Manuscript Oai283-A@1 · Oai283-A.tex · theorem · 6fb75895

Theorem: Exact phase-family pilot

An explicit nonempty pilot and process-matrix insertion › Uniform phase-oracle pilot · lines 678–699 · proof included or linked

Oai283-A@1#thm:phasepilot

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For every $n \geq 1$ , there is a circuit skeleton generated from $n$ alone, with $n + 1$ qubits, one $n$ -bit Boolean-oracle query, and $O (1)$ ordinary gates, such that every supplied $f$ implements

U_{f} ∣ x ⟩ = (- 1)^{f (x)} ∣ x ⟩

and returns its additional qubit to zero exactly.

Proof. Prepare the answer qubit in $∣ - ⟩ = (∣0 ⟩ - ∣1 ⟩) / 2$ . Then

O_{f} (∣ x ⟩ ∣ - ⟩) = (- 1)^{f (x)} ∣ x ⟩ ∣ - ⟩ .

Reverse the answer preparation. The resulting isometry is $U_{f} \otimes ∣0 ⟩$ . Preparation uses $X$ and $H$ , and $X = H T^{4} H$ .

Displayed formulas in this passage

unlabeled · line 683
$U_{f} ∣ x ⟩ = (- 1)^{f (x)} ∣ x ⟩$
unlabeled · line 693
$O_{f} (∣ x ⟩ ∣ - ⟩) = (- 1)^{f (x)} ∣ x ⟩ ∣ - ⟩ .$

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section: Uniform phase-oracle pilot