QSVT signal and reflection query counts

The loop in [F1] executes once for each integer $j=d-1,\dots ,0$, hence it makes exactly $d$ signal calls. The side starts at $R$, so the first call is $U$ and directions alternate; therefore the counts are $\lceil d\mathrm{/}2\rceil$ forward calls and $\lfloor d\mathrm{/}2\rfloor$ inverse calls. If $d=0$, the loop is empty.

For a controlled sequence against identity, use the externally controlled projector-phase implementation in [F2] at every phase slot. On the zero branch each phase is then the identity, so the alternating signal calls cancel in adjacent $U{U}^{\ast }$ or $U^{\ast }U$ pairs. If $d$ is even all signal calls pair and cancel; if $d$ is odd, one forward $U$ call remains, which is made controlled while the other signal calls pair and cancel. On the zero branch that controlled call is also identity, while the one branch executes the original sequence. Thus this variant requires zero extra controlled signal calls for even $d$ and one controlled $U$ call for odd $d$, whose availability is explicit by [F3].

For a branch selector between $\mathrm{\Phi }$ and $-\mathrm{\Phi }$, let control value $0$ select $\mathrm{\Phi }$ and value $1$ select $-\mathrm{\Phi }$. At a slot with phase $\varphi$ about $\mathrm{\Pi }$, use the clean-ancilla toggle $X_{\mathrm{\Pi }}$ from step 1.1 of [F2], then unconditional $R_z(2\varphi )$, then a control-one-controlled $R_z(-4\varphi )$, and then $X_{\mathrm{\Pi }}$ again. The middle rotation is $R_z(2\varphi )$ in branch $0$ and $R_z(-2\varphi )$ in branch $1$; because $X_{\mathrm{\Pi }}$ flips the ancilla on $\mathrm{ran}\mathrm{\Pi }$ and leaves it unchanged on $\ker \mathrm{\Pi }$, the system phases are respectively $e^{i\varphi (2\mathrm{\Pi }-I)}$ and $e^{-i\varphi (2\mathrm{\Pi }-I)}$. The same unconditioned signal call occurs between each pair of phase slots, so the signal list is shared and this selector adds no controlled signal query.

There is one phase at the initial $R$ side and one after each of the $d$ signal calls, for $d+1$ slots. Their sides alternate beginning at $R$, giving $\lceil (d+1)\mathrm{/}2\rceil$ right-projector slots and $\lfloor (d+1)\mathrm{/}2\rfloor$ left-projector slots.

Apply [F2] to each slot counted in step 2.2. The total controlled-reflection calls are $2\lceil (d+1)\mathrm{/}2\rceil$ for ${\mathrm{\Pi }}_R$ and $2\lfloor (d+1)\mathrm{/}2\rfloor$ for ${\mathrm{\Pi }}_L$, unchanged for the paired selector. A single unconditioned sequence has $d+1$ variable rotations; the $\mathrm{\Phi }\mathrm{/}-\mathrm{\Phi }$ selector instead uses $d+1$ unconditional rotations $R_z(2{\varphi }_j)$ and $d+1$ separately charged branch-controlled rotations $R_z(-4{\varphi }_j)$, with constant one-qubit overhead per slot.

The counts above describe only the listed operations. By [F4], the cost of supplied $U$, $U^{\ast }$, controlled calls, phase gates, ancilla wires, classical phase generation, and finite-gate compilation are recorded in their applicable separate coordinates; none is inferred from the matrix $A$ alone.