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The periodic toric code encodes two qubits

Statement

For the periodic $L \times L$ toric checks with $L \geq 3$ , the common $+ 1$ eigenspace is a valid stabilizer code on $n = 2 L^{2}$ qubits. Its independent check rank is $r = 2 L^{2} - 2$ , so its dimension is $2^{n - r} = 4 = 2^{2}$ and it encodes two logical qubits.

Facts & Assumptions

Given: An integer $L \geq 3$ , the periodic toric checks, and the common $+ 1$ eigenspace convention.

[F1]

Toric checks commute and have two global relations: the checks are commuting Hermitian involutions; the product of all stars and the product of all plaquettes are both $I$ .

[F2]

Toric incidence maps have exactly one relation each: the star supports have rank $L^{2} - 1$ and their only relation is the total star sum; the face boundaries have rank $L^{2} - 1$ and their only relation is the total face sum.

[F3]

Stabilizer code space: a stabilizer code is the common $+ 1$ eigenspace of an abelian subgroup of Hermitian Pauli operators that does not contain $- I$ .

[F4]

Stabilizer syndromes and logical Pauli operators: the binary Pauli label records the X and Z support vectors separately.

[F5]

Stabilizer projector and code dimension: an $n$ -qubit stabilizer subgroup with $r$ independent generators has a nonzero code space of dimension $2^{n - r}$ .

[F6]

Periodic square cellulation of the torus: the periodic lattice has $L^{2}$ vertices and faces and $2 L^{2}$ edges.

Proof

1.1F1F2F4givenalgebra

Omit the fixed star $A_{0, 0}$ and fixed plaquette $B_{0, 0}$ . By [F2], the remaining $L^{2} - 1$ stars are independent, and the remaining $L^{2} - 1$ plaquettes are independent; [F1] expresses each omitted check as the product of the retained checks of its type. Their combined binary supports are independent because an X-support and a Z-support occupy separate coordinates by [F4]. Thus these checks generate all checks with $r = 2 L^{2} - 2$ independent generators.

2.1F1F2F3F4step 1.1algebra

Let $S$ be their generated subgroup. By [F1] it is abelian and generated by Hermitian Paulis. If a product of selected generators had zero binary label, its X-support and Z-support would each be zero by [F4]. The two independent families in [F2] then force both selected subsets to be empty, so that product is $I$ . In particular, no product is $- I$ , and $- I \notin S$ . Hence $S$ satisfies the stabilizer hypotheses in [F3].

3.1F3F5F6step 1.1step 2.1algebra∎

There are $n = 2 L^{2}$ edge qubits by [F6]. The dimension theorem [F5] therefore gives $\dim C (S) = 2^{2 L^{2} - (2 L^{2} - 2)} = 4 = 2^{2}$ ; [F3] and step 2.1 ensure that its hypotheses hold, and step 1.1 supplies the independent rank. The code is nonzero and encodes exactly two qubits.

Depends on

Used by

A finite toric gap does not imply arbitrary perturbation stability Counterexample
Planar and periodic surface-code distances differ Counterexample
Noncontractible toric logical loops Definition
Pauli-path energy barrier Definition
L=3 torus checks and syndrome calculation Example
Exact finite-volume toric-code gap Theorem
Exact indistinguishability in a contractible disk Theorem
Toric code corrects Pauli errors below half the distance Theorem
Toric code distance is L Theorem
Two anticommuting toric logical Pauli pairs Theorem

Dependency tree · two levels

21 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)
Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill, Topological quantum memory (standard reference, not scraped)