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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].

Depends on

Used by

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Dependency tree · two levels

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Sources

Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill, Topological quantum memory (standard reference, not scraped)
John Preskill, Quantum Computation, Chapter 9: Topological Quantum Computation (standard reference, not scraped)