The modern concept of phases of matter has undergone tremendous developments since the first observation of topologically ordered states in fractional quantum Hall systems in the 1980s. In this paper, we explore the following question: in principle, how much detail of the physics of topological orders can be observed using state of the art technologies? We find that using surprisingly little data, namely the toric code Hamiltonian in the presence of generic disorders and detuning from its exactly solvable point, the modular matrices—characterizing anyonic statistics that are some of the most fundamental fingerprints of topological orders—can be reconstructed with very good accuracy solely by experimental means. This is an experimental realization of these fundamental signatures of a topological order, a test of their robustness against perturbations, and a proof of principle—that current technologies have attained the precision to identify phases of matter and, as such, probe an extended region of phase space around the soluble point before its breakdown. Given the special role of anyonic statistics in quantum computation, our work promises myriad applications both in probing and realistically harnessing these exotic phases of matter.

自从1980年代在分数量子霍尔系统中首次观察到拓扑有序状态以来，现代的物相概念已经发生了巨大的发展。在本文中，我们探讨了以下问题：原则上，使用最新技术可以观察到多少拓扑次序物理细节？我们发现，使用令人惊讶的少量数据，即存在泛型障碍的复曲面代码哈密顿量，并从其确切可解点失谐，可以使用以下方法重构模块化矩阵（表征某些拓扑结构最基本指纹的非音调统计数据）仅通过实验手段就可以获得非常好的准确性。这是对拓扑顺序的这些基本特征的实验性实现，测试它们对干扰的鲁棒性，并证明原理—当前技术已达到识别物质相的精度，并因此在分解之前在可溶点周围探测了相空间的扩展区域。鉴于非调子统计在量子计算中的特殊作用，我们的工作有望在探测和实际利用这些奇异的物质相中得到无数的应用。