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Utilizing the framework of\mathbb{Z}_2${\mathbb{Z}}_2$lattice gauge theories in the context of Pauli stabilizer codes, we present methodologies for simulating fermions via qubit systems on a two-dimensional square lattice. We investigate the symplectic automorphisms of the Pauli module over the Laurent polynomial ring. This enables us to systematically increase the code distances of stabilizer codes while fixing the rate between encoded logical fermions and physical qubits. We identify a family of stabilizer codes suitable for fermion simulation, achieving code distances of d=2,3,4,5,6,7, allowing correction of any\lfloor \frac{d-1}{2} \rfloor$\lfloor \frac{d-1}{2}\rfloor$-qubit error. In contrast to the traditional code concatenation approach, our method can increase the code distances without decreasing the (fermionic) code rate. In particular, we explicitly show all stabilizers and logical operators for codes with code distances of d=3,4,5. We provide syndromes for all Pauli errors and invent a syndrome-matching algorithm to compute code distances numerically.

 

(3+1)D topological phases of matter can host a broad class of non-trivial topological defects of codimension-1, 2, and 3, of which the well-known point charges and flux loops are special cases. The complete algebraic structure of these defects defines a higher category, and can be viewed as an emergent higher symmetry. This plays a crucial role both in the classification of phases of matter and the possible fault-tolerant logical operations in topological quantum error-correcting codes. In this paper, we study several examples of such higher codimension defects from distinct perspectives. We mainly study a class of invertible codimension-2 topological defects, which we refer to as twist strings. We provide a number of general constructions for twist strings, in terms of gauging lower dimensional invertible phases, layer constructions, and condensation defects. We study some special examples in the context of \mathbb{Z}_2 ℤ 2 gauge theory with fermionic charges, in \mathbb{Z}_2 \times \mathbb{Z}_2 ℤ 2 × ℤ 2 gauge theory with bosonic charges, and also in non-Abelian discrete gauge theories based on dihedral ( D_n D n ) and alternating ( A_6 A 6 ) groups. The intersection between twist strings and Abelian flux loops sources Abelian point charges, which defines an H^4 H 4 cohomology class that characterizes part of an underlying 3-group symmetry of the topological order. The equations involving background gauge fields for the 3-group symmetry have been explicitly written down for various cases. We also study examples of twist strings interacting with non-Abelian flux loops (defining part of a non-invertible higher symmetry), examples of non-invertible codimension-2 defects, and examples of the interplay of codimension-2 defects with codimension-1 defects. We also find an example of geometric, not fully topological, twist strings in (3+1)D A_6 A 6 gauge theory. 

Invented in 2010, NanoCluster Beacons (NCBs) (1) are an emerging class of turn-on probes that show unprecedented capabilities in single-nucleotide polymorphism (2) and DNA methylation (3) detection. As the activation colors of NCBs can be tuned by a near-by, guanine-rich activator strand, NCBs are versatile, multicolor probes suitable for multiplexed detection at low cost. Whereas a variety of NCB designs have been explored and reported, further diversification and optimization of NCBs require a full scan of the ligand composition space. However, the current methods rely on microarray and multi-well plate selection, which only screen tens to hundreds of activator sequences (4, 5). Here we take advantage of the next-generation-sequencing (NGS) platform for high-throughput, large-scale selection of activator strands. We first generated a ~104 activator sequence library on the Illumina MiSeq chip. Hybridizing this activator sequence library with a common nucleation sequence (which carried a nonfluorescent silver cluster) resulted in hundreds of MiSeq chip images with millions of bright spots (i.e. light-up polonies) of various intensities and colors. With a method termed Chip-Hybridized Associated Mapping Platform (CHAMP) (6), we were able to map these bright spots to the original DNA sequencing map, thus recovering the activator sequence behind each bright spot. After assigning an “activation score” to each “light-up polony”, we used a computational algorithm to select the best activator strands and validate these strands using the traditional in-solution preparation and fluorometer measurement method. By exploring a vast ligand composition space and observing the corresponding activation behaviors of silver clusters, we aim to elucidate the design rules of NCBs. 

NanoCluster Beacons (NCBs) are multicolor silver nanocluster probes whose fluorescence can be activated or tuned by a proximal DNA strand called the activator. While a single‐nucleotide difference in a pair of activators can lead to drastically different activation outcomes, termed polar opposite twins (POTs), it is difficult to discover new POT‐NCBs using the conventional low‐throughput characterization approaches. Here, a high‐throughput selection method is reported that takes advantage of repurposed next‐generation‐sequencing chips to screen the activation fluorescence of ≈40 000 activator sequences. It is found that the nucleobases at positions 7–12 of the 18‐nucleotide‐long activator are critical to creating bright NCBs and positions 4–6 and 2–4 are hotspots to generate yellow–orange and red POTs, respectively. Based on these findings, a “zipper‐bag” model is proposed that can explain how these hotspots facilitate the formation of distinct silver cluster chromophores and alter their chemical yields. Combining high‐throughput screening with machine‐learning algorithms, a pipeline is established to design bright and multicolor NCBs in silico.