TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generated

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A single primal-dual crossing anticommutes

Statement

Let $P$ be a primal path and $Q$ a dual path whose edge supports share exactly one physical qubit. Then their string Paulis anticommute: $Z (P) X (Q) = - X (Q) Z (P)$ . This is the algebraic mutual e/m string sign.

Facts & Assumptions

Given: A primal Z string and a dual X string with exactly one shared edge-qubit.

[F1]

Open toric strings have endpoint syndromes: $Z (P)$ is Z on the primal path edges and $X (Q)$ is X on the primal edge-qubits crossed by the dual path.

[F2]

Pauli commutator is symplectic pairing: two Pauli strings commute with sign $(- 1)^{ω}$ , where $ω$ is the parity of their X/Z support overlap.

Proof

1.1givenF1F2algebra

The X and Z supports meet at exactly one qubit, so their binary symplectic pairing is $1 \in F_{2}$ . By [F2], $Z (P) X (Q) = (- 1)^{1} X (Q) Z (P) = - X (Q) Z (P)$ .

2.1givenstep 1.1∎

The sign is obtained from the finite Pauli commutator alone. No trajectory, Hamiltonian evolution, or dynamical exchange is part of the hypotheses, so the result asserts the mutual string algebra and no braiding experiment.

Remarks

The sign matches the conventional e/m mutual-statistics label for the toric code. The theorem proves the finite operator identity only.

Depends on

Used by

Two anticommuting toric logical Pauli pairs Theorem

Dependency tree · two levels

7 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)