DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-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.

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Toric e and m syndromes

Definition

For a nonzero vector $ψ$ that is a simultaneous eigenvector of all toric checks, its e syndrome is the set of vertices $q$ for which $A_{q} ψ = - ψ$ , and its m syndrome is the set of faces $f$ for which $B_{f} ψ = - ψ$ . A violated star is called an e defect and a violated plaquette is called an m defect. These are finite stabilizer-syndrome labels; the names do not assume that a material contains localized physical quasiparticles.

These eigenvalue labels agree with the commutation-sign syndrome of Stabilizer syndromes and logical Pauli operators when a Pauli error acts on a common $+ 1$ check eigenvector, as verified below.

The checks are Hermitian involutions, as verified below; thus their eigenvalues on any nonzero simultaneous eigenvector are indeed signs.

Facts & Assumptions

Given: The toric checks on the finite edge register and their simultaneous-eigenvalue convention.

[F1]

Toric star and plaquette checks: every toric check is a Hermitian involution.

[F2]

Stabilizer syndromes and logical Pauli operators: a syndrome bit is $0$ for commutation sign $+ 1$ and $1$ for sign $- 1$ .

Verification

1.1givenF1algebra

A Hermitian involution has only eigenvalues $+ 1$ and $- 1$ . Therefore each check eigenvalue on the given nonzero simultaneous eigenvector is one of these two signs, so the two sets in the Definition are well-defined finite subsets of vertices and faces.

2.1F1F2step 1.1algebra∎

Let $ϕ \neq 0$ be a common $+ 1$ check eigenvector, and let $E$ be a Pauli error. For any check $g$ , write $E g = (- 1)^{s} g E$ with $s \in F_{2}$ . Then $g (E ϕ) = (- 1)^{s} E (g ϕ) = (- 1)^{s} E ϕ$ . Since $E$ is invertible, $E ϕ \neq 0$ ; its negative-check sets therefore consist exactly of the checks with commutation bit $s = 1$ . On an independent generating list these are the syndrome bits of [F2]. Thus the eigenvalue labels defined here are compatible with that error-syndrome convention, with stars and plaquettes kept separate.

Depends on

Used by

L=3 torus checks and syndrome calculation Example
Open toric strings have endpoint syndromes Lemma
Toric checks decompose into joint syndrome eigenspaces Lemma
Exact finite-volume toric-code gap Theorem

Dependency tree · two levels

11 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

Alexei Kitaev, Fault-tolerant quantum computation by anyons (standard reference, not scraped)
John Preskill, Quantum Computation, Chapter 9: Topological Quantum Computation (standard reference, not scraped)