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2D QEC Overhead From Logical Error Target

Claim/Theorem

For a 2D-local error-correction procedure on \(m\) physical qubits protecting \(k\) logical qubits, with arbitrary error-free classical computation and constant depolarizing noise strength \(p>0\), Baspin, Fawzi, and Shayeghi prove that achieving logical error target \(\delta=p^f\) requires

\[ \frac{m}{k}\;\in\;\Omega(\sqrt{f}) \;=\; \Omega\!\left(\sqrt{\frac{\log(1/\delta)}{\log(1/p)}}\right). \]

Equivalently, constant overhead in 2D forbids arbitrarily strong logical-error suppression. This is the closest existing rigorous theorem on the current graph to the “threshold consequence” half of Conjecture 3.

Dependencies

  • None.

Conflicts/Gaps

  • The theorem concerns a noisy error-correction module, not one ideal syndrome-extraction round. It therefore supports the consequence part of Conjecture 3 more directly than the CD(T_n,\mathfrak G) lower bound itself.
  • The statement is proved for geometrically local 2D operations with free classical computation. Extending it to arbitrary hardware graph families would require a replacement for the Euclidean geometric step.
  • The result lower-bounds space overhead as a function of target logical error, but it does not yet identify the correct architecture-dependent constant \delta_\star from the conjecture.

Sources

  • 10.48550/arXiv.2302.04317