LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

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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.
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Operator error adds under finite unitary products

Statement

Let $U_{1}, \dots, U_{m}$ and $V_{1}, \dots, V_{m}$ be unitary operators on the same finite-dimensional Hilbert space, and suppose $∥ U_{j} - V_{j} ∥_{o p} \leq ϵ_{j}$ with $ϵ_{j} \geq 0$ . With empty products equal to the identity and empty sums equal to zero, $∥ U_{m} \dots U_{1} - V_{m} \dots V_{1} ∥_{o p} \leq \sum_{j = 1}^{m} ϵ_{j} .$

Facts & Assumptions

Given: Two finite lists of unitaries on one finite-dimensional Hilbert space and their individual error bounds.

[F1]

The operator norm is induced by the Hilbert-space norm, and a unitary preserves that vector norm. The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv|| Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces

[F2]

The operator norm is submultiplicative. On a nonzero Hilbert space, for a unitary $Q$ , its norm and the norm of $Q^{*}$ are both $1$ by [F1], so submultiplicativity gives $∥ Q X ∥_{o p} = ∥ X ∥_{o p} = ∥ X Q ∥_{o p}$ by applying the same upper bound to $Q$ and the reverse bound to $Q^{*}$ . The operator norm is submultiplicative and satisfies ||T^*T|| = ||T||^2

Proof

1.1givenalgebra

If $m = 0$ , both products are $I$ , so their difference is zero and the empty sum is zero. This is the claimed bound in the empty case.

2.1F1F2givenalgebra∎

If the Hilbert space is zero-dimensional, its unique endomorphism has norm zero, so the product difference is zero and the claimed bound holds. Otherwise, for $m \geq 1$ , insert and subtract mixed products to obtain $U_{m} \dots U_{1} - V_{m} \dots V_{1} = \sum_{j = 1}^{m} U_{m} \dots U_{j + 1} (U_{j} - V_{j}) V_{j - 1} \dots V_{1} .$ Expansion cancels adjacent terms and leaves exactly the left side. Each outer product is unitary and has norm one by [F1]; [F2] and the unitary invariance derived there show that each summand has norm at most $ϵ_{j}$ . The operator-norm triangle inequality then gives the stated sum.

Depends on

Used by

Dependency tree · two levels

8 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

Ronald de Wolf, Quantum Computing: Lecture Notes (standard reference, not scraped)