On Topology of the Moduli Space of Gapped Hamiltonians for Topological Phases

The moduli space of gapped Hamiltonians that are in the same topological phase is an intrinsic object that is associated to the topological order. The topology of these moduli spaces is used recently in the construction of Floquet codes. We propose a systematical program to study the topology of these moduli spaces. In particular, we use effective field theory to study the cohomology classes of these spaces, which includes and generalizes the Berry phase. We discuss several applications to studying phase transitions. We show that nontrivial family of gapped systems with the same topological order can protect isolated phase transitions in the phase diagram, and we argue that the phase transitions are characterized by screening of topological defects. We argue that family of gapped systems obey a version of bulk-boundary correspondence. We show that family of gapped systems in the bulk with the same topological order can rule out family of gapped systems on the boundary with the same topological boundary condition, constraining phase transitions on the boundary.Comment: 37 pages, 2 figure

Noise-Resilient Group Testing with Order-Optimal Tests and Fast-and-Reliable Decoding

Group testing (GT) is the Boolean counterpart of compressed sensing and the marketplace of new ideas for related problems such as cognitive radio and heavy hitter. A GT scheme is considered good if it is nonadaptive, uses $O(k \log n)$ tests, resists noise, can be decoded in $O(k \operatorname{poly}(\log n))$ time, and makes nearly no mistakes. In this paper, we propose "Gacha GT", an elementary, self-contained, and unified randomized scheme that, for the first time, satisfies all criteria for a fairly large region of parameters, namely when $\log k < \log(n)^{1-1/O(1)}$. Outside this parameter region, Gacha can be specialized to outperform the state-of-the-art partial-recovery GTs, exact-recovery GTs, and worst-case GTs. The new idea that runs through this paper, using an analogy, is to ask every person to break her $9$-digit "phone number" into three $3$-digit numbers $x$, $y$, and $z$ and write $(b, x)$, $(b, y)$, and $(b, z)$ on three pieces of sticky notes, where $b$ is her "birthday". This way, one can sort the sticky notes by birthday to reassemble the phone numbers. This birthday--number code and other coded constructions can be stacked like a multipartite graph pyramid. Gacha's encoder will synthesize the test results from the bottom up; and Gacha's decoder will reassemble the phone numbers from the top down.Comment: 23 pages, 8 figure

Complexity and second moment of the mathematical theory of communication

The performance of an error correcting code is evaluated by its block error probability, code rate, and encoding and decoding complexity. The performance of a series of codes is evaluated by, as the block lengths approach infinity, whether their block error probabilities decay to zero, whether their code rates converge to channel capacity, and whether their growth in complexities stays under control. Over any discrete memoryless channel, I build codes such that: for one, their block error probabilities and code rates scale like random codes’; and for two, their encoding and decoding complexities scale like polar codes’. Quantitatively, for any constants π, ρ > 0 such that π+2ρ < 1, I construct a series of error correcting codes with block length N approaching infinity, block error probability exp(−Nπ), code rate N−ρ less than the channel capacity, and encoding and decoding complexity O(N logN) per code block. Over any discrete memoryless channel, I also build codes such that: for one, they achieve channel capacity rapidly; and for two, their encoding and decoding complexities outperform all known codes over non-BEC channels. Quantitatively, for any constants τ, ρ > 0 such that 2ρ < 1, I construct a series of error correcting codes with block length N approaching infinity, block error probability exp(−(logN)τ ), code rate N−ρ less than the channel capacity, and encoding and decoding complexity O(N log(logN)) per code block. The two aforementioned results are built upon two pillars—a versatile framework that generates codes on the basis of channel polarization, and a calculus–probability machinery that evaluates the performances of codes. The framework that generates codes and the machinery that evaluates codes can be extended to many other scenarios in network information theory. To name a few: lossless compression with side information, lossy compression, Slepian–Wolf problem, Wyner–Ziv Problem, multiple access channel, wiretap channel of type I, and broadcast channel. In each scenario, the adapted notions of block error probability and code rate approach their limits at the same paces as specified above