Manuscript En04-A@1 · En04-A.tex · section · bab77211

An exact non-Clifford responding program

A genuine Clifford criterion and a responding deformation › An exact non-Clifford responding program · lines 1080–1112

En04-A@1#eq:rotationisometry

On one logical qubit and one allocated residual qubit, define

R_{y} (2 θ) = e^{- i θ Y},

U_{θ} = ∣ 0 ⟩ ⟨ 0 ∣ \otimes I + ∣ 1 ⟩ ⟨ 1 ∣ \otimes R_{y} (2 θ), 0 \leq θ \leq \frac{π}{2} .

Initialize $B$ in $∣ 0 ⟩$ and return the first qubit as $R$ . Then

V_{θ} ∣ 0 ⟩ = ∣ 0 ⟩_{R} ∣ 0 ⟩_{B}, V_{θ} ∣ 1 ⟩ = ∣ 1 ⟩_{R} ∣ b_{θ} ⟩_{B}, ∣ b_{θ} ⟩ = cos θ ∣ 0 ⟩ + sin θ ∣ 1 ⟩ .

The typed body is

b := new; (R,B) := ControlledRy_theta(C,b); returnQ(R,B);

It is a finite semantic program with a specified unitary constant. No exact compilation into a fixed finite gate library is asserted.

Displayed formulas in this passage

En04-A@1#eq:rotationisometry · line 1096
$V_{θ} ∣ 0 ⟩ = ∣ 0 ⟩_{R} ∣ 0 ⟩_{B}, V_{θ} ∣ 1 ⟩ = ∣ 1 ⟩_{R} ∣ b_{θ} ⟩_{B}, ∣ b_{θ} ⟩ = cos θ ∣ 0 ⟩ + sin θ ∣ 1 ⟩ .$
unlabeled · line 1083
$R_{y} (2 θ) = e^{- i θ Y},$
unlabeled · line 1086
$U_{θ} = ∣ 0 ⟩ ⟨ 0 ∣ \otimes I + ∣ 1 ⟩ ⟨ 1 ∣ \otimes R_{y} (2 θ), 0 \leq θ \leq \frac{π}{2} .$