DefinitionDefinition: AI-adaptedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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.
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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Stabilizer distance

Definition

Let $n \geq 1$ and let $S$ be a stabilizer code on $n$ qubits whose code space has dimension $2^{k}$ . The stabilizer distance is

$d (S) : = \min {wt (E) : E \in N (S) ∖ Z_{n} S} if k > 0.$

This is a set difference: it excludes scalar-phase multiples of stabilizers. If $k = 0$ , define $d (S) = + \infty$ , since there is no nontrivial logical Pauli class.

Facts & Assumptions

Given: A finite stabilizer group $S$ on $n$ qubits, its logical Pauli quotient, and the quotient weight inherited from Pauli support.

[F1]

Dimension of the logical Pauli quotient: when the code encodes $k$ qubits, $N (S) / (Z_{n} S)$ has binary dimension $2 k$ .

[F2]

Support and weight of an operator on a finite register: every Pauli has a well-defined integer weight between $0$ and $n$ .

Verification

1.1F1F2construct

If $k > 0$ , [F1] gives a quotient of positive dimension, so it contains a nonidentity class. Any representative is a Pauli in the finite group $P_{n}$ , and [F2] assigns it an integer weight in ${0, \dots, n}$ . Thus the set in the minimum is nonempty and finite, so its minimum exists.

2.1F1algebra∎

If $k = 0$ , [F1] gives a zero-dimensional binary quotient, hence the quotient is trivial and $N (S) = Z_{n} S$ . Therefore the set difference defining the finite-distance branch is empty, which is why the definition assigns $+ \infty$ .

Remarks

The weight is the number of physical qubits on which the Pauli is nonidentity. Scalar phases do not change it. The product set $N (S) S$ is not the definition of logical nontriviality; the relevant set is $N (S) ∖ Z_{n} S$ .

Depends on

Used by

Fault-Tolerant Gadget and Proper-Code Contract Definition
Twice-Concatenated Steane Code and Its Transversal Clifford Gates Lemma
CSS syndrome correction Theorem
Square planar surface-code distance and path count Theorem
Stabilizer distance detection and correction Theorem
Toric code distance is L Theorem

Dependency tree · two levels

20 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

John Preskill, Quantum Computation, Chapter 7: Quantum Error Correction (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, §20.1 (standard reference, not scraped)