Implementation and strictness of Q-RAM relative to QACZ

Determine whether the QRAM indexing gate can be implemented in QACZ, and determine whether either containment in QACZ ⊆ QACZ[QRAM] ⊆ QACZ_f is strict.

Background

The paper removes the need for QRAM from a construction showing that arbitrary logarithmic-qubit states can be prepared in QACZ with polynomially many ancillas. Nevertheless, the relationship between plain QACZ, QACZ augmented with QRAM, and QACZ_f remains unresolved. In particular, it is unknown whether QRAM is already realizable within QACZ and whether the displayed containments represent genuine increases in computational power.

References

We emphasize that it is remains unknown whether $\QRAM$ itself can be implemented in $\QACZ$, or whether either containment in $\QACZ$ $\subseteq$ $\QACZ[\QRAM]$ $\subseteq$ $\QACZF$ is strict.

— QAC0 Can Prepare Every Logarithmic-Qubit State  (2609.17408 - Gretta et al., 15 Sep 2026) in Section 1, Introduction

While the map $\ket{e_i}\to\ket{e_i}\ket{\operatorname{bin}(i)}$ is in $\QAC0$, as used in the above proof, we do not know if it entails $\ket{e_i}\to\ket i$ in $\QAC0$, which would put the indexing function into $\QAC0$.

— The Robustness of QAC0  (2610.02154 - Grier et al., 1 Oct 2026) in Appendix, Section “Random selectors in QAC^0,” final remark of the binary-selector subsection