量子回路
{\displaystyle \left\langle {\vec {x}},0{\big |}U^{*}\operatorname {E} _{F(x)}U{\big |}{\vec {x}},0\right\rangle =\left\langle \operatorname {E} _{F(x)}U(|{\vec {x}},0\rangle ){\big |}U(|{\vec {x}},0\rangle )\right\rangle \geq 1-\epsilon .}
ここで、
| Tr  ( S U ∗ E F ( x )  U ) − ⟨ x → , 0 | U ∗ E F ( x )  U | x → , 0 ⟩ | ≤ Tr  ( | | x → , 0 ⟩ ⟨ x → , 0 | − S | ) ‖ U ∗ E F ( x )  U ‖ ≤ δ {\displaystyle \left|\operatorname {Tr} (SU^{*}\operatorname {E} _{F(x)}U)-\left\langle {\vec {x}},0{\big |}U^{*}\operatorname {E} _{F(x)}U{\big |}{\vec {x}},0\right\rangle \right|\leq \operatorname {Tr} ({\big |}|{\vec {x}},0\rangle \langle {\vec {x}},0|-S{\big |})\|U^{*}\operatorname {E} _{F(x)}U\|\leq \delta }