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Quantum Tanner Diagonal Expansion Structure

Claim/Theorem

Leverrier and Zemor's quantum Tanner codes are explicitly CSS pairs of Tanner codes on the two diagonal graphs of a left-right Cayley complex:

\[ C_0=T(G^\square_0,C_0^\perp), \qquad C_1=T(G^\square_1,C_1^\perp). \]

If the left and right Cayley graphs \(G_A\) and \(G_B\) are Ramanujan, then their Lemma 4 shows that the diagonal graphs \(G^\square_0\) and \(G^\square_1\) are \(\Delta^2\)-regular and satisfy

\[ \lambda(G^\square_0),\;\lambda(G^\square_1)\;\le\;4\Delta. \]

Hence explicit quantum Tanner families are built from bounded-degree classical Tanner ingredients whose underlying graphs are genuine spectral expanders. This sharpens the Conjecture-3 anchor family: the remaining issue is no longer whether the construction contains explicit expanders, but how this auxiliary expansion transfers to the Tanner graph of a chosen stabilizer presentation in the precise sense needed by [[tanner-to-contracted-expansion-transfer.md]].

Dependencies

  • None.

Conflicts/Gaps

  • The expansion here is for the auxiliary diagonal graphs \(G^\square_0\) and \(G^\square_1\), not directly for the stabilizer-generator Tanner graph of a fixed presentation of the quantum code.
  • Spectral expansion of the diagonal graphs does not by itself identify the small-set Tanner expansion hypothesis used in [[tanner-to-contracted-expansion-transfer.md]].
  • The good-parameter theorem for the full quantum code still depends on robust local component codes and left-right Cayley-complex assumptions beyond pure spectral expansion.

Sources

  • 10.48550/arXiv.2202.13641