TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Crossing strings and local scalar compression

Physical scope

Exact finite square-torus stabilizer algebra only; no perturbation-stability or thermodynamic-limit claim.

Statement

Assume the Axiom of Choice and use the square $L \times L$ torus cellulation, $L \geq 3$ . A closed primal $Z$ string and a closed dual $X$ string commute or anticommute according to the parity of their edge crossings: their commutator sign is $(- 1)^{I}$ , where $I$ is the mod-two intersection number. Let $P_{0}$ be the orthogonal projection onto the common $+ 1$ eigenspace of all vertex and face stabilizers. For any operator $O$ supported on an edge set contained in a disk whose complement contains the coordinate cycles $p_{1}, p_{2}, q_{1}, q_{2}$ specified in the definition, there is a scalar $λ (O) \in C$ such that $P_{0} O P_{0} = λ (O) P_{0}$ . This is exact local indistinguishability for the unperturbed finite Hamiltonian.

Statement references

A. Yu. Kitaev, “Fault-tolerant quantum computation by anyons,” §1

John Preskill, Lecture Notes for Physics 219: Quantum Computation, §§9.6–9.7

Facts & Assumptions

Given: A square $L \times L$ torus with $L \geq 3$ , the toric-code operators of the definition, the common ground projection $P_{0}$ , and the Axiom of Choice inherited by the ground-space theorem.

[A1]

The Axiom of Choice (The Axiom of Choice), used only through the surface Euler-characteristic input in [F2].

[F1]

Toric-code edge spins on closed-surface cellulations: $X (a) Z (b)$ is the canonical Pauli basis; $A_{v} = X (\partial_{1}^{T} v)$ , $B_{f} = Z (\partial_{2} f)$ ; primal and dual crossing parity is the mod-two intersection number.

[F2]

Commuting stabilizers and topological degeneracy: under [A1], the common $+ 1$ eigenspace is nonzero, is the ground space, and has dimension $2^{2 g}$ .

[F3]

Disk-supported Pauli centralizers are stabilizers: a Pauli string supported on the stated disk-local set and commuting with every stabilizer is a product of stabilizers.

Proof

1.1F1algebra

On each edge, $X Z = - Z X$ , while Pauli operators on distinct edges commute. Moving the $X$ factors of a dual string past the $Z$ factors of a primal string therefore contributes one minus sign for each shared crossed edge. The total sign is $(- 1)^{I}$ , with $I$ the number of crossings modulo two.

1.2givenA1F1F2algebra

Let $T \subseteq E$ be a finite edge support for $O$ contained in the disk from the statement. On one edge, $I, X, Z, X Z$ are four linearly independent matrices in the four-dimensional space $M_{2} (C)$ ; their tensor products indexed by $a, b \in F_{2}^{T}$ form a basis of the matrix algebra on $(C^{2})^{\otimes T}$ . Extend $a, b$ by zero on $E ∖ T$ and expand $O = \sum_{a, b} c_{a, b} X (a) Z (b)$ in that basis. For a Pauli term $Q$ that anticommutes with some stabilizer $S$ , the ground projection satisfies $P_{0} S = S P_{0} = P_{0}$ and $S Q = - Q S$ . Hence $P_{0} Q P_{0} = P_{0} S Q P_{0} = - P_{0} Q S P_{0} = - P_{0} Q P_{0}$ , so $P_{0} Q P_{0} = 0$ .

2.1givenA1F2F3step 1.2algebra∎

Every Pauli term in the expansion either anticommutes with a stabilizer, and has zero compression by step 1.2, or commutes with every stabilizer. In the latter case [F3] writes it as a product of stabilizers; each acts as the identity on the range of $P_{0}$ , so its compression is $P_{0}$ . Summing the finitely many compressed terms gives $P_{0} O P_{0} = λ (O) P_{0}$ for the sum of their coefficients.

Remarks

The proof is independent of the diagram. The picture shows one representative pair of fundamental cycles on the square fundamental domain; opposite sides are identified, the blue solid path is primal, and the red dashed path is dual.

At the displayed crossing the primal and dual strings meet on one edge, so their intersection parity is one and their Pauli strings anticommute. A diagram illustrates the convention; it supplies no proof step. No perturbation-stability claim follows from this exact compression identity.

Crossing strings and local scalar compression

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources