Small incremental magic conjecture

Let an $n$-qudit system have local dimension $d$, and let $H$ be a Hamiltonian acting on a $k$-qudit subsystem. Let $R_M(H,O)$ denote the magic-rate quantity for $H$ and a linear operator $O$.

Small incremental magic conjecture. One should have

$$\left|R_M(H,O)\right|\leq c k\log(d)\left\lVert H\right\rVert_{\infty},$$

where $c$ is independent of $k$, $d$, and $n$.

The conjecture seeks logarithmic rather than polynomial dependence on the local dimension, in analogy with the coherence bound discussed in the paper. Its resolution is not established in the supplied text.