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Quantum Tanner Expander Anchor

Claim/Theorem

Within the left-right Cayley-complex construction of quantum Tanner codes, Leverrier and Zemor prove a theorem-level good-family anchor rather than just a heuristic expander analogy. In the arXiv preprint, Theorem 17 (p. 23) shows that if the component codes C_A, C_B satisfy explicit rate, distance, and dual-tensor robustness hypotheses, then the resulting CSS code Q = (C_0, C_1) has parameters n, k >= (1 - 2rho)^2 n, and d >= (delta / (4 Delta^(3/2 + epsilon))) n. Theorem 18 (p. 24) then shows that these hypotheses are simultaneously achievable with nonzero probability for random component-code choices, yielding an infinite asymptotically good quantum Tanner QLDPC family. This supplies the concrete expander-style family anchor for Conjecture 3, while leaving the static-2D depth barrier to downstream nodes.

Dependencies

  • [[quantum-tanner-diagonal-expansion-structure.md]]
  • [[quantum-tanner-incidence-spectral-gap.md]]

Conflicts/Gaps

  • The paper proves good code parameters, not a lower bound on static-2D implementation depth.
  • The expansion is stated at the level of the left-right Cayley complex and its associated Tanner constructions; [[quantum-tanner-diagonal-expansion-structure.md]] and [[quantum-tanner-incidence-spectral-gap.md]] sharpen this to explicit spectral expansion of the diagonal graphs and their one-parity incidence graph, but not yet to the exact small-set expansion hypothesis required by [[tanner-to-contracted-expansion-transfer.md]].
  • This node is an anchor family for Conjecture 3, not yet a proof that every good QLDPC family has the same routing obstruction.

Sources

  • 10.48550/arXiv.2202.13641 (Theorem 17, p. 23; Theorem 18, p. 24; verified against the local PDF bundle built from the Zotero attachment)
  • 10.1109/FOCS54457.2022.00117 (FOCS publication record for the same construction/result family)