Motivated by limitations on the depth of near-term quantum devices, we study the depth-computation trade-off in the query model, where the depth corresponds to the number of adaptive query rounds and the computation per layer corresponds to the number of parallel queries per round. We achieve the strongest known separation between quantum algorithms with $r$ versus $r-1$ rounds of adaptivity. We do so by using the $k$-fold Forrelation problem introduced by Aaronson and Ambainis (SICOMP'18). For $k=2r$, this problem can be solved using an $r$ round quantum algorithm with only one query per round, yet we show that any $r-1$ round quantum algorithm needs an exponential (in the number of qubits) number of parallel queries per round. Our results are proven following the Fourier analytic machinery developed in recent works on quantum-classical separations. The key new component in our result are bounds on the Fourier weights of quantum query algorithms with bounded number of rounds of adaptivity. These may be of independent interest as they distinguish the polynomials that arise from such algorithms from arbitrary bounded polynomials of the same degree.

翻译：受限于近近期量子设备深度的限制，我们研究了查询模型中的深度-计算权衡，其中深度对应于自适应查询轮数，每层计算对应于每轮并行查询次数。我们实现了量子算法在$r$轮与$r-1$轮自适应性之间已知的最强分离。为此，我们采用了由Aaronson和Ambainis（SICOMP'18）引入的$k$重Forrelation问题。当$k=2r$时，该问题可以通过一个每轮仅需一次查询的$r$轮量子算法解决，然而我们证明任何$r-1$轮量子算法每轮都需要指数级（在量子比特数量上）的并行查询次数。我们的结果基于近期量子-经典分离研究中发展的傅里叶分析框架进行证明。我们结果的关键新要素是对于具有有限轮次自适应性的量子查询算法的傅里叶权重界限。这些界限可能具有独立意义，因为它们能够区分这类算法产生的多项式与相同次数的任意有界多项式。