LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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.

Rescaling a matrix-function target for QSVT

Statement

Let $H$ be Hermitian on a nonzero finite-dimensional complex inner-product space, with $∥ H ∥ \leq α$ , where $α > 0$ , and let $f : σ (H) \to C$ . If $β > 0$ , $β \geq ∥ f (H) ∥$ , and a polynomial $p$ satisfies $\sup_{x \in σ (H / α)} ∣ p (x) - \frac{f (α x)}{β} ∣ \leq η,$ then $∥ β p (H / α) - f (H) ∥ \leq β η$ . Consequently, if a unitary has top-left block $p (H / α)$ , it is a $β$ -block encoding of $f (H)$ with compression error at most $η$ under the block-encoding convention.

Facts & Assumptions

Given: $H = H^{*}$ on a nonzero finite-dimensional space, $0 < α$ , $∥ H ∥ \leq α$ , $β > 0$ , $β \geq ∥ f (H) ∥$ , and the displayed scalar approximation.

[F1]

Uniform polynomial approximation transfers to a Hermitian matrix: a uniform scalar approximation on the spectrum transfers to the operator norm by finite spectral calculus.

[F2]

The spectral functional calculus f(T) for a normal endomorphism: $g (H) = \sum_{λ} g (λ) P_{λ}$ , so scalar multiplication of $g$ multiplies its matrix function by the same scalar.

[F3]

Finite block encoding: a unitary is a $β$ -block encoding of $A$ when its designated compression equals $A / β$ (or is within the separately stated error).

Proof

1.1F1givenalgebra

The spectrum of $H / α$ is ${λ / α : λ \in σ (H)}$ , since each eigenvector of $H$ remains an eigenvector with its eigenvalue divided by $α$ . For each such eigenvalue, the hypothesis gives $∣ β p (λ / α) - f (λ) ∣ \leq β η$ . Apply [F1] to $H$ and the scalar function $λ \mapsto β p (λ / α) - f (λ)$ to get the asserted operator-norm inequality.

2.1F2F3step 1.1algebra∎

The top-left block equals $p (H / α)$ by hypothesis. Dividing the proved inequality by $β$ gives $∥ p (H / α) - f (H) / β ∥ \leq η$ . The additional hypothesis $β \geq ∥ f (H) ∥$ supplies the normalization condition in the block-encoding definition [F3], so this is a valid $β$ -block encoding with compression error at most $η$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe, Quantum singular value transformation and beyond, §§2–3 (standard reference, not scraped)
Andrew M. Childs, Lecture Notes on Quantum Algorithms, Chapter 29 §29.3 (standard reference, not scraped)