AI 中文总结
研究随机量子\(k\)-SAT阈值定量位置，结合几何表述与维度衰减分析，证明新上界，将已知渐近上界提高\(k\)倍，小\(k\)值时有显著改进，关键输入是张量积子空间的乘法希勒型不等式。
AI 中文摘要
随机量子可满足性是随机约束满足的自然量子类似物，也是无挫折局部哈密顿量的基本模型。尽管对其可满足和不可满足情况进行了大量研究，但随机量子\(k\)-SAT阈值的定量位置仍知之甚少，已知最佳通用上界与下界差距很大。本文证明了随机量子\(k\)-SAT可满足性阈值的新上界，将先前已知的渐近上界提高了\(k\)倍，得到\(2^k/k\)的界。对于小\(k\)值也有显著改进，如随机量子3-SAT得到比之前更小的显式上界。证明结合了通用量子可满足性的几何表述和全满足子空间的维度衰减分析，关键输入是张量积子空间的乘法希勒型不等式。
英文摘要
Random quantum satisfiability is a natural quantum analogue of random constraint satisfaction and a basic model for frustration-free local Hamiltonians. Despite extensive work on its satisfiable and unsatisfiable regimes, the quantitative location of the random quantum \(k\)-SAT threshold has remained poorly understood, with the best general upper bounds leaving a large gap to the known lower bounds. In this paper we prove a new upper bound on the satisfiability threshold of random quantum \(k\)-SAT. Our result improves the previously known asymptotic upper bound by a factor of order \(k\), giving a bound of order \(2^k/k\). The improvement is also significant at small values of \(k\); in particular, for random quantum \(3\)-SAT we obtain a substantially smaller explicit upper bound than the one previously available. The proof combines the geometric formulation of generic quantum satisfiability with a dimension-decay analysis of the full satisfying subspace. The key input is a multiplicative Shearer-type inequality for tensor-product subspaces, which quantifies how global dimension forces nontrivial local dimension on typical sets of qubits.
