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1. Explicit Abelian Lifts and Quantum LDPC Codes

   https://doi.org/10.4230/LIPIcs.ITCS.2022.88  

   Jeronimo, Fernando Granha (January 2022, Leibniz international proceedings in informatics)

   Braverman, Mark (Ed.)

   For an abelian group H acting on the set [𝓁], an (H,𝓁)-lift of a graph G₀ is a graph obtained by replacing each vertex by 𝓁 copies, and each edge by a matching corresponding to the action of an element of H. Expanding graphs obtained via abelian lifts, form a key ingredient in the recent breakthrough constructions of quantum LDPC codes, (implicitly) in the fiber bundle codes by Hastings, Haah and O'Donnell [STOC 2021] achieving distance Ω̃(N^{3/5}), and in those by Panteleev and Kalachev [IEEE Trans. Inf. Theory 2021] of distance Ω(N/log(N)). However, both these constructions are non-explicit. In particular, the latter relies on a randomized construction of expander graphs via abelian lifts by Agarwal et al. [SIAM J. Discrete Math 2019]. In this work, we show the following explicit constructions of expanders obtained via abelian lifts. For every (transitive) abelian group H ⩽ Sym(𝓁), constant degree d ≥ 3 and ε > 0, we construct explicit d-regular expander graphs G obtained from an (H,𝓁)-lift of a (suitable) base n-vertex expander G₀ with the following parameters: ii) λ(G) ≤ 2√{d-1} + ε, for any lift size 𝓁 ≤ 2^{n^{δ}} where δ = δ(d,ε), iii) λ(G) ≤ ε ⋅ d, for any lift size 𝓁 ≤ 2^{n^{δ₀}} for a fixed δ₀ > 0, when d ≥ d₀(ε), or iv) λ(G) ≤ Õ(√d), for lift size "exactly" 𝓁 = 2^{Θ(n)}. As corollaries, we obtain explicit quantum lifted product codes of Panteleev and Kalachev of almost linear distance (and also in a wide range of parameters) and explicit classical quasi-cyclic LDPC codes with wide range of circulant sizes. Items (i) and (ii) above are obtained by extending the techniques of Mohanty, O'Donnell and Paredes [STOC 2020] for 2-lifts to much larger abelian lift sizes (as a byproduct simplifying their construction). This is done by providing a new encoding of special walks arising in the trace power method, carefully "compressing" depth-first search traversals. Result (iii) is via a simpler proof of Agarwal et al. [SIAM J. Discrete Math 2019] at the expense of polylog factors in the expansion. 

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2. Boundary algebras of the Kitaev quantum double model

   https://doi.org/10.1063/5.0212164  

   Chuah, Chian Yeong; Hungar, Brett; Kawagoe, Kyle; Penneys, David; Tomba, Mario; Wallick, Daniel; Wei, Shuqi (October 2024, Journal of Mathematical Physics)

   The recent article by Jones et al. [arXiv:2307.12552 (2023)] gave local topological order (LTO) axioms for a quantum spin system, showed they held in Kitaev’s Toric Code and in Levin-Wen string net models, and gave a bulk boundary correspondence to describe bulk excitations in terms of the boundary net of algebras. In this article, we prove the LTO axioms for Kitaev’s Quantum Double model for a finite group G. We identify the boundary nets of algebras with fusion categorical nets associated to (Hilb(G),C[G]) or (Rep(G),CG) depending on whether the boundary cut is rough or smooth, respectively. This allows us to make connections to the work of Ogata [Ann. Henri Poincaré 25, 2353–2387 (2024)] on the type of the cone von Neumann algebras in the algebraic quantum field theory approach to topological superselection sectors. We show that the boundary algebras can also be calculated from a trivial G-symmetry protected topological phase (G-SPT), and that the gauging map preserves the boundary algebras. Finally, we compute the boundary algebras for the (3 + 1)D Quantum Double model associated to an Abelian group. 

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3. Rigid automorphisms of linking systems

   https://doi.org/10.1017/fms.2021.17  

   Glauberman, George; Lynd, Justin (January 2021, Forum of Mathematics, Sigma)

   null (Ed.)

   Abstract A rigid automorphism of a linking system is an automorphism that restricts to the identity on the Sylow subgroup. A rigid inner automorphism is conjugation by an element in the center of the Sylow subgroup. At odd primes, it is known that each rigid automorphism of a centric linking system is inner. We prove that the group of rigid outer automorphisms of a linking system at the prime $$2$$ is elementary abelian and that it splits over the subgroup of rigid inner automorphisms. In a second result, we show that if an automorphism of a finite group G restricts to the identity on the centric linking system for G , then it is of $p'$ -order modulo the group of inner automorphisms, provided G has no nontrivial normal $p'$ -subgroups. We present two applications of this last result, one to tame fusion systems. 

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4. Codimension-2 defects and higher symmetries in (3+1)D topological phases

   https://doi.org/10.21468/SciPostPhys.14.4.065  

   Barkeshli, Maissam; Chen, Yu-An; Huang, Sheng-Jie; Kobayashi, Ryohei; Tantivasadakarn, Nathanan; Zhu, Guanyu (January 2023, SciPost Physics)

   (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. 

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5. Crossed Product Equivalence of Quantum Automorphism Groups of Finite Dimensional C*-Algebras

   https://doi.org/10.1093/imrn/rnad060  

   Brannan, Michael; Elzinga, Floris; Harris, Samuel J; Yamashita, Makoto (March 2023, International Mathematics Research Notices)

   Abstract We compare the algebras of the quantum automorphism group of finite-dimensional C$$^\ast $$-algebra $$B$$, which includes the quantum permutation group $$S_N^+$$, where $$N = \dim B$$. We show that matrix amplification and crossed products by trace-preserving actions by a finite Abelian group $$\Gamma $$ lead to isomorphic $$\ast $$-algebras. This allows us to transfer various properties such as inner unitarity, Connes embeddability, and strong $$1$$-boundedness between the various algebras associated with these quantum groups. 

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