多线性公式与量子计算的怀疑论
Multilinear Formulas and Skepticism of Quantum Computing
摘要：
包括列昂尼德·莱文（Leonid Levin）、杰拉德·特胡夫特（Gerard 't Hooft）和斯蒂芬·沃尔夫勒姆（Stephen Wolfram）在内的几位研究人员认为，在大数的因式分解成为可能之前，量子力学就会崩溃。如果这是真的，那么应该有一组自然的量子态，可以解释迄今为止进行的所有实验，但不能解释肖尔的因子分解算法。我们调查作为一个候选人的一组状态表示的多项式数量的加法和张量积。使用最近的下限多线性公式大小由于拉兹，我们然后表明，在量子纠错中产生的状态需要n^{Ω（log n）}添加和张量积，甚至近似，这顺便产生的第一个超多项式之间的差距一般和多线性公式大小的功能。更广泛地说，我们介绍了一个复杂性分类的纯量子态，并证明了许多基本事实，这种分类。我们的目标是将关于量子力学分解的模糊想法提炼成在不久的将来可能通过实验检验的具体假设。
英文摘要：
Several researchers, including Leonid Levin, Gerard 't Hooft, and Stephen Wolfram, have argued that quantum mechanics will break down before the factoring of large numbers becomes possible. If this is true, then there should be a natural set of quantum states that can account for all experiments performed to date, but not for Shor's factoring algorithm. We investigate as a candidate the set of states expressible by a polynomial number of additions and tensor products. Using a recent lower bound on multilinear formula size due to Raz, we then show that states arising in quantum error-correction require n^{Omega(log n)} additions and tensor products even to approximate, which incidentally yields the first superpolynomial gap between general and multilinear formula size of functions. More broadly, we introduce a complexity classification of pure quantum states, and prove many basic facts about this classification. Our goal is to refine vague ideas about a breakdown of quantum mechanics into specific hypotheses that might be experimentally testable in the near future.
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