Hierarchical Memories 2D Threshold Escape¶

Claim/Theorem¶

Pattison, Krishna, and Preskill construct hierarchical codes by concatenating a constant-rate quantum LDPC outer code with a rotated surface-code inner code of size \(d_\ell=\Theta(\log n)\). If the outer code has \(k=\rho n\) and \(d=\Theta(n^\delta)\) with \(\delta\ge 1/2\), then the concatenated family has

\[ K(N)=\Omega\!\left(\frac{N}{\log^2 N}\right), \qquad D(N)=\Omega\!\left(\frac{N^\delta}{\log^{2\delta-1}(N/\log N)}\right), \]

and admits explicit syndrome-extraction circuits using only local Clifford operations and range-\(R\) SWAP gates with width \(O(N)\) and depth \(O(\sqrt{N}/R)\). Repeating those circuits for \(d(n)\) rounds yields a threshold under the paper's local stochastic noise model. This is a concrete 2D-local escape route: threshold behavior can be recovered by hierarchical concatenation, but only after abandoning direct constant-rate QLDPC implementation.

Dependencies¶

None.

Conflicts/Gaps¶

This is not a direct implementation of the original expander-style QLDPC family. It uses concatenation with an inner surface code and therefore changes both the code family and the operational target.
The encoded rate drops from constant to \(K(N)=\Omega(N/\log^2 N)\), so this does not contradict lower bounds aimed at direct constant-rate QLDPC embedding.
The threshold statement is tied to repeated syndrome-extraction rounds and a specific local stochastic noise model, not a single ideal extraction round.

Sources¶

10.22331/q-2025-05-05-1728
10.48550/arXiv.2303.04798