We study the number of copies needed to estimate the root Uhlmann fidelity between an unknown quantum state and a classically known reference, under collective measurements. If the unknown state has rank at most $s$, we give an estimator using $O(s^2/\varepsilon^2)$ copies, uniformly in the ambient dimension and refere...
Let $R_r(k)$ denote the diagonal $r$-colour Ramsey number. We prove that there exist absolute constants $c,K>0$ such that $R_r(k)\le r^{rk}\exp\!\left(-c\frac{k}{r\log^2(2r)}\right)$ for every $r\ge2$ and every $k\ge Kr^2\log^6(2r)$. This improves the exponential saving in a recent bound of Yang and Mao by a factor of...
We settle the sample complexity of estimating the root Uhlmann fidelity $F(\rho,\sigma)=\operatorname{tr}\sqrt{\sqrt{\sigma}\rho\sqrt{\sigma}}$ between an unknown state $\rho$ and a known rank-$r$ reference state $\sigma$. Writing $S(r,\varepsilon)$ for the sample complexity at additive error $\varepsilon$, we resolve...
This paper considers the design of optimal fixed-step first-order methods for high-dimensional minimization of $L$-smooth convex functions. For optimizing worst-case performance measured via suboptimality of the final function value (relative to the initial squared distance to a minimizer), we provide an algebraic proo...
Benjamin Grimmer, Sunghyeon Jo, Chanwoo Park· 1 citation
Let $G\in\mathbb{R}^{M\times N}$ have independent standard Gaussian entries. For a fixed margin $\kappa\in\mathbb{R}$, the asymmetric binary perceptron asks for $\sigma\in\{\pm1\}^N$ such that $G\sigma/\sqrt{N}\ge\kappa\mathbf{1}_M$. We study the online version of this problem, in which the columns of $G$ arrive sequen...
Sunghyeon Jo, Taekyun Lee· 1 citation
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