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Toric Code Anyons, Logical Operators, and Protection — Examples

1 · Prerequisites

2 · Summary

The periodic square fundamental domain

The picture shows a $3 \times 3$ fundamental square for the periodic cellulation. Opposite boundary vertices and edge segments with matching transverse index are identified. Indices are read modulo $3$ ; the exact vertex, edge and face incidences are given by Periodic square cellulation of the torus. The endpoint notation in that definition specifies each edge, while chains and check supports use coefficients in $F_{2}$ and hence do not depend on orientation.

\begin{tikzpicture}[scale=0.78, every node/.style={font=\small}]
  \draw[gray!65] (0,0) grid (3,3);
  \foreach \i in {0,1,2,3} {
    \foreach \j in {0,1,2,3} {
      \fill (\i,\j) circle (1.5pt);
    }
  }
  \node[below left] at (0,0) {$q_{0,0}$};
  \node[above right] at (3,3) {$q_{0,0}$};
  \node[below] at (1.5,0) {$h_{i,0}$};
  \node[left] at (0,1.5) {$v_{0,j}$};
  \draw[->,thick,blue] (-0.20,3.38) -- (3.20,3.38)
    node[midway,above] {identify left/right, same $j$};
  \draw[->,thick,red] (3.43,-0.18) -- (3.43,3.18)
    node[midway,right,align=left] {identify bottom/top,\ same $i$};
  \node[below] at (1.5,-0.50) {$i=0,1,2$};
  \node[rotate=90] at (-0.55,1.5) {$j=0,1,2$};
\end{tikzpicture}

The two arrows show the quotient identifications only; they are not extra edges. The four corner vertices represent one torus vertex after the paired identifications. The example below lists all eighteen edge qubits and checks the length-three logical loop directly, so no proof depends on the drawing.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: AI-adaptedVerification: AI-generatedOpen item page →

L=3 torus checks and syndrome calculation

Example

Take $L = 3$ , with all indices in $Z / 3 Z$ . The edge qubits are the eighteen distinct labels ${h_{i, j} : i, j \in {0, 1, 2}} ⊔ {v_{i, j} : i, j \in {0, 1, 2}} .$ There are eighteen checks $A_{i, j}, B_{i, j}$ , given by the star and plaquette formulas in Toric star and plaquette checks. The displayed horizontal primal loop is ${\overset{ˉ}{Z}}_{1} = Z_{h_{0, 0}} Z_{h_{1, 0}} Z_{h_{2, 0}}$ .

Facts & Assumptions

Given: The periodic $3 \times 3$ cellulation and the toric check, code-space, string-syndrome, and Hamiltonian conventions.

[F1]

Periodic square cellulation of the torus: each edge label is distinct modulo $3$ , and the incidence formulas use indices in $Z / 3 Z$ .

[F2]

Toric star and plaquette checks: $A_{i, j}$ is X on the four edges at $q_{i, j}$ and $B_{i, j}$ is Z on the four boundary edges of $f_{i, j}$ .

[F3]

Toric checks commute and have two global relations: all checks commute, and the products of all stars and of all plaquettes are each $I$ .

[F4]

Toric incidence maps have exactly one relation each: each check type has rank $3^{2} - 1 = 8$ and only its total relation.

[F5]

The periodic toric code encodes two qubits: the code has $18$ physical qubits, check rank $16$ , and dimension $2^{18 - 16} = 4$ .

[F6]

Noncontractible toric logical loops: ${\overset{ˉ}{Z}}_{1}$ is a nontrivial logical Pauli supported on $h_{i, 0}$ for $i \in Z_{3}$ .

[F7]

Open toric strings have endpoint syndromes: an edge-simple open primal Z trail or dual X trail has the respective defects at exactly its two endpoints; a closed one has no defects.

[F8]

Exact finite-volume toric-code gap: for couplings $J_{e}, J_{m} > 0$ , the exact gap is $4 \min (J_{e}, J_{m})$ and the ground energy is $- 9 (J_{e} + J_{m})$ .

[F9]

Finite toric-code Hamiltonian: for $L = 3$ the Hamiltonian is $- J_{e} \sum_{i, j} A_{i, j} - J_{m} \sum_{i, j} B_{i, j}$ .

[F10]

Toric checks decompose into joint syndrome eigenspaces: every nonzero syndrome sector has an even number of e defects and an even number of m defects.

[F11]

Toric e and m syndromes: e and m defects are respectively negative star and plaquette eigenvalues.

Verification

1.1givenF1F2algebra

There are $3 \cdot 3 = 9$ horizontal and nine vertical edge labels; they are disjoint by [F1], so there are $18$ qubits. Likewise there are nine stars and nine plaquettes, hence $18$ checks in the displayed list.

1.2F3F4F5algebra

Every edge occurs in two stars, so the product of all nine stars is $I$ ; every edge also occurs in two plaquettes, so their total product is $I$ , as [F3] records. By [F4], deleting $A_{0, 0}$ and $B_{0, 0}$ leaves $8 + 8 = 16$ independent checks, with no other relation. Thus [F5] gives a four-dimensional code space.

1.3F1F6F7constructalgebra

The three edges $h_{0, 0}, h_{1, 0}, h_{2, 0}$ form a closed horizontal primal loop. Applying $Z_{h_{0, 0}}$ first gives e defects at $q_{0, 0}$ and $q_{1, 0}$ . Applying $Z_{h_{1, 0}}$ next leaves only the two endpoints $q_{0, 0}$ and $q_{2, 0}$ . Applying $Z_{h_{2, 0}}$ closes the trail, leaving no defects. These are precisely the endpoint and closed-trail cases of [F7]; [F6] says the final loop is logically nontrivial.

2.1F3F7F8F9F10F11algebra∎

For a joint sector with $n_{e}$ negative stars and $n_{m}$ negative plaquettes, [F9] gives energy $- 9 (J_{e} + J_{m}) + 2 J_{e} n_{e} + 2 J_{m} n_{m}$ , using [F11] to count the negative signs. By [F10], any nonzero excited sector has $n_{e} \geq 2$ or $n_{m} \geq 2$ , so it lies at least $4 \min (J_{e}, J_{m})$ above the code. A one-edge Z string attains $4 J_{e}$ and a one-edge X string attains $4 J_{m}$ ; hence the smaller value is attained, in agreement with [F8].

CounterexampleConstruction: AI-adaptedVerification: AI-generatedOpen item page →

Planar and periodic surface-code distances differ

Statement

At linear size three, the square planar surface code has asymmetric Z and X distances $3$ and $4$ , whereas the periodic $3 \times 3$ toric code has distance $3$ for both logical species and encodes two qubits. Thus the planar surface-code calculation cannot be used as the periodic torus calculation.

Facts & Assumptions

Given: The relative square-patch convention and the periodic toric-code convention, both with linear size three.

[F1]

Square planar surface code: the planar patch has left and right rough boundaries; its physical qubits and checks use relative boundary maps, not periodic incidence.

[F2]

Square planar surface-code distance and path count: for a $d \times d$ planar patch, the code encodes one qubit, has Z distance $d$ , and has X distance $d + 1$ .

[F3]

Toric code distance is L: the periodic $L \times L$ toric code has distance $L$ for both primal and dual nontrivial cycles.

[F4]

The periodic toric code encodes two qubits: the periodic toric code encodes two qubits.

[F5]

Noncontractible toric logical loops: the periodic code has nontrivial primal Z and dual X loop representatives, each supported on $L$ edges.

Counterexample

1.1givenF2F3F4F5algebra

Set $d = L = 3$ . By [F2], the planar code has Z distance $3$ and X distance $4$ , and encodes one qubit. For the toric code, [F3] gives the lower bound $3$ for every nontrivial logical Pauli, while [F5] supplies a nontrivial three-edge representative of each pure species. Thus both species have distance $3$ ; [F4] gives code dimension four. The distance profiles and code dimensions differ at the same linear size.

2.1F1F2F3algebra∎

The reason is already present in the hypotheses: [F1] uses relative boundary maps and rough edges, while [F3] uses periodic incidence on a closed torus. Therefore planar boundary paths and their asymmetric distance theorem do not establish torus winding classes, check relations, or toric distance. The distinction is a finite combinatorial one and does not require a physical interpretation.

Toric Code Anyons, Logical Operators, and Protection — Examples

1 · Prerequisites

2 · Summary

The periodic square fundamental domain

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

L=3 torus checks and syndrome calculation

Example

Facts & Assumptions

Verification

Planar and periodic surface-code distances differ

Statement

Facts & Assumptions

Counterexample

Sources