A recent work of Schmidhuber et al. (QIP, SODA, & Phys. Rev. X 2025) exhibited a quantum algorithm for the noisy planted \(k\)xor problem running quartically faster than all known classical algorithms. In this work, we design a new classical algorithm that is quadratically faster than the best previous one, in the case of large constant \(k\). Thus for such \(k\), the quantum speedup of Schmidhuber et al. becomes only quadratic (though it retains a space advantage). Our algorithm, which also works in the semirandom case, combines tools from sublinear-time algorithms (essentially, the birthday paradox) and polynomial anticoncentration.

A fundamental task in quantum information science is state certification: testing whether a lab-prepared n-qubit state is close to a given hypothesis state. In this work, we show that every pure hypothesis state can be certified using only O(n^2) single-qubit measurements applied to O(n) copies of the lab state. Prior to our work, it was not known whether even subexponentially many single-qubit measurements could suffice to certify arbitrary states. This resolves the main open question of Huang, Preskill, and Soleimanifar (FOCS 2024, QIP 2024).

We prove that the permutation computed by a reversible circuit with Õ (nk · log(1/ε )) random 3-bit gates is ε-approximately k-wise independent. Our bound improves on currently known bounds in the regime when the approximation error ε is not too small and is optimal up to logarithmic factors when ε is a constant. We obtain our results by analyzing the log-Sobolev constants of appropriate Markov chains rather than their spectral gaps.

Karger (STOC 1995) gave the first FPTAS for the network (un)reliability problem, setting in motion research over the next three decades that obtained increasingly faster running times, eventually leading to a Õ(n2)-time algorithm (Karger, STOC 2020). This represented a natural culmination of this line of work because the algorithmic techniques used can enumerate Θ(n2) (near)-minimum cuts. In this paper, we go beyond this quadratic barrier and obtain a faster FPTAS for the network unreliability problem. Our algorithm runs in m1+o(1) + Õ)(n1.5) time.

We study the formula complexity of the word problem $\mathsf{Word}_{S_n,k} : \{0,1\}^{kn^2} \to \{0,1\}$: given $n$-by-$n$ permutation matrices $M_1,\dots,M_k$, compute the $(1,1)$-entry of the matrix product $M_1\cdots M_k$. An important feature of this function is that it is invariant under action of $S_n^{k-1}$ given by \[ (π_1,\dots,π_{k-1})(M_1,\dots,M_k) = (M_1π_1^{-1},π_1M_2π_2^{-1},\dots,π_{k-2}M_{k-1}π_{k-1}^{-1},π_{k-1}M_k). \] This symmetry is also exhibited in the smallest known unbounded fan-in $\{\mathsf{AND},\mathsf{OR},\mathsf{NOT}\}$-formulas for $\mathsf{Word}_{S_n,k}$, which have size $n^{O(\log k)}$. In this paper we prove a matching $n^{Ω(\log k)}$ lower bound for $S_n^{k-1}$-invariant formulas computing $\mathsf{Word}_{S_n,k}$. This result is motivated by the fact that a similar lower bound for unrestricted (non-invariant) formulas would separate complexity classes $\mathsf{NC}^1$ and $\mathsf{Logspace}$. Our more general main theorem gives a nearly tight $n^{d(k^{1/d}-1)}$ lower bound on the $G^{k-1}$-invariant depth-$d$ $\{\mathsf{MAJ},\mathsf{AND},\mathsf{OR},\mathsf{NOT}\}$-formula size of $\mathsf{Word}_{G,k}$ for any finite simple group $G$ whose minimum permutation representation has degree~$n$. We also give nearly tight lower bounds on the $G^{k-1}$-invariant depth-$d$ $\{\mathsf{AND},\mathsf{OR},\mathsf{NOT}\}$-formula size in the case where $G$ is an abelian group.

We give an almost-linear time algorithm for the Steiner connectivity augmentation problem: given an undirected graph, find a smallest (or minimum weight) set of edges whose addition makes a given set of terminals τ-connected (for any given τ > 0). The running time of our algorithm is dominated by polylogarithmic calls to any maximum flow subroutine; using the recent almost-linear time maximum flow algorithm (Chen et al., FOCS 2022), we get an almost-linear running time for our algorithm as well. This is tight up to the polylogarithmic factor even for just two terminals. Prior to our work, an almost-linear (in fact, near-linear) running time was known only for the special case of global connectivity augmentation, i.e., when all vertices are terminals (Cen et al., STOC 2022). We also extend our algorithm to the closely related Steiner splitting-off problem, where the edges incident on a vertex have to be split-off while maintaining the (Steiner) connectivity of a given set of terminals. Prior to our work, a nearly-linear time algorithm was known only for the special case of global connectivity (Cen et al., STOC 2022). The only known generalization beyond global connectivity was to preserve all pairwise connectivities using a much slower algorithm that makes n calls to an all-pairs maximum flow (or Gomory-Hu tree) subroutine (Lau and Yung, SICOMP 2013), as against polylog(n) calls to a (single-pair) maximum flow subroutine in this work. * Ruoxu Cen and Debmalya Panigrahi were supported in part by NSF grants CCF-1750140 (CAREER Award) and CCF-1955703.