The Power of Adaptivity in Quantum Query Algorithms

TL;DR

This paper demonstrates a significant separation in quantum query complexity based on the number of adaptive rounds, showing that fewer rounds require exponentially more queries, highlighting the importance of adaptivity in quantum algorithms.

Contribution

The paper establishes the strongest known separation between quantum algorithms with r and r-1 rounds of adaptivity using Fourier analysis and the k-fold Forrelation problem.

Findings

r-round algorithms solve the problem efficiently with one query per round

r-1 round algorithms require exponential queries, even with many parallel queries

Fourier weight bounds distinguish adaptive quantum algorithms from arbitrary polynomials

Abstract

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.