Manuscript En01-A@1 · En01-A.tex · theorem · 558b972b

Theorem: Branch normal form

Recorded circuits and the two statistical models › Compiled towers · lines 873–918 · proof included or linked

En01-A@1#thm:bnf, En01-A@1#eq:bnf

Every compiled tower has the form

∣Ψ ⟩ = r \in {0, 1}^{N} \sum (j = 1 ⨂^{N} ∣ r_{j} ⟩^{\otimes K_{j}}) \otimes ∣ ξ_{r} ⟩, p_{r} = ∥ ξ_{r} ∥^{2}, r \sum p_{r} = 1.

For supported branches $p_{r} > 0$ , define

∣ ξ^_{r} ⟩ = ∣ ξ_{r} ⟩ / p_{r}, ∣ Φ_{r} ⟩ = (j ⨂ ∣ r_{j} ⟩^{\otimes K_{j}}) \otimes ∣ ξ^_{r} ⟩ .

Then

∣Ψ ⟩ = r \sum p_{r} ∣ Φ_{r} ⟩, ⟨ Φ_{r} ∣ Φ_{r^{'}} ⟩ = δ_{r r^{'}} .

Phases are included in $ξ^_{r}$ . All subsequent branch sums range over supported branches.

Proof. Induct on the event sequence. With no designated classes the assertion is immediate. For (T1), expand the residual vector in the computational basis of the new root, creating an additional branch bit. For (T2), a source in class $K_{j}$ is $∣ r_{j} ⟩$ in branch $r$ ; the fresh target becomes another factor $∣ r_{j} ⟩$ . A (T3) gate acts on the residual vector by the unitary selected by $r$ . A (T4) gate acts on each residual vector without touching the record factors. These exhaust the allowed events. Distinct branch labels differ on a class carrying orthogonal computational factors, proving orthonormality.

Displayed formulas in this passage

En01-A@1#eq:bnf · line 875
$∣Ψ ⟩ = r \in {0, 1}^{N} \sum (j = 1 ⨂^{N} ∣ r_{j} ⟩^{\otimes K_{j}}) \otimes ∣ ξ_{r} ⟩, p_{r} = ∥ ξ_{r} ∥^{2}, r \sum p_{r} = 1.$
unlabeled · line 887
$∣ ξ^_{r} ⟩ = ∣ ξ_{r} ⟩ / p_{r}, ∣ Φ_{r} ⟩ = (j ⨂ ∣ r_{j} ⟩^{\otimes K_{j}}) \otimes ∣ ξ^_{r} ⟩ .$
unlabeled · line 896
$∣Ψ ⟩ = r \sum p_{r} ∣ Φ_{r} ⟩, ⟨ Φ_{r} ∣ Φ_{r^{'}} ⟩ = δ_{r r^{'}} .$

Theorem: Branch normal form

Displayed formulas in this passage

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