Tight Fourier-growth bounds below the two highest levels

Determine tight upper bounds for the Fourier-growth quantities \(\|\widehat p_s\|_{\ell_1}\) of acceptance probabilities of t-query quantum algorithms for every level \(s\leq 2t−2\), extending the bounds established at levels \(2t\) and \(2t−1\).

Background

The paper proves tight, or nearly tight, Fourier-growth bounds for the highest level $2t$, and also obtains a bound at the second-highest level $2t−1$. The proof uses matrices whose action isolates multilinear terms corresponding to sets of distinct indices.

For lower levels, repeated indices occur in the relevant multi-indices, causing the constructed matrices to annihilate terms needed for the argument. The authors explicitly state that their technique does not yield tight bounds for levels $s\leq 2t−2$, leaving the problem unresolved.