DefinitionDefinition: Literature-sourcedProof: AI-adaptedPipeline-generated

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.
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Logical block error and leakage amplitude

Definition

Let $J : C \to V$ be a fixed-charge encoding, $P = J J^{*}$ its code projector, $U \in U (C)$ a logical target, and $W \in U (V)$ an ambient unitary. Its zero-on-the-complement target extension is $\tilde{U} : = J U J^{*} : V \to V$ . Define the logical block error and leakage amplitude by $ϵ_{l o g} (W, U) : = ∥ P W P - \tilde{U} ∥_{o p}, ϵ_{l e a k} (W) : = ∥ (I - P) W P ∥_{o p} .$ Both are finite operator norms. The first measures the compressed block against the target; the second measures transfer from the code into its orthogonal complement. They diagnose different errors.

Facts & Assumptions

Given: A finite-dimensional encoding $J : C \to V$ , its projector $P$ , a logical unitary $U$ , and an ambient unitary $W$ .

[F1]

$P$ is the orthogonal projection onto $J C$ . Fixed-charge encoding and code projector

[F2]

The operator norm is finite for every linear map between finite-dimensional inner-product spaces. The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||

Proof

1.1givenF1F2

The maps $P W P - \tilde{U}$ and $(I - P) W P$ are endomorphisms of $V$ . Because $V$ is finite dimensional, [F2] assigns a finite operator norm to each map; the two quantities are therefore well defined.

2.1givenF1F2algebra∎

For a zero-leakage but logically incorrect boundary case, take $V = C = C^{2}$ , $J = P = I$ , $W = I$ , and $U = diag (1, - 1)$ . Then $ϵ_{l e a k} = 0$ and $ϵ_{l o g} = ∥ I - U ∥_{o p} = 2$ . This proves the two quantities are not interchangeable.

Depends on

Used by

Dependency tree · two levels

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Sources

Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma, Non-Abelian Anyons and Topological Quantum Computation (standard reference, not scraped)
John Preskill, Lecture Notes for Physics 219: Quantum Computation, Chapter 9: Topological Quantum Computation (standard reference, not scraped)