基于量子论证的矩阵刚性下界
Lower Bounds on Matrix Rigidity via a Quantum Argument
摘要：
矩阵的刚性度量了为了将其秩降低到某个值，需要改变多少个元素。一个显式矩阵的刚性的好的下界意味着算术电路和通信复杂性的好的下界。在这里，我们reprove最有名的边界上的刚性阿达玛矩阵，由于卡申和拉兹博罗夫，使用量子计算的工具。我们的证明比以前的证明要简单一些（至少对那些熟悉量子的人来说是这样），并且给出了稍微好一点的常数。更重要的是，它们提供了一种新的方法来解决这个长期存在的开放问题。
英文摘要：
The rigidity of a matrix measures how many of its entries need to be changed in order to reduce its rank to some value. Good lower bounds on the rigidity of an explicit matrix would imply good lower bounds for arithmetic circuits as well as for communication complexity. Here we reprove the best known bounds on the rigidity of Hadamard matrices, due to Kashin and Razborov, using tools from quantum computing. Our proofs are somewhat simpler than earlier ones (at least for those familiar with quantum) and give slightly better constants. More importantly, they give a new approach to attack this longstanding open problem.
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