Manuscript Oai283-A@1 · Oai283-A.tex · lemma · 6fb75895

Lemma: Telescoping on an ideal clean prefix

Composition, workspace, and implementation allowances › Clean-subspace estimates · lines 535–561 · proof included or linked

Oai283-A@1#lem:clean-telescope

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Let $A_{j}, V_{j}$ be unitaries on a common enlarged space, let $P_{0} : H_{M} \to H_{all}$ be the initial clean isometry, and put $P_{j - 1} = V_{j - 1} \dots V_{1} P_{0}$ . If

∥ (A_{j} - V_{j}) P_{j - 1} ∥_{\infty} \leq δ_{j},

then

∥ (A_{L} \dots A_{1} - V_{L} \dots V_{1}) P_{0} ∥_{\infty} \leq j = 1 \sum^{L} δ_{j} .

Proof. Use

A_{L} \dots A_{1} - V_{L} \dots V_{1} = j = 1 \sum^{L} A_{L} \dots A_{j + 1} (A_{j} - V_{j}) V_{j - 1} \dots V_{1} .

Each error is evaluated after an ideal prefix. Its actual suffix is unitary and preserves norm.

Displayed formulas in this passage

unlabeled · line 541
$∥ (A_{j} - V_{j}) P_{j - 1} ∥_{\infty} \leq δ_{j},$
unlabeled · line 545
$∥ (A_{L} \dots A_{1} - V_{L} \dots V_{1}) P_{0} ∥_{\infty} \leq j = 1 \sum^{L} δ_{j} .$
unlabeled · line 553
$A_{L} \dots A_{1} - V_{L} \dots V_{1} = j = 1 \sum^{L} A_{L} \dots A_{j + 1} (A_{j} - V_{j}) V_{j - 1} \dots V_{1} .$

Lemma: Telescoping on an ideal clean prefix

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