We study the quantum-classical polynomial hierarchy, QCPH, which is the class of languages solvable by a constant number of alternating classical quantifiers followed by a quantum verifier. Our main result is that QCPH is infinite relative to a random oracle (previously, this was not even known relative to any oracle). We further prove that higher levels of PH are not contained in lower levels of QCPH relative to a random oracle; this is a strengthening of the somewhat recent result that PH is infinite relative to a random oracle (Rossman, Servedio, and Tan 2016). The oracle separation requires lower bounding a certain type of low-depth alternating circuit with some quantum gates. To establish this, we give a new switching lemma for quantum algorithms which may be of independent interest. Our lemma says that for any $d$, if we apply a random restriction to a function $f$ with quantum query complexity $\mathrm{Q}(f)\le n^{1/3}$, the restricted function becomes exponentially close (in terms of $d$) to a depth-$d$ decision tree. Our switching lemma works even in a "worst-case" sense, in that only the indices to be restricted are random; the values they are restricted to are chosen adversarially. Moreover, the switching lemma also works for polynomial degree in place of quantum query complexity.

翻译：我们研究量子-经典多项式层级（QCPH），即由常数个交替的经典量词后接一个量子验证器所解决的语言类。我们的主要结果表明，相对于随机预言机，QCPH是无限的（此前，甚至相对于任何预言机都未得知这一结论）。我们进一步证明，相对于随机预言机，PH的更高层级不包含于QCPH的较低层级；这是对近期结果（Rossman、Servedio和Tan，2016）——即PH相对于随机预言机是无限的——的强化。该预言机分离需要下界估计一种包含某些量子门的低深度交替电路。为此，我们提出了一种新的量子算法切换引理，这可能具有独立的研究价值。我们的引理表明，对于任意$d$，若对具有量子查询复杂度$\mathrm{Q}(f)\le n^{1/3}$的函数$f$施加随机限制，受限函数将以指数速度（关于$d$）逼近深度为$d$的决策树。我们的切换引理甚至在“最坏情况”意义下成立，即仅被限制的索引是随机的；而限制所赋予的值由对抗方选择。此外，该切换引理同样适用于以多项式次数替代量子查询复杂度的情形。