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01 - Current Frontier and Static 2D Barrier

Learning Target

After this chapter, you should be able to state what is already solved for Conjecture 3 and why the remaining frontier is not the basic static 2D lower bound.

The Current Shape Of The Problem

The simplest Conjecture 3 story is:

expander-style QLDPC codes create large nonlocal syndrome demand, while static near-square 2D hardware has small separators.

That story remains correct as intuition. The research graph now makes it sharper:

  • static near-square 2D syndrome extraction is already theorem-backed for theorem-level Quantum Tanner families;
  • the chosen-presentation expansion route explains the mechanism but is not the final invariant language;
  • the live frontier is now about compiler-native CD(T_n,G) semantics, explicit deterministic packaging, and intrinsic cut-rank mechanisms.

This distinction matters. If you keep trying to prove "some 2D lower bound," you are behind the graph. The better question is what exact model and demand object the lower bound should be stated in.

The Static 2D Parameter Route

The cleanest theorem-backed route uses code parameters. The relevant node is 2d-syndrome-depth-from-code-parameters.md.

For a stabilizer code with parameters \([n,k,d]\), the 2D local syndrome-depth theorem gives a lower bound of the form:

\[ \Delta=\Omega\left(\frac{k\sqrt d}{m}\right), \]

where \(m\) is the total number of qubits used by the circuit, including ancillas.

For theorem-level good Quantum Tanner families:

\[ k=\Theta(n), \qquad d=\Theta(n). \]

In linear space, \(m=\Theta(n)\), so:

\[ \Delta = \Omega\left(\frac{n\sqrt n}{n}\right) = \Omega(\sqrt n). \]

This is the core reason the static near-square 2D theorem is no longer the main open point.

Why This Route Is Presentation-Invariant

A stabilizer code is a stabilizer space, not a single preferred list of checks. If a proof depends on a particular generating set, then one has to ask whether a different generating set can hide or rearrange the apparent cross-cut demands.

The parameter route avoids that issue. It uses \(k\) and \(d\), not a chosen Tanner graph. That is why quantum-tanner-good-family-presentation-invariant-2d-barrier.md is the clean theorem-level anchor for the minimal static 2D statement.

Separator Meta-Theorem

The square grid is only one low-separator hardware family. The broader hardware-side pattern is captured by weighted-separator-function-to-syndrome-depth.md:

if a hardware family has sublinear weighted separators and the code family has the relevant local-expander demand, then syndrome extraction requires superconstant depth.

For square-like 2D hardware, the separator scale is \(O(\sqrt N)\). With \(N=\Theta(n)\) hardware qubits, this recovers the \(\Omega(\sqrt n)\) scale.

This is why the static-grid theorem is best viewed as one planar corollary of a broader separator barrier.

What Remains Open

The remaining frontier is not empty. It has shifted:

  • make the family statement as explicit and deterministic as desired;
  • replace chosen-presentation crossing checks with intrinsic stabilizer cut rank;
  • formulate the right compiler-native demand object;
  • understand when, if ever, that demand object has classical routing semantics.

The next chapters follow those shifts.

Active Recall

  1. Derive \(\Omega(\sqrt n)\) from \(\Omega(k\sqrt d/m)\).
  2. Why does this derivation not depend on a chosen generating set?
  3. What does the separator meta-theorem add beyond square grids?
  4. Name one current open frontier that remains after the static theorem.

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