LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Projected block identity for a supplied LCU circuit

Statement

Let $A_{j} : C^{n} \to C^{n}$ have exact $(α_{j}, a_{j}, 0)$ block encodings $U_{j}$ , let $β_{j} \neq 0$ , and put $Λ = \sum_{j} ∣ β_{j} ∣ α_{j} > 0$ . Set $a = \max_{j} a_{j}$ and extend each component to the common ordered ancilla/system register by ${\tilde{U}}_{j} = I_{2^{a - a_{j}}} \otimes U_{j}$ , with unused ancillas preceding its original ancillas. Let $P$ be a supplied unitary with $P ∣ 0 ⟩ = \sum_{j} \sqrt{∣ β_{j} ∣ α_{j} / Λ} ∣ j ⟩$ , and let the separately supplied unitary SELECT act as ${\tilde{U}}_{j}$ with phase $β_{j} / ∣ β_{j} ∣$ on index $j$ . With $W = P^{†} S E L E C T P$ , the all-zero coefficient and component-ancilla block of $W$ is $\sum_{j} \frac{β_{j}}{Λ} A_{j} .$ This is a conditional matrix identity. It does not construct SELECT from an XOR-valued function oracle or assign a cost to any supplied operation.

Facts & Assumptions

Given: Finite square matrices $A_{j}$ , their exact block encodings, nonzero coefficients $β_{j}$ , and the supplied unitaries $P$ and SELECT.

[A1]

$P ∣ 0 ⟩ = \sum_{j} \sqrt{∣ β_{j} ∣ α_{j} / Λ} ∣ j ⟩$ , SELECT applies $(β_{j} / ∣ β_{j} ∣) {\tilde{U}}_{j}$ on index $j$ , and $W = P^{†} S E L E C T P$ .

[F1]

Finite block encoding: for an exact $(α_{j}, a_{j}, 0)$ block encoding, the all-zero ancilla compression of $U_{j}$ is $A_{j} / α_{j}$ .

[F2]

Block-encoding oracle and implementation cost model: access to SELECT and its component circuits is separately supplied and costed.

[F3]

Dirac Notation Conversion: the ket $∣ x ⟩$ denotes vector $x$ , and the bra $⟨ x ∣$ denotes the functional $z \mapsto ⟨ z, x ⟩_{r o o t}$ .

Proof

1.1givenA1F2F3algebra

Put $w_{j} = ∣ β_{j} ∣ α_{j} / Λ$ and $ζ_{j} = β_{j} / ∣ β_{j} ∣$ . The weights are nonnegative and sum to one by the definition of $Λ$ . By [A1], the initial coefficient register is $\sum_{j} \sqrt{w_{j}} ∣ j ⟩$ , and SELECT applies $ζ_{j} {\tilde{U}}_{j}$ in the orthogonal index- $j$ subspace, so $⟨ 0 ∣ P^{†} S E L E C T P ∣ 0 ⟩ = \sum_{j} w_{j} ζ_{j} {\tilde{U}}_{j}$ on the component and data registers. This is an identity on already supplied operations; [F2] keeps SELECT construction and component costs separate.

2.1A1F1F2F3step 1.1algebra∎

Project the common $a$ -ancilla register onto its all-zero input and output. The identity extension contributes one on the unused zero ancillas, so by [F1] each extended component still compresses to $A_{j} / α_{j}$ . Hence the resulting data block is $\sum_{j} w_{j} ζ_{j} A_{j} / α_{j} = \sum_{j} (∣ β_{j} ∣ α_{j} / Λ) (β_{j} / ∣ β_{j} ∣) (A_{j} / α_{j}) = \sum_{j} β_{j} A_{j} / Λ$ . The coefficient and component registers were both projected onto their specified zero states, so this is exactly the claimed all-zero block.

Depends on

Used by

Finite CKS sparse QLSP with variable-time inverse estimation Algorithm

Dependency tree · two levels

21 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Andrew M. Childs, Lecture Notes on Quantum Algorithms (standard reference, not scraped)
András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe, Quantum singular value transformation and beyond (standard reference, not scraped)