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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.

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A projected unitary block is a contraction

Statement

For a finite system space, an ancilla injection $J_{a} x = e_{0} \otimes x$ , and a unitary $U$ on the joint space, the projected block $B_{U} = J_{a}^{†} U J_{a}$ satisfies $∥ B_{U} ∥_{o p} \leq 1$ . Consequently, every exact $(α, a, 0)$ block encoding of $A$ satisfies $∥ A ∥_{o p} \leq α$ .

Facts & Assumptions

Given: The compression and exact-block conventions of Finite block encoding, a unitary $U$ , and the finite-dimensional operator norm.

[F1]

Finite block encoding: $B_{U} = J_{a}^{†} U J_{a}$ , with $J_{a}$ the isometric inclusion of the all-zero ancilla subspace; in the exact case $B_{U} = \hat{A} / α$ , where $\hat{A} = E_{o u t} A E_{i n}^{†}$ and $∥ \hat{A} ∥_{o p} = ∥ A ∥_{o p}$ by its Proof, step 3.1.

[F2]

The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||: the operator norm is the maximum of $∥ T x ∥$ over unit vectors, and the zero-domain norm is zero.

[F3]

Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces: a unitary preserves every vector norm.

Proof

1.1givenF1F2F3algebra

The zero-ancilla coordinate projection $J_{a}^{†}$ has norm at most one: it discards all coordinates orthogonal to $e_{0} \otimes C^{d}$ , so the squared length of the retained coordinates is at most the squared length of the full vector. Hence for each unit $x$ , $∥ B_{U} x ∥ = ∥ J_{a}^{†} U J_{a} x ∥ \leq ∥ U J_{a} x ∥ = ∥ J_{a} x ∥ = 1$ , using [F3] and the isometry of $J_{a}$ from [F1]. Taking the maximum gives $∥ B_{U} ∥_{o p} \leq 1$ ; the zero-dimensional case follows from [F2].

2.1step 1.1F1F2algebra∎

In the exact case [F1] gives $\hat{A} = α B_{U}$ and $∥ A ∥_{o p} = ∥ \hat{A} ∥_{o p}$ . The unit-vector definition [F2] gives $∥ α B_{U} ∥_{o p} = α ∥ B_{U} ∥_{o p}$ for positive $α$ , hence $∥ A ∥_{o p} = α ∥ B_{U} ∥_{o p} \leq α$ .

Depends on

Used by

Product block-encoding error bound Theorem

Dependency tree · two levels

13 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, Chapter 29 §29.1 (17 April 2025) (standard reference, not scraped)
András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe, Quantum singular value transformation and beyond, Definition 43 (standard reference, not scraped)