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

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Uniform polynomial approximation transfers to a Hermitian matrix

Statement

Let $V$ be a finite-dimensional complex inner-product space, let $H : V \to V$ be Hermitian, and let $I \subseteq R$ contain its spectrum. For functions $f : I \to C$ , a polynomial $p \in C [x]$ , and $η \geq 0$ , if $∣ p (λ) - f (λ) ∣ \leq η$ for every $λ \in σ (H)$ , then $∥ p (H) - f (H) ∥ \leq η .$ In particular, a uniform bound on all of $I$ suffices. Here $f (H)$ is the finite spectral functional calculus.

Facts & Assumptions

Given: A finite-dimensional complex inner-product space $V$ , a Hermitian endomorphism $H$ , an interval $I \supseteq σ (H)$ , functions $f : I \to C$ and $p \in C [x]$ , and $η \geq 0$ with $∣ p (λ) - f (λ) ∣ \leq η$ on $σ (H)$ .

[F1]

A normal endomorphism is a sum of its eigenvalues times pairwise orthogonal projections, and each spectral projection is a polynomial in the endomorphism: $H = \sum_{λ \in σ (H)} λ P_{λ}$ , where the distinct-eigenvalue projections are pairwise orthogonal and sum to the identity.

[F2]

The spectral functional calculus f(T) for a normal endomorphism and For normal endomorphisms, the spectral functional calculus respects sums, products, adjoints, and composition of scalar functions: $g (H) = \sum_{λ} g (λ) P_{λ}$ for every function $g$ on the finite spectrum, and polynomial evaluation agrees with this calculus.

[F3]

The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||: $∥ T ∥ = \sup_{∥ v ∥ = 1} ∥ T v ∥$ .

[F4]

Pythagoras and finite orthogonal sums: the squared norm of a finite sum of pairwise orthogonal vectors is the sum of their squared norms.

Proof

1.1F1F2F3F4given

If $V = 0$ , [F3] defines the operator norm as zero, so the result is immediate. Otherwise [F1]–[F2] give $p (H) - f (H) = \sum_{λ} (p (λ) - f (λ)) P_{λ}$ . For any $v \in V$ , the vectors $P_{λ} v$ are pairwise orthogonal by [F1]; hence [F4] gives $∥ (p (H) - f (H)) v ∥^{2} = \sum_{λ} ∣ p (λ) - f (λ) ∣^{2} ∥ P_{λ} v ∥^{2} \leq η^{2} \sum_{λ} ∥ P_{λ} v ∥^{2} = η^{2} ∥ v ∥^{2}$ , with the last equality following from $\sum_{λ} P_{λ} = I$ and [F4].

2.1F3step 1.1algebra∎

For every unit vector $v$ , step 1.1 yields $∥ (p (H) - f (H)) v ∥ \leq η$ . Taking the supremum over unit vectors using [F3] proves the asserted operator-norm bound. A bound on all of $I$ implies the required bound on its subset $σ (H)$ .

Depends on

Used by

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Sources

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