发表机构
Stony Brook University; Paderborn University（石溪大学; 帕德博恩大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究承诺问题的相对化，区分稳健与宽松查询语义，构造预言机分离语言与承诺结果，并加强量子-经典多项式层次上界，引入新类证明其低性。
AI 中文摘要
相对化涉及比较具有对预言机黑盒访问的计算模型。对于承诺问题，由于承诺之外的输入不受约束，黑盒访问并非规范性的。我们研究了这种访问的两种语义。在稳健查询下，无论问题的完成情况如何，机器都必须正确回答，而宽松访问要求机器的内部选择不基于承诺外查询而改变。我们的第一个结果区分了语言和承诺设置。即，我们构造了一个预言机$O$，使得$\mathsf{P}^O = \mathsf{BQP}^O = \mathsf{AWPP}^O$，但$\mathsf{PromiseBQP}^O\not\subseteq\mathsf{PromiseP}^O_{\mathsf{/poly}}$。特别地，$\mathsf{BPP}^O = \mathsf{BQP}^O$，但$\mathsf{PromiseBQP}^O \neq \mathsf{PromiseBPP}^O$，表明语言的结果不一定转移到承诺上。接下来，我们使用宽松查询将量子-经典多项式层次的上界从$\mathsf{P}^{\mathsf{PP}^{\mathsf{PP}}}$加强到$\mathsf{QCPH} \subseteq \mathsf{BP\cdot PP} \subseteq \mathsf{PromiseBPP}^{\mathsf{PP}}$。同样的证明也表明$\mathsf{PP}^\mathsf{PromiseBQP} = \mathsf{PP}$。此外，我们表明$\mathsf{PromiseBQP}$，即使在给定量子建议时，在稳健查询下也是自低的。最后，我们展示了将语言级计数结果转移到承诺类的障碍。尽管$\mathsf{AWPP}$和$\mathsf{APP}$对$\mathsf{PP}$是低的，但相应的承诺类似物将导致计数层次崩溃为$\mathsf{GapP} \subseteq \mathsf{FP}^{\mathsf{PromiseAWPP}}$。这促使我们引入$\mathsf{PromisePostBQP^*}$，它将$\mathsf{PostBQP}$限制为输入无关的后选择。通过证明它对$\mathsf{PP}$是低的，我们得到$\mathsf{PP}^{\mathsf{PromiseYQP^*}} = \mathsf{PP}$。
英文摘要
Relativization is concerned with comparing computational models with black-box access to an oracle. For promise problems, black-box access is not canonical due to inputs outside of the promise being unconstrained. We study two semantics for such access. Under robust queries, a machine must correctly answer regardless of the completion of the problem,, while loose access requires that the internal choices of a machine do not change based on off-promise queries. Our first result separates the language and promise settings. Namely, we construct an oracle $O$ such that $\mathsf{P}^O = \mathsf{BQP}^O = \mathsf{AWPP}^O$, but $\mathsf{PromiseBQP}^O\not\subseteq\mathsf{PromiseP}^O_{\mathsf{/poly}}$. In particular, $\mathsf{BPP}^O = \mathsf{BQP}^O$, but $\mathsf{PromiseBQP}^O \neq \mathsf{PromiseBPP}^O$, showing that results for languages need not transfer to promises. Next, we use loose queries to strengthen the upper bound on the Quantum-Classical Polynomial Hierarchy from $\mathsf{P}^{\mathsf{PP}^{\mathsf{PP}}}$ to $\mathsf{QCPH} \subseteq \mathsf{BP\cdot PP} \subseteq \mathsf{PromiseBPP}^{\mathsf{PP}}$. The same proof also shows $\mathsf{PP}^\mathsf{PromiseBQP} = \mathsf{PP}$. Additionally, we show that $\mathsf{PromiseBQP}$, even when given quantum advice, is self-low under robust queries. Finally, we exhibit an obstruction to transferring language-level counting results to promise classes. Although $\mathsf{AWPP}$ and $\mathsf{APP}$ are low for $\mathsf{PP}$, a corresponding promise analogue would collapse the counting hierarchy as $\mathsf{GapP} \subseteq \mathsf{FP}^{\mathsf{PromiseAWPP}}$. This motivates the introduction of $\mathsf{PromisePostBQP^*}$, which restricts $\mathsf{PostBQP}$ to input-indepencent postselection. By showing that it is low for \PP, we obtain $\mathsf{PP}^{\mathsf{PromiseYQP^*}} = \mathsf{PP}$.
Comments21 pages, 1 figure
