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1. Companions and an Essential Motion of a Reaction System

   https://doi.org/10.3233/FI-2020-1953  

   Genova, Daniela; Hoogeboom, Hendrik Jan; Jonoska, Nataša (September 2020, Fundamenta Informaticae)

   ter Beek, Maurice; Koutny, Maciej; Rozenberg, Grzegorz (Ed.)

   For a family of sets we consider elements that belong to the same sets within the family as companions. The global dynamics of a reactions system (as introduced by Ehrenfeucht and Rozenberg) can be represented by a directed graph, called a transition graph, which is uniquely determined by a one-out subgraph, called the 0-context graph. We consider the companion classes of the outsets of a transition graph and introduce a directed multigraph, called an essential motion, whose vertices are such companion classes. We show that all one-out graphs obtained from an essential motion represent 0-context graphs of reactions systems with isomorphic transition graphs. All such 0-context graphs are obtained from one another by swapping the outgoing edges of companion vertices. 

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2. 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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3. Minimum L_∞ Hausdorff distance of point sets under translation: Generalizing Klee's measure problem

   https://doi.org/10.4230/LIPIcs.SoCG.2023.24  

   Chan, Timothy M. (June 2023, Proc. 39th Sympos. Computational Geometry (SoCG))

   Chambers, Erin W.; Gudmundsson, Joachim (Ed.)

   We present a (combinatorial) algorithm with running time close to O(n^d) for computing the minimum directed L_∞ Hausdorff distance between two sets of n points under translations in any constant dimension d. This substantially improves the best previous time bound near O(n^{5d/4}) by Chew, Dor, Efrat, and Kedem from more than twenty years ago. Our solution is obtained by a new generalization of Chan’s algorithm [FOCS'13] for Klee’s measure problem. To complement this algorithmic result, we also prove a nearly matching conditional lower bound close to Ω(n^d) for combinatorial algorithms, under the Combinatorial k-Clique Hypothesis. 

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4. On the large charge sector in the critical O(N) model at large N

   https://doi.org/10.1007/JHEP09(2021)184  

   Giombi, Simone; Hyman, Jonah (September 2021, Journal of High Energy Physics)

   A bstract We study operators in the rank- j totally symmetric representation of O ( N ) in the critical O ( N ) model in arbitrary dimension d , in the limit of large N and large charge j with j/N ≡ $$ \hat{j} $$ j ̂ fixed. The scaling dimensions of the operators in this limit may be obtained by a semiclassical saddle point calculation. Using the standard Hubbard-Stratonovich description of the critical O ( N ) model at large N , we solve the relevant saddle point equation and determine the scaling dimensions as a function of d and $$ \hat{j} $$ j ̂ , finding agreement with all existing results in various limits. In 4 < d < 6, we observe that the scaling dimension of the large charge operators becomes complex above a critical value of the ratio j/N , signaling an instability of the theory in that range of d . Finally, we also derive results for the correlation functions involving two “heavy” and one or two “light” operators. In particular, we determine the form of the “heavy-heavy-light” OPE coefficients as a function of the charges and d . 

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5. Quantum algorithms and approximating polynomials for composed functions with shared inputs

   https://doi.org/10.22331/q-2021-09-16-543  

   Bun, Mark; Kothari, Robin; Thaler, Justin (September 2021, Quantum)

   We give new quantum algorithms for evaluating composed functions whose inputs may be shared between bottom-level gates. Let f be an m -bit Boolean function and consider an n -bit function F obtained by applying f to conjunctions of possibly overlapping subsets of n variables. If f has quantum query complexity Q ( f ) , we give an algorithm for evaluating F using O ~ ( Q ( f ) ⋅ n ) quantum queries. This improves on the bound of O ( Q ( f ) ⋅ n ) that follows by treating each conjunction independently, and our bound is tight for worst-case choices of f . Using completely different techniques, we prove a similar tight composition theorem for the approximate degree of f .By recursively applying our composition theorems, we obtain a nearly optimal O ~ ( n 1 − 2 − d ) upper bound on the quantum query complexity and approximate degree of linear-size depth- d AC 0 circuits. As a consequence, such circuits can be PAC learned in subexponential time, even in the challenging agnostic setting. Prior to our work, a subexponential-time algorithm was not known even for linear-size depth-3 AC 0 circuits.As an additional consequence, we show that AC 0 ∘ ⊕ circuits of depth d + 1 require size Ω ~ ( n 1 / ( 1 − 2 − d ) ) ≥ ω ( n 1 + 2 − d ) to compute the Inner Product function even on average. The previous best size lower bound was Ω ( n 1 + 4 − ( d + 1 ) ) and only held in the worst case (Cheraghchi et al., JCSS 2018). 

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