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Toric code corrects Pauli errors below half the distance

Statement

Let $t \in Z_{\geq 0}$ satisfy $2 t < L$ . Every finite Pauli error channel whose Kraus operators lie in the span of Pauli operators of weight at most $t$ has an exact recovery on the periodic $L \times L$ toric code. In particular, every unknown Pauli error of weight at most $t$ is exactly correctable.

Facts & Assumptions

Given: The periodic toric code on $n = 2 L^{2}$ qubits and an integer $t \geq 0$ with $2 t < L$ .

[F1]

Toric code distance is L: the code has stabilizer distance $d (S) = L$ .

[F2]

Stabilizer distance detection and correction: a finite stabilizer error channel with Kraus operators in the span of Paulis of weight at most $t$ has an exact recovery whenever $2 t < d (S)$ .

[F3]

The periodic toric code encodes two qubits: the toric code is a nonzero finite-dimensional stabilizer code.

Proof

1.1givenF3

By [F3], the common toric code space is nonzero and finite-dimensional; it is the stabilizer code defined by the commuting toric checks.

2.1givenF1F2step 1.1algebra∎

By [F1], its distance is $d (S) = L$ . The hypothesis $2 t < L$ is therefore exactly the strict condition $2 t < d (S)$ in [F2]. Applying [F2] gives one finite exact recovery for every error channel in the stated Pauli span, and hence for each Pauli error of weight at most $t$ . No circuit-level noisy procedure or hardware threshold is asserted.

Depends on

Used by

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Dependency tree · two levels

18 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

Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill, Topological quantum memory (standard reference, not scraped)
John Preskill, Quantum Computation, Chapter 7: Quantum Error Correction (standard reference, not scraped)