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Quantum Tanner Random-Model Parity Expander

Claim/Theorem

For the existence-style random local-code model of Leverrier and Zemor's Theorem 18, the chosen parity Tanner graphs are local expanders with nonzero probability.

This is now an immediate corollary of [[quantum-tanner-theorem17-parity-expander.md]]. Theorem 18 states that with nonzero probability the random component codes \(C_A,C_B\) satisfy all hypotheses of Theorem 17. Whenever that event occurs, the deterministic theorem node [[quantum-tanner-theorem17-parity-expander.md]] applies and yields local expansion for the chosen parity Tanner graphs.

Hence the random model no longer needs a separate gadget-specific argument. It inherits the parity-expander conclusion from the stronger deterministic conditional theorem on the graph.

Dependencies

  • [[quantum-tanner-theorem17-parity-expander.md]]

Conflicts/Gaps

  • This remains an existence-style corollary because Theorem 18 is probabilistic. The stronger deterministic conditional content now lives in [[quantum-tanner-theorem17-parity-expander.md]].
  • The node still does not supply an explicit deterministic component-code construction satisfying Theorem 17.
  • As elsewhere on the graph, the conclusion is about a chosen parity Tanner presentation rather than every stabilizer presentation of the code.

Sources

  • 10.48550/arXiv.2202.13641
  • 10.1007/BF02579166
  • 10.48550/arXiv.2109.14599