Aaronson-Ambainis猜想的分布变体
Distributional Variants of the Aaronson-Ambainis Conjecture
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- University of Toronto（多伦多大学）
- Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）
- Columbia University（哥伦比亚大学）
- Princeton University（普林斯顿大学）
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中文总结 AI 辅助
该论文研究量子查询算法经典模拟猜想在非均匀输入分布下的变体，证明乘积分布与切片均匀分布下的变体均等价于原始均匀分布猜想，并推广了Aaronson-Ambainis猜想至这些分布。
中文摘要 AI 辅助
量子复杂性理论中一个长期存在的猜想断言，在均匀输入分布下，量子查询算法可以被经典查询算法多项式模拟。更精确地说，任何量子查询算法的接受概率，在均匀随机输入的平均意义上，可以被一个经典查询算法近似，且仅需多项式查询开销。该猜想对于理解决策问题的指数级量子优势是否必然依赖于额外结构至关重要。我们研究了在其他自然输入分布下该猜想的类似形式，并证明它们都等价于原始的均匀分布猜想。我们首先考虑乘积分布 $\mu_p$，其中输入比特是独立伯努利变量，具有固定偏置 $p$。我们证明，对于每个固定的 $p \in (0, 1)$，在 $\mu_p$ 分布下的量子查询算法允许多项式开销的经典模拟，当且仅当在均匀分布下同样成立。其次，我们考虑分布 $\nu_p$，它是在汉明权重为 $\lfloor pn \rfloor$ 的字符串切片上的均匀分布，并证明 $\nu_p$ 分布与均匀分布之间具有类似的等价性。Aaronson-Ambainis猜想是一个更强的陈述，它蕴含上述猜想，并以布尔超立方体上的有界低次多项式表述。它断言在均匀分布下，任何具有不可忽略方差的多项式必须有一个有影响力的变量。我们提出了该猜想的类似形式，其中底层分布是偏置乘积分布或切片上的均匀分布，并证明所有这些变体都等价于原始的Aaronson-Ambainis猜想。
英文摘要
A longstanding conjecture in quantum complexity theory asserts that, under the uniform input distribution, quantum query algorithms can be polynomially simulated by classical query algorithms. More precisely, the acceptance probability of any quantum query algorithm can be approximated, on average over uniformly random inputs, by a classical query algorithm, with only polynomial query overhead. The conjecture is central to understanding whether exponential quantum advantages for decision problems necessarily rely on additional structure. We study analogues of this conjecture under other natural input distributions and prove that they are all equivalent to the original uniform-distribution conjecture. We first consider the product distribution $μ_p$, where the input bits are independent Bernoulli variables with fixed bias $p$. We show that for every fixed $p \in (0, 1)$, quantum query algorithms under the $μ_p$ distribution admit polynomial-overhead classical simulations if and only if the same holds under the uniform distribution. Second, we consider the distribution $ν_p$ that is uniform over the slice of strings with Hamming weight $\lfloor pn \rfloor$ and prove a similar equivalence for the $ν_p$ distribution and the uniform distribution. The Aaronson-Ambainis conjecture is a stronger statement that implies the above-mentioned conjecture and is formulated in terms of bounded low-degree polynomials on the Boolean hypercube. It asserts that under the uniform distribution, any such polynomial with nonnegligible variance must have an influential variable. We formulate analogues of this conjecture, where the underlying distribution is a biased product distribution or a uniform distribution over a slice, and prove that all these variants are equivalent to the original Aaronson-Ambainis conjecture.
