QLDPC CSS Constituent Codes Are Not Good

Claim/Theorem

For a CSS quantum LDPC code presented by sparse parity-check matrices

\[ C_0=\ker H_0, \qquad C_1=\ker H_1, \qquad C_0^\perp \subseteq C_1, \]

the constituent classical codes \(C_0\) and \(C_1\) are not asymptotically good.

Reason:

1. Sparsity of \(H_0\) and \(H_1\) means that the dual codes \(C_0^\perp\) and \(C_1^\perp\) contain many constant-weight words, namely the LDPC parity checks themselves.
2. The CSS inclusion implies

\[ C_1 \supseteq C_0^\perp, \qquad C_0 \supseteq C_1^\perp. \]

1. Therefore both \(C_0\) and \(C_1\) themselves contain constant-weight words, so their classical minimum distances are bounded by the LDPC check weight rather than growing linearly with blocklength.

In particular, any route to Conjecture 3 that tries to lower-bound depth through classical-code parameters of the measured constituent codes \(C_0\) or \(C_1\) cannot by itself recover the sharp static-2D Omega(sqrt(n)) barrier for Quantum Tanner codes. One needs either:

• a different invariant than classical distance for those constituent codes,
• or a direct stabilizer-space argument such as the expander-cut route already on the graph,
• or a theorem controlling intrinsic cut-rank for the full stabilizer space rather than for one constituent classical code.

Dependencies

• None.

Conflicts/Gaps

• This only rules out using asymptotic goodness of the constituent classical codes as the input to Wolf-type trellis-width bounds. It does not show that those constituent codes have small trellis width, only that linear distance is unavailable as a proof route.
• The statement is phrased for CSS qLDPC constructions. Translating it to more general non-CSS stabilizer presentations would require additional bookkeeping.
• The node explains why the newer cutwidth and branchwidth routes are conditional rather than decisive for Quantum Tanner families, but it does not weaken the older expander-based lower bounds already on the graph.

Sources

• 10.48550/arXiv.2202.13641