Matrix Samplers and Quantum-Proof Extractors: Their Correspondence and Almost-Optimal Constructions

Abstract

We establish an equivalence between matrix samplers and quantum-proof randomness extractors. The classical sampler--extractor equivalence is established by a seminal work of Zuckerman (Random Structures & Algorithms, 1997). Using this equivalence, we construct explicit strong matrix samplers with optimal randomness overhead and near-optimal sample complexity via constructing quantum-proof extractors. Specifically for matrix dimension $D$, accuracy $\varepsilon$ and failure probability $δ\leq\varepsilon$, our matrix sampler uses $m+O_α(\log(D/δ))$ random bits and $O_α(\varepsilon^{-2-α}\log^{1+α}(D/δ))$ samples, for any constant $α>0$. Compared with the matrix sampler of Xun and Zuckerman (CCC 2025), our construction has improvements in two aspects. Under the same optimal random seed length, we can replace the $\operatorname{poly} D$ dependence in the number of samples by $O(\log^{1-γ} D)$. Under the same scale of sample size, we can remove their $\log\log D$ factor in the length of the random seed. Our underlying strong quantum-proof extractor has output length $n-O_α(a+\log(1/η))$ and seed length at most $(1+α)\log a+(2+α)\log(1/η)+O_α(1)$ for $n$-bit sources with conditional min-entropy $n-a$ and error $η$. We also obtain strong quantum-proof extractors for every constant entropy rate, for any error as small as $2^{-Ω(n)}$, having optimal seed length $O(\log(n/η))$ and output length $Ω(n)$, improving previous constructions.