Controlled T using a clean ancilla

Input two data bits $a,b$ and a third clean work bit $t=0$. On the ordered register $(a,b,t)$, apply the following chronological word:

1. $H_t$.
2. ${\mathrm{CNOT}}_{b,t},T_t^{†},{\mathrm{CNOT}}_{a,t},T_t,{\mathrm{CNOT}}_{b,t},T_t^{†},{\mathrm{CNOT}}_{a,t}$.
3. $T_b,T_t,H_t$.
4. ${\mathrm{CNOT}}_{a,b},T_a,T_b^{†},{\mathrm{CNOT}}_{a,b}$.
5. $T_t$.
6. Apply the same 15-gate word from steps 1–4 again.

The 15-gate word in steps 1–4 is the specified three-wire block. The output is the complete three-wire circuit, with no implicit fourth wire.

The source's controlled-$T$ construction computes $ab$ into the clean bit, applies $T$ there, and uncomputes. The target operation $C(T)$ is the control-first block in Controlled Unitary Access. Its exact correctness and restoration claim is the separate proposition above.

The actual QC09 syntax/denotational-semantics and QC10 Toffoli/ancilla carriers were inspected. Their ordered-wire and clean-return interfaces agree with this circuit. This is interface reconciliation, not acceptance of the supplier drafts.