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Noncontractible toric logical loops

Definition

On the periodic torus, define four Pauli representatives

${\overset{ˉ}{Z}}_{1} = \prod_{i \in Z_{L}} Z_{h_{i, 0}}, {\overset{ˉ}{Z}}_{2} = \prod_{j \in Z_{L}} Z_{v_{0, j}}, {\overset{ˉ}{X}}_{1} = \prod_{j \in Z_{L}} X_{h_{0, j}}, {\overset{ˉ}{X}}_{2} = \prod_{i \in Z_{L}} X_{v_{i, 0}} .$

The $\overset{ˉ}{Z}$ supports are the two fixed fundamental primal cycles. The $\overset{ˉ}{X}$ supports are the dual cycles crossing them. These operators are called the toric logical loops. As verified below, each is a nontrivial logical Pauli class in the periodic toric code.

Facts & Assumptions

Given: The periodic toric checks and the four specified Pauli loop representatives.

[F1]

Toric star and plaquette checks: stars are X on the four incident edges and plaquettes are Z on the four boundary edges.

[F2]

Pauli commutator is symplectic pairing: a Pauli commutes with a check exactly when its binary symplectic pairing with that check is zero; for these pure X/Z checks this is the parity of the shared support.

[F3]

Toric cycle classes are determined by two winding parities: the two primal fundamental loops have winding pairs $(1, 0)$ and $(0, 1)$ ; the two dual loops have the corresponding nonzero dual winding classes; a cycle is a boundary exactly when both winding parities vanish.

[F4]

Toric incidence maps have exactly one relation each: the only star-support relation and the only face-boundary relation are their respective total sums.

[F5]

The periodic toric code encodes two qubits: the common check eigenspace is nonzero and has dimension four.

[F6]

Stabilizer syndromes and logical Pauli operators: logical Pauli classes are normalizer elements modulo scalar phases and stabilizers.

[F7]

Periodic square cellulation of the torus: indices are modulo $L$ ; horizontal edges join consecutive vertices in their first coordinate and vertical edges join consecutive vertices in their second coordinate. The dual edges cross the corresponding primal edges and join the adjacent faces.

[F8]

Toric checks commute and have two global relations: the product of all stars and the product of all plaquettes are identity, and all checks commute.

Verification

1.1givenF1F2F3F4F6F7F8algebra

The supports of ${\overset{ˉ}{Z}}_{1}$ and ${\overset{ˉ}{Z}}_{2}$ close because their successive horizontal and vertical edges meet at vertices modulo $L$ , by [F7]. The supports of ${\overset{ˉ}{X}}_{1}$ and ${\overset{ˉ}{X}}_{2}$ close on the dual graph because the crossed edges join successive adjacent faces modulo $L$ , also by [F7]. By [F1] and [F2], a primal Z cycle commutes with every star and plaquette, and a dual X cycle commutes with every plaquette and star. Their winding pairs are nonzero by [F3], so none is a boundary. If a pure Z loop were in the stabilizer, its X support would be zero; by [F4] the selected star factors would either be empty or the full star family, and [F8] makes that full product $I$ . The loop would therefore be a product of plaquettes, contrary to its nonzero winding. Interchanging X and Z gives the same conclusion for the dual loops. Hence all four lie in $N (S) ∖ Z_{n} S$ and define nontrivial logical classes by [F6].

2.1F5step 1.1∎

The code is nonzero by [F5], so these normalizer classes act on its four-dimensional code space. The four formulas fix the representatives whose pairings are calculated in the subsequent logical-algebra result.

Remarks

The commutation pairings for these fixed representatives are listed in Two anticommuting toric logical Pauli pairs. This is an orientation-only forward reference; the verification above does not depend on that theorem.

Depends on

Used by

Planar and periodic surface-code distances differ Counterexample
Pauli-path energy barrier Definition
L=3 torus checks and syndrome calculation Example
A toric logical string has constant energy barrier Theorem
Toric code distance is L Theorem
Two anticommuting toric logical Pauli pairs Theorem

Dependency tree · two levels

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Sources

Alexei Kitaev, Fault-tolerant quantum computation by anyons (standard reference, not scraped)
Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill, Topological quantum memory (standard reference, not scraped)