任何主动干预。结果，存储的量子信息可以在存储时间内被忠实地检索，该存储时间随着温度的倒数的平方而指数地增长。相比之下，对于先前已知的在三个或更少空间维度中的拓扑量子存储类型，存储时间与温度的倒数呈指数关系，而不是其平方。这个自旋模型表现出一个以前意想不到的拓扑量子秩序，其中基态是局部不可区分的，点状激发是固定的，不动性不受小扰动的哈密顿量。基态的简并度虽然对微扰也不敏感，但却是系统大小的一个复杂的数论函数，并且系统在实空间重整化群变换下分叉成多个自身的非相互作用副本。简并度、激发和重整化群流可以利用自旋模型的对称性和多项式代数上模的一些相关自由分解的框架来分析。
英文摘要：
This thesis addresses whether it is possible to build a robust memory device for quantum information. A three-dimensional gapped lattice spin model is found which demonstrates for the first time that a reliable quantum memory at finite temperature is possible, at least to some extent. When quantum information is encoded into a highly entangled ground state of this model and subjected to thermal errors, the errors remain easily correctable for a long time without any active intervention, because a macroscopic energy barrier keeps the errors well localized. As a result, stored quantum information can be retrieved faithfully for a memory time which grows exponentially with the square of the inverse temperature. In contrast, for previously known types of topological quantum storage in three or fewer spatial dimensions the memory time scales exponentially with the inverse temperature, rather than its square. This spin model exhibits a previously unexpected topological quantum order, in which ground states are locally indistinguishable, pointlike excitations are immobile, and the immobility is not affected by small perturbations of the Hamiltonian. The degeneracy of the ground state, though also insensitive to perturbations, is a complicated number-theoretic function of the system size, and the system bifurcates into multiple noninteracting copies of itself under real-space renormalization group transformations. The degeneracy, the excitations, and the renormalization group flow can be analyzed using a framework that exploits the spin model's symmetry and some associated free resolutions of modules over polynomial algebras.
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