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Quantum Tanner Local Generator Blowup

Claim/Theorem

For the explicit stabilizer presentation used by Leverrier and Zemor, the quantum Tanner code is built from local generators rooted at vertices of the left-right Cayley complex. Choosing bases \(\beta_0\) of \(C_0=C_A\otimes C_B\) and \(\beta_1\) of \(C_1=C_A^\perp\otimes C_B^\perp\), the construction assigns:

  • for each vertex \(v\in V_0\), exactly \(\dim C_0\) Z-type generators supported inside the square-neighborhood \(Q(v)\),
  • for each vertex \(v\in V_1\), exactly \(\dim C_1\) X-type generators supported inside \(Q(v)\).

Each generator has weight at most \(\Delta^2\), and each qubit is contained in at most \(\Delta^2\) generators. Consequently, the stabilizer Tanner graph of this chosen presentation is a constant-size local blow-up of the vertex-square incidence graph of the left-right Cayley complex: every check node is attached to one vertex neighborhood \(Q(v)\), and no generator reaches outside its root neighborhood.

This identifies the exact missing transfer step for the explicit-family route to Conjecture 3: even after upgrading the underlying one-parity incidence graph to the spectral statement in [[quantum-tanner-incidence-spectral-gap.md]], one still needs a theorem that turns this incidence-level expansion into the small-set Tanner expansion hypothesis required by [[tanner-to-contracted-expansion-transfer.md]] for the chosen local blow-up.

Dependencies

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

Conflicts/Gaps

  • A bounded local blow-up does not automatically preserve the required small-set Tanner expansion for arbitrary choices of the local bases \(\beta_0,\beta_1\).
  • The node isolates the structure of the chosen stabilizer presentation, but it does not prove that this presentation is locally expanding in the sense needed by [[tanner-to-contracted-expansion-transfer.md]].
  • Different choices of generator basis inside the same local code can change the literal Tanner graph while preserving the code.

Sources

  • 10.48550/arXiv.2202.13641