Adaptive feedforward constant

Determine whether there exists a universal constant \(c\) with \(1<c\le4/3\) such that every adaptive feedforward implementation of a diagonal level-three gate uses at least \(c\nu\) T gates, where \(\nu\) is stabilizer nullity.

Background

Adaptive feedforward is proved to satisfy only the stabilizer-nullity lower bound t≥νt\ge\nu. The Jones gadget implements CCZCCZ using four T gates, while ν(CCZ)=3\nu(CCZ)=3, ruling out any universal multiplicative constant greater than $4/3$.

The unresolved range is therefore strictly between the nullity bound and the Jones-gadget ratio.

References

Does the bound hold for some $c$ with $1<c\le4/3$?

— Exact $T$-counts of Toffoli layers from an isotropy bound  (2610.01024 - Mazumder, 1 Oct 2026) in Open Problem 3, Section 4.2

Whether an adaptive construction can close the gap between Gidney's $4m$ and the adaptive floor $2m+1$ is open.

— Exact $T$-counts of Toffoli layers from an isotropy bound  (2610.01024 - Mazumder, 1 Oct 2026) in Appendix I, Remark 8.1