Clean ancilla controlled T circuit is exact

Fix the controls $a,b$ and let $q$ be the target bit immediately after the first $H_t$. The seven phase gates in [F1] contribute exponent $E_{a,b}(q)=-(q\oplus b)+(q\oplus a\oplus b)-(q\oplus a)+b+q(\mathrm{m}\mathrm{o}\mathrm{d}8)$. For $(a,b)=00,01,10,11$, the pairs $(E(0),E(1))$ are respectively $(0,0),(1,1),(0,0),(7,3)$. Conjugating these diagonal phases by the final $H_t$ therefore gives, for the four control values, $I,\omega I,I,{\omega }^{7}X$ on the target. The last CNOT–$T_a$–$T_b^{†}$–CNOT block contributes data phase ${\omega }^{a-(a\oplus b)}$, which is $1,{\omega }^{-1},1,\omega$ in the same order. The combined action is exactly $\mathrm{\mid }a,b,t⟩\mapsto \mathrm{\mid }a,b,t\oplus ab⟩$ with phase $1$ in all eight cases, proving the displayed word is Toffoli in the sense of [F7]. By [F3,F4], these basis actions give the denotational matrix of the displayed chronological circuit.

Starting with $t=0$, the first Toffoli sends $\mathrm{\mid }a,b,0⟩$ to $\mathrm{\mid }a,b,ab⟩$. The middle $T_t$ multiplies this vector by ${\omega }^{ab}$; applying the Toffoli action [F7] again gives target $ab\oplus ab=0$. Thus the full circuit sends each basis vector to ${\omega }^{ab}\mathrm{\mid }a,b,0⟩$. The matrix product is the chronological circuit action by [F3,F4].

Every vector on data and a finite-dimensional reference with a clean ancilla has the finite expansion ${\sum }_{a,b,r}{c}_{abr}\mathrm{\mid }a,b,r⟩\mathrm{\mid }0⟩$. Applying step 2.1 termwise gives ${\sum }_{a,b,r}{c}_{abr}{\omega }^{ab}\mathrm{\mid }a,b,r⟩\mathrm{\mid }0⟩$, which is $C(T)$ on the data and identity on the reference, tensored with the restored zero state, by [F5,F6]. This is the asserted correlated-input action.