Qubitization query and gate resources

Let $\tau =\alpha \mathrm{\mid }t\mathrm{\mid }$ and $L=\ln (1\mathrm{/}ϵ)$, with $0<ϵ<1$. In the nontrivial branch $\tau >ϵ$, put $S=\tau +L\mathrm{/}\ln (e+L\mathrm{/}\tau )$. The small-time-aware simulator of Small-time-aware qubitized simulator uses zero signal queries when $\tau \le ϵ$. When $\tau >ϵ$, the OP4 degree satisfies $d=O(S)$. The complete OAA circuit uses at most $3(d_E+d_O)\le 6d$ ordinary $U\mathrm{/}{U}^{\ast }$ signal calls, where $d_E,d_O\le d$ are the even and odd QSP degrees. It uses $3(d_E+d_O+2)$ projector-phase slots and therefore $6(d_E+d_O+2)$ controlled-reflection calls, by QSVT signal and reflection query counts. Coherent selection of each real sequence against identity uses exactly one controlled signal call when that sequence degree is odd and none when it is even; because the selection occurs in each of the three OAA blocks $W,W^{\ast },W$, the total is $3({\mathbf{1}}_{d_E\text{ odd}}+{\mathbf{1}}_{d_O\text{ odd}})\le 6$ controlled signal calls. The $\mathrm{\Phi }\mathrm{/}-\mathrm{\Phi }$ branch itself controls phases, not signal calls. Any external control of the complete simulator controls and charges the whole block.

Gate, depth, width, ancilla, classical bit, precision, PREPARE/SELECT, phase-list-generation, and sparse-location/value costs remain separate coordinates. For a $d_H$-row-sparse input with supplied-permutation bound $h\ge h_{\max }$, the $h=0$ branch encodes $H=0$ with normalization $1$ and uses no location or value calls. If $h>0$, Correctness and access cost of sparse block encoding gives normalization $\alpha =d_{H}h$; choosing the tight bound $h=h_{\max }$ gives $\alpha =d_H{h}_{\max }$. Multiply the signal-query count by the actual supplied sparse-encoding cost, including any location-permutation completion. The optional sorted-XOR mode has the same normalization $d_{H}h$ for a supplied positive entry bound $h$ and multiplies the signal count by $O(\log (d_H+1))$ location calls and $O(1)$ value calls per signal, including inverses and coherent controls. No cost-free conversion from XOR row-location access is claimed. If $\tau$ is bounded above and below by positive constants as $ϵ\to 0$, the query bound specializes to $O(\tau +L\mathrm{/}\ln L)=O(\tau +\ln (1\mathrm{/}ϵ)\mathrm{/}\ln \ln (1\mathrm{/}ϵ))$. This is an upper bound; no matching lower bound for arbitrary block-encoding access or for the additive precision term is asserted.