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：受近量子设备深度限制的启发，我们研究了查询模型中的深度-计算权衡问题，其中深度对应自适应查询轮数，每层计算对应每轮并行查询数。我们实现了已知最强的量子算法在$r$轮与$r-1$轮自适应能力之间的分离。为此，我们采用了Aaronson和Ambainis（SICOMP'18）提出的$k$重Forrelation问题。当$k=2r$时，该问题可通过每轮仅需一次查询的$r$轮量子算法求解，然而我们证明任何$r-1$轮量子算法每轮需要指数级（量子比特数）的并行查询数。我们的结果遵循近期关于量子-经典分离研究中发展起来的傅里叶分析工具得以证明。结果中的关键新组件是对具有有限自适应轮数的量子查询算法傅里叶权重的界限。这些结论可能具有独立意义，因为它们将此类算法产生的多项式与同次数的任意有界多项式区分开来。