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1. Quantum No-signalling Correlations and Non-local Games

   https://doi.org/10.1007/s00220-024-05001-x  

   Todorov, Ivan G; Turowska, Lyudmila (June 2024, Communications in Mathematical Physics)

   Abstract We introduce and examine three subclasses of the family of quantum no-signalling (QNS) correlations introduced by Duan and Winter: quantum commuting, quantum and local. We formalise the notion of a universal TRO of a block operator isometry, define an operator system, universal for stochastic operator matrices, and realise it as a quotient of a matrix algebra. We describe the classes of QNS correlations in terms of states on the tensor products of two copies of the universal operator system and specialise the correlation classes and their representations to classical-to-quantum correlations. We study various quantum versions of synchronous no-signalling correlations and show that they possess invariance properties for suitable sets of states. We introduce quantum non-local games as a generalisation of non-local games. We define the operation of quantum game composition and show that the perfect strategies belonging to a certain class are closed under channel composition. We specialise to the case of graph colourings, where we exhibit quantum versions of the orthogonal rank of a graph as the optimal output dimension for which perfect classical-to-quantum strategies of the graph colouring game exist, as well as to non-commutative graph homomorphisms, where we identify quantum versions of non-commutative graph homomorphisms introduced by Stahlke. 

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2. CONTINUITY OF UNIVERSALLY MEASURABLE HOMOMORPHISMS

   https://doi.org/10.1017/fmp.2019.5  

   ROSENDAL, CHRISTIAN (January 2019, Forum of Mathematics, Pi)

   null (Ed.)

   Answering a longstanding problem originating in Christensen’s seminal work on Haar null sets [ Math. Scand.   28 (1971), 124–128; Israel J. Math.   13 (1972), 255–260; Topology and Borel Structure. Descriptive Topology and Set Theory with Applications to Functional Analysis and Measure Theory , North-Holland Mathematics Studies, 10 (Notas de Matematica, No. 51). (North-Holland Publishing Co., Amsterdam–London; American Elsevier Publishing Co., Inc., New York, 1974), iii+133 pp], we show that a universally measurable homomorphism between Polish groups is automatically continuous. Using our general analysis of continuity of group homomorphisms, this result is used to calibrate the strength of the existence of a discontinuous homomorphism between Polish groups. In particular, it is shown that, modulo $$\text{ZF}+\text{DC}$$ , the existence of a discontinuous homomorphism between Polish groups implies that the Hamming graph on $$\{0,1\}^{\mathbb{N}}$$ has finite chromatic number. 

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3. 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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4. A Dichotomy for Bounded Degree Graph Homomorphisms with Nonnegative Weights

   Govorov, Artem Cai (July 2020, 47th International Colloquium on Automata, Languages, and Programming)

   Czumaj, Artur Dawar (Ed.)

   We consider the complexity of counting weighted graph homomorphisms defined by a symmetric matrix A. Each symmetric matrix A defines a graph homomorphism function Z A (·), also known as the partition function. Dyer and Greenhill [10] established a complexity dichotomy of Z A (·) for symmetric {0, 1}-matrices A, and they further proved that its #P-hardness part also holds for bounded degree graphs. Bulatov and Grohe [4] extended the Dyer-Greenhill dichotomy to nonnegative symmetric matrices A. However, their hardness proof requires graphs of arbitrarily large degree, and whether the bounded degree part of the Dyer-Greenhill dichotomy can be extended has been an open problem for 15 years. We resolve this open problem and prove that for nonnegative symmetric A, either Z A (G) is in polynomial time for all graphs G, or it is #P-hard for bounded degree (and simple) graphs G. We further extend the complexity dichotomy to include nonnegative vertex weights. Additionally, we prove that the #P-hardness part of the dichotomy by Goldberg et al. [12] for Z A (·) also holds for simple graphs, where A is any real symmetric matrix. 

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5. List-Decoding Homomorphism Codes with Arbitrary Codomains

   Babai, Laszlo; Black, Timothy; Wuu, Angela (August 2018, Leibniz international proceedings in informatics)

   The codewords of the homomorphism code aHom(G,H) are the affine homomorphisms between two finite groups, G and H, generalizing Hadamard codes. Following the work of Goldreich-Levin (1989), Grigorescu et al. (2006), Dinur et al. (2008), and Guo and Sudan (2014), we further expand the range of groups for which local list-decoding is possible up to mindist, the minimum distance of the code. In particular, for the first time, we do not require either G or H to be solvable. Specifically, we demonstrate a poly(1/epsilon) bound on the list size, i. e., on the number of codewords within distance (mindist-epsilon) from any received word, when G is either abelian or an alternating group, and H is an arbitrary (finite or infinite) group. We conjecture that a similar bound holds for all finite simple groups as domains; the alternating groups serve as the first test case. The abelian vs. arbitrary result permits us to adapt previous techniques to obtain efficient local list-decoding for this case. We also obtain efficient local list-decoding for the permutation representations of alternating groups (the codomain is a symmetric group) under the restriction that the domain G=A_n is paired with codomain H=S_m satisfying m < 2^{n-1}/sqrt{n}. The limitations on the codomain in the latter case arise from severe technical difficulties stemming from the need to solve the homomorphism extension (HomExt) problem in certain cases; these are addressed in a separate paper (Wuu 2018). We introduce an intermediate "semi-algorithmic" model we call Certificate List-Decoding that bypasses the HomExt bottleneck and works in the alternating vs. arbitrary setting. A certificate list-decoder produces partial homomorphisms that uniquely extend to the homomorphisms in the list. A homomorphism extender applied to a list of certificates yields the desired list. 

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