Previously, building quantum error correction codes achieving optimal performance demanded circuit depths scaling as O(log³ n). Now, an alternative protocol demonstrates equivalent results, the same asymptotic rate-distance tradeoff, with substantially shallower circuits operating at depth O(log n), provided that $frac kn < 1, H(frac{d}{n}), frac{d}{n}log_2 3, δ$. This reduction in gate complexity opens new avenues towards practical fault tolerance.

**Authors:** Hongmin Li We study fixed-layout local verification with target-dependent local tests. Let $M$ be a random variable on $\\{0,1\\}^K$, and let $S$ record the test selected at each coordinate. For each $s\in\operatorname{supp}(S)$, let $F_s$ be the corresponding target fiber and set $D_{\mathrm{fib}}=\max_s\operatorname{VCdim}(F_s)$. We prove $H(M\mid S)\le \log_2\\!\left(\sum_{j=0}^{D_{\mathrm{fib}}}\binom Kj\right)$. A fiber that shatters $d$ coordinates yields a weak relaxed locally decodable code with message length $d$ and block length $d+P$ over the original proof alphabet. For a uniform $K$-bit target and fixed proof alphabet, $Q$, and $σ$, the Goldberg--Gur--Saraogi lower bound implies that $I(M;S)\leγK$, for fixed $γ<1$, forces $P=Ω\\!\left(K^{1+1/a}/(\log K)^{2+2/a}\right)$, where $a=\lceil Q/σ\rceil$. If $P\le K(\log K)^c$, then $I(M;S)\ge K-O\\!\left(K^{a/(a+1)}(\log K)^{3+ac/(a+1)}\right)=K-o(K)$. Any discrete verifier state $T$ determining $S$ satisfies the same information lower bound. Bounded-randomness adaptive branches can be simulated nonadaptively by exposing their decision trees. A branch using at most $r$ random bits and $q$ adaptive proof queries yields a decoder with perfect completeness and at most $1+2^{r+1}\sum_{j""

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DJI has officially unveiled the Osmo Pocket 4P, a groundbreaking portable camera that redefines cinematic capabilities with its innovative features.

**Authors:** Anders Aamand, Mikkel Abrahamsen, Simon Bartlmae, Arindam Khan, Linda Kleist, Csaba D. Tóth We consider the problem of online packing of convex polygons into a strip by translations. While online algorithms with a constant competitive ratio have been known for rectangles for decades [Baker and Schwarz, SICOMP 1983], the current best algorithm for convex polygons has competitive ratio $O(n^{\log_2 3-1}\log n) = O(n^{0.59})$, where $n$ is the number of polygons. This algorithm was described by Aamand, Abrahamsen, Beretta, and Kleist [SODA 2023], who also proved a lower bound of $Ω(\sqrt{\log n/\log\log n})$ on the competitive ratio of any algorithm. Their lower bound is obtained via a reduction from \emph{online sorting}, a problem introduced in the same paper, for which they established a lower bound on the competitive ratio. We introduce a new, natural online problem that we call online TSP scheduling. Here, points $x_1,\ldots,x_n$ arrive online from a metric space $(M,d)$, and upon arrival each $x_i$ must be assigned a visit time $p_i\in[0,\infty)$ satisfying $|p_i-p_j|\ge d(x_i,x_j)$ for all $j __

お久しぶりです。 引っ越しが終わってやっと Qiita を思い出して気づいたら 2 ヵ月くらい放置してました。 どうも復活した Xu です。 たまにはちょっと変わった話をしたくて、今日は数学に近い話をしようと思います。 早速ですが、 log_2(1+\frac{M}{2^{...

**Authors:** Leo Zanotti The complexity of continuous piecewise affine (CPA) functions can be measured by the number of pieces $p$ or the number of distinct affine functions $n$. For CPA functions on $\mathbb{R}^d$, this paper shows an upper bound of $p=O(n^{d+1})$ and constructs a family of functions achieving a lower bound of $p=\Omega(n^{d+1-\frac{c}{\sqrt{\log_2(n)}}})$.

**Authors:** Leo Zanotti The complexity of continuous piecewise affine (CPA) functions can be measured by the number of pieces $p$ or the number of distinct affine functions $n$. For CPA functions on $\mathbb{R}^d$, this paper shows an upper bound of $p=O(n^{d+1})$ and constructs a family of functions achieving a lower bound of $p=\Omega(n^{d+1-\frac{c}{\sqrt{\log_2(n)}}})$.