Statistical mechanical study of disordered systems by the method of stochastic information processing

我们首先从纠错码与自旋玻璃理论的关系出发，总结了以往经典信息科学的相关知识，希望阐明在自旋玻璃的背景下如何解释信息科学的结果。在此基础上，我们从信息学的角度澄清了其重要性与自旋玻璃理论的意义的区别。例如，香农提出的信道编码给出了解码成功的条件，然而，如果在自旋玻璃理论的背景下解释，同样的定理给出了存在完美铁磁相位的条件，除非解码的信息与原始信息完全一致，否则对噪声信号的解码就失去了意义；但是在磁性物理学中，非最大值的磁化强度在物理上是普遍存在的。我们还通过求解一个无限范围模型，研究了自旋系统中量子效应和随机性之间的竞争。结果表明，量子效应在定性确定相图结构时并不十分显著。然而，我们在该模型的低温阶段观察到许多典型的临界现象，这表明随机性和挫败感的竞争，没有量子效应，导致了许多非平凡现象。我们进一步提出了有限维自旋玻璃多临界点精确位置的猜想。利用自旋玻璃理论与量子纠错之间的形式关系，推导出了在量子存储器环面码中不可能纠错的噪声强度极限。我们将这一猜想推广到非自对偶情况，从而得到了具有时间依赖性环面码在错误检测中的修正极限。

We first summarized previous knowledge in classical information science with an eye on the relation between error correcting codes and the theory of spin glasses, hoping to clarify how the results in information science are interpreted in the context of spin glasees. Based on this study, we have clarified the difference in the importance from the view point of information science and the significance as spin glass theory. For example, channel coding formulated by Shannon gives the condition for decoding to be successful, whereas, if interpreted in the context of the theory of spin glasses, the same theorem yields the condition for the perfect ferromagnetic phase to exist Decoding of noisy signal loses its significance unless the decoded message is in perfect agreement with the original message; but in the physics of magnetism non-maximum values of magnetization are of course physically prevalent.We have also studied the competition between quantum effects and randomness in spin systems by solving an infinite-range model. The result shows that quantum effects are not very significant in qualitative determination of the structure of the phase diagram. Nevertheless we observed many typical critical phenomena in the low-temperature phase of this model, which suggests that competition of randomness and frustration, without quantum effects, causes many non-trivial phenomena. We have further presented a conjecture on the exact location of the multicritical point of finite-dimensional spin glasses.We have derived the limit of noise strength above which error correction becomes impossible in toric code, a quantum memory, by exploiting the formal relation between the spin glass theory and quantum error correction. We have generalized this conjecture to non-self dual cases, thus yielding the correction limit for torus code with time dependence in error detection.