发表机构
Université Paris Cité, CNRS, IRIF; Chennai Mathematical Institute（巴黎西岱大学; 金奈数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出鲁棒子空间设计，给出概率与显式构造，并应用于量子空间有界Valiant-Vazirani定理，隔离唯一量子见证者，同时改进经典零化检验的包含证明。
AI 中文摘要
子空间设计的概念由Guruswami和Xing引入（STOC'13），Guruswami和Kopparty给出了显式构造（FOCS'13）。这些子空间族与任何给定固定维数子空间的交集很小。我们引入鲁棒子空间设计。非正式地说，这是一种定量扩展，要求族中不能有太多子空间包含靠近任何给定固定维数子空间的方向。我们给出了一个多项式大小的此类鲁棒子空间设计的概率构造，以及一个非平凡的、超多项式大小的显式构造。我们这一新概念的主要应用是Valiant-Vazirani定理（STOC'86）的量子空间有界变体，该定理表明将NP完全问题限制为至多有一个接受见证者的实例，在随机归约下保持困难性。对于量子见证者，类似的数量是接受见证者子空间的维数。我们使用鲁棒子空间设计的概率构造，为具有完美完备性和接受间隙的量子Merlin-Arthur协议隔离出其完美接受子空间之外的唯一见证者。作为进一步的应用，我们给出了良条件零化检验到具有完美完备性的空间有界量子Merlin-Arthur协议的随机归约。利用类似的思想，我们发现普通子空间设计允许我们通过更简单的证明恢复Allender、Beals和Ogihara（STOC'96）关于一般零化检验的经典C_=L包含结果。
英文摘要
The concept of subspace designs was introduced by Guruswami and Xing (STOC'13), and explicit constructions were given by Guruswami and Kopparty (FOCS'13). These are families of subspaces that have small intersection with any given subspace of a fixed dimension. We introduce \emph{robust subspace designs}. Informally, these are a quantitative extension in which we demand that not too many subspaces of the family contain directions that lie close to any given subspace of a fixed dimension. We give a probabilistic construction of such a robust subspace design of polynomial size, as well as a non-trivial explicit construction of superpolynomial size. Our main application of this new concept is a quantum space-bounded variant of the Valiant-Vazirani theorem (Theor.Comput.Sci.'86), which shows that restricting $\mathsf{NP}$-complete problems to instances with at most one accepting witness preserves hardness under randomized reductions. For quantum witnesses, the analogous quantity is the dimension of an accepting witness subspace. We use our probabilistic construction of robust subspace designs to isolate a unique witness for space-bounded quantum Merlin-Arthur protocols with perfect completeness and an acceptance gap outside their perfectly accepting subspace. As further applications, we give a randomized reduction of well-conditioned nullity testing to space-bounded quantum Merlin-Arthur protocols with perfect completeness. Using a similar idea, we find that ordinary subspace designs allow us to recover the classical $\mathsf{C_= L}$ containment of Allender, Beals, and Ogihara (STOC'96) for general nullity testing through a simpler proof.
