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Abstract Given a $$k$$-uniform hypergraph $$H$$ on $$n$$ vertices, an even cover in $$H$$ is a collection of hyperedges that touch each vertex an even number of times. Even covers are a generalization of cycles in graphs and are equivalent to linearly dependent subsets of a system of linear equations modulo $$2$$. As a result, they arise naturally in the context of well-studied questions in coding theory and refuting unsatisfiable $$k$$-SAT formulas. Analogous to the irregular Moore bound of Alon, Hoory, and Linial [3], Feige conjectured [8] an extremal trade-off between the number of hyperedges and the length of the smallest even cover in a $$k$$-uniform hypergraph. This conjecture was recently settled up to a multiplicative logarithmic factor in the number of hyperedges [12, 13]. These works introduce the new technique that relates hypergraph even covers to cycles in the associated Kikuchi graphs. Their analysis of these Kikuchi graphs, especially for odd $$k$$, is rather involved and relies on matrix concentration inequalities. In this work, we give a simple and purely combinatorial argument that recovers the best-known bound for Feige’s conjecture for even $$k$$. We also introduce a novel variant of a Kikuchi graph which together with this argument improves the logarithmic factor in the best-known bounds for odd $$k$$. As an application of our ideas, we also give a purely combinatorial proof of the improved lower bounds [4] on 3-query binary linear locally decodable codes. 

We investigate the problem of predicting the output behavior of unknown quantum channels. Given query access to an n-qubit channel E and an observable O, we aim to learn the mapping ρ↦Tr(OE[ρ]) to within a small error for most ρ sampled from a distribution D. Previously, Huang, Chen, and Preskill proved a surprising result that even if E is arbitrary, this task can be solved in time roughly nO(log(1/ϵ)), where ϵ is the target prediction error. However, their guarantee applied only to input distributions D invariant under all single-qubit Clifford gates, and their algorithm fails for important cases such as general product distributions over product states ρ. In this work, we propose a new approach that achieves accurate prediction over essentially any product distribution D, provided it is not "classical" in which case there is a trivial exponential lower bound. Our method employs a "biased Pauli analysis," analogous to classical biased Fourier analysis. Implementing this approach requires overcoming several challenges unique to the quantum setting, including the lack of a basis with appropriate orthogonality properties. The techniques we develop to address these issues may have broader applications in quantum information. 

In [Saunderson, 2011; Saunderson et al., 2013], Saunderson, Parrilo, and Willsky asked the following elegant geometric question: what is the largest m = m(d) such that there is an ellipsoid in ℝ^d that passes through v_1, v_2, …, v_m with high probability when the v_is are chosen independently from the standard Gaussian distribution N(0,I_d)? The existence of such an ellipsoid is equivalent to the existence of a positive semidefinite matrix X such that v_i^⊤ X v_i = 1 for every 1 ⩽ i ⩽ m - a natural example of a random semidefinite program. SPW conjectured that m = (1-o(1)) d²/4 with high probability. Very recently, Potechin, Turner, Venkat and Wein [Potechin et al., 2022] and Kane and Diakonikolas [Kane and Diakonikolas, 2022] proved that m ≳ d²/polylog(d) via a certain natural, explicit construction. In this work, we give a substantially tighter analysis of their construction to prove that m ≳ d²/C for an absolute constant C > 0. This resolves one direction of the SPW conjecture up to a constant. Our analysis proceeds via the method of Graphical Matrix Decomposition that has recently been used to analyze correlated random matrices arising in various areas [Barak et al., 2019; Bafna et al., 2022]. Our key new technical tool is a refined method to prove singular value upper bounds on certain correlated random matrices that are tight up to absolute dimension-independent constants. In contrast, all previous methods that analyze such matrices lose logarithmic factors in the dimension. 

We give efficient algorithms for finding power-sum decomposition of an input polynomial with component s. The case of linear s is equivalent to the well-studied tensor decomposition problem while the quadratic case occurs naturally in studying identifiability of non-spherical Gaussian mixtures from low-order moments. Unlike tensor decomposition, both the unique identifiability and algorithms for this problem are not well-understood. For the simplest setting of quadratic s and , prior work of [GHK15] yields an algorithm only when . On the other hand, the more general recent result of [GKS20] builds an algebraic approach to handle any components but only when is large enough (while yielding no bounds for or even ) and only handles an inverse exponential noise. Our results obtain a substantial quantitative improvement on both the prior works above even in the base case of and quadratic s. Specifically, our algorithm succeeds in decomposing a sum of generic quadratic s for and more generally the th power-sum of generic degree- polynomials for any . Our algorithm relies only on basic numerical linear algebraic primitives, is exact (i.e., obtain arbitrarily tiny error up to numerical precision), and handles an inverse polynomial noise when the s have random Gaussian coefficients.