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02 - Quantum Tanner Explicit-Family Stack

Learning Target

After this chapter, you should understand why the chosen Quantum Tanner presentation remains important even though the minimal static 2D theorem can be proved from code parameters.

Why The Explicit Stack Still Matters

The parameter route proves a clean lower bound, but it hides the syndrome-demand geometry. The conjectured CD(T_n,G) route needs a more structural object: something like the demand that the code places on the hardware.

Quantum Tanner codes provide a concrete construction where that demand can be studied. The local graph organizes the construction as a stack:

  1. diagonal graphs from the left-right Cayley complex;
  2. spectral or incidence expansion;
  3. local tensor-code generators;
  4. a constant-size local blow-up from incidence structure to chosen stabilizer checks;
  5. a parity-Tanner local-expander statement for the chosen presentation.

This stack is not just background. It is where the intrinsic and compiler-native routes try to extract a theorem-sized demand object.

Worked Picture

Think of an expander graph whose vertices are replaced by constant-size gadgets. If the original graph expands and each gadget has a positive boundary property, then small nontrivial sets inside many gadgets should expose many boundary edges.

The Quantum Tanner stack follows the same shape:

  • global expansion comes from the diagonal or incidence graph;
  • local tensor codes define constant-size check gadgets;
  • no-zero-coordinate and connected-basis conditions prevent local checks from becoming invisible on one side of a cut;
  • the local-generator blow-up transfers expansion from the incidence structure to the chosen parity-Tanner presentation.

The important word is "chosen." This is a theorem about a presentation, not automatically about every generator set for the same stabilizer space.

The Local Graph Chain

The graph-backed stack is:

The high-level implication is:

diagonal or incidence expansion plus local gadget conditions gives parity-Tanner local expansion for the chosen generator presentation.

What This Buys

The chosen-presentation theorem gives a direct version of the expander-versus-grid story:

  • balanced cuts in the Tanner-like demand structure are large;
  • low-separator hardware cannot service them quickly;
  • the proof shape resembles the original intuitive CD story.

It also provides concrete local blocks \(H(v)\) that later become relevant for local quotient-image accumulation.

What This Does Not Buy

It does not immediately prove the full invariant frontier:

  • a different stabilizer generator set can change the Tanner graph;
  • local expansion does not automatically imply large cross-cut stabilizer rank;
  • tester-side expansion does not automatically become original-qubit matroid connectivity;
  • many local crossed blocks can cancel or overlap in the global quotient.

Those limitations are why the next chapter introduces stabilizer cut rank.

Active Recall

  1. Why does the Quantum Tanner explicit stack remain useful after the static theorem?
  2. What role do no-zero-coordinate and connected-basis conditions play?
  3. Why is a chosen-presentation theorem not the same as a stabilizer-space invariant theorem?
  4. Which later frontier uses the local blocks \(H(v)\)?

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