Intrinsic characterization of bounded Fourier depth

Characterize the additional algebraic or operator-theoretic structure imposed on a quantum query form by a bound of k on the number of global Hadamard layers, and determine whether the resulting FH_k class differs from the (k-1)-round parallel-query model into which phase-query FH_k circuits embed.

Background

Quantum query algorithms admit characterizations as completely bounded forms of bounded degree, but those characterizations measure query complexity and do not capture the placement or number of Hadamard layers. The paper proves only a one-way simulation from phase-query Fourier depth k to k-1 rounds of parallel queries. It remains unclear whether the larger round model has strictly greater computational power or whether an intrinsic characterization of Fourier depth can be formulated within the language of query forms.

References

What additional structure on the form corresponds to a bound of $k$ on the number of Hadamard layers? A related question is whether $FH_k$ and the $(k-1)$-round parallel-query model of Lemma~\ref{lem:round-embedding} differ as classes.

— Oracle Separations in the Fourier Hierarchy  (2609.11830 - Mantri, 10 Sep 2026) in Section 7, Discussion and open problems

Whether $\mathrm{DQC}_k$ or $\tfrac12BQP$ is in turn contained in a fixed Fourier level, and more generally whether some explicit problem separates mixedness from Fourier depth, remains open.

— Oracle Separations in the Fourier Hierarchy  (2609.11830 - Mantri, 10 Sep 2026) in Section 2, Related work, paragraph "Restricted quantum models and Fourier growth"

For $k\ge3$ the expression contains adjacent oracle positions, and we do not know how to control the variance of the corresponding estimator. The higher levels correspond to the case $t>1$ of that conjecture, which is open.

— Oracle Separations in the Fourier Hierarchy  (2609.11830 - Mantri, 10 Sep 2026) in Remark 5.7, Section "Classical simulation at the second level"