分布型量子查询复杂度

中文摘要 AI 辅助

量子查询复杂度享有多种令人满意的联合计算性质：例如，一个组合定理断言对于所有布尔函数$f$和$g$，$Q(f\circ g)=\Theta(Q(f)Q(g))$；一个直和定理断言计算一个函数（或搜索问题）的$k$个副本的成本至少是计算一个副本成本的$\Omega(k)$倍；以及一个直积定理断言对于布尔函数，即使以指数小的概率成功解决直和问题，仍需要$\Omega(k)$倍于计算一个副本到有界误差的成本。然而，所有这些结果都严格针对最坏情况下的量子查询复杂度。例如，如果我们对输入有一个固定的分布$\mu$，直和定理对于当输入来自乘积分布$\mu^k$而非最坏情况时计算$f$的$k$个副本的量子查询复杂度没有任何说明。（注意，虽然标准的Yao型极小极大定理保证了直和问题存在一个困难分布，但无法保证这个困难分布是乘积分布。）直积定理和组合定理也存在类似的问题：这些结果都不考虑分布。在这项工作中，我们给出了组合、直和和直积问题的分布型联合计算下界。在此过程中，我们引入了一些处理量子查询下界的新工具，包括（a）$\gamma_2$范数的一个新的“乘法”变体（我们用它代替乘法对手方法来证明直积定理），以及（b）一种新的“无Shaltiel”量子查询复杂度度量，我们证明它刻画了分布型量子查询复杂度的组合行为并满足令人满意的性质。

英文摘要

Quantum query complexity enjoys a variety of pleasing joint computation properties: for example, a composition theorem asserting $Q(f\circ g)=Θ(Q(f)Q(g))$ for all Boolean functions $f$ and $g$; a direct sum theorem asserting that computing $k$ copies of a function (or search problem) costs $Ω(k)$ times as much as the cost of computing one copy; and a direct product theorem asserting that for Boolean functions, even succeeding at the direct sum problem with exponentially small probability still requires $Ω(k)$ times the cost of computing one copy to bounded error. However, all of these results are strictly for worst-case quantum query complexity. For example, if we have a fixed distribution $μ$ over inputs, the direct sum theorem says nothing about the quantum query complexity of computing $k$ copies of $f$ when the input comes from the product distribution $μ^k$ instead of being worst-case. (Note that while a standard Yao-type minimax theorem guarantees a hard distribution for the direct sum problem, there's no guarantee that this hard distribution is a product distribution.) A similar problem occurs for the direct product theorem and the composition theorem: none of these results respect distributions. In this work, we give distributional joint computation lower bounds for the composition, direct sum, and direct product problems. Along the way, we introduce some new tools for handling quantum query lower bounds, including (a) a new ``multiplicative'' variant of the $γ_2$ norm (which we use in place of the multiplicative adversary method for proving the direct product theorem), and (b) a new ``Shaltiel-free'' measure of quantum query complexity, which we show characterizes the composition behavior of distributional quantum query complexity and satisfies pleasing properties.

发表机构

• Institute for Quantum Computing（量子计算研究所）
• University of Waterloo（滑铁卢大学）

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