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Thin Planar Connectivity Escape

Claim/Theorem

Tremblay, Delfosse, and Beverland show that for any CSS code with degree-\(\delta\) Tanner graph, one can design stabilizer-measurement circuits of depth at most \((2\delta+2)\) using at most \(\lceil \delta/2 \rceil\) planar layers of noncrossing long-range connections. Thus constant-depth syndrome extraction is possible for bounded-degree QLDPC codes once the hardware is upgraded from a single static 2D nearest-neighbor grid to a small number of planar long-range wiring layers.

Dependencies

  • None.

Conflicts/Gaps

  • This is an explicit escape route from the static-grid SWAP-only regime, not a counterexample to Conjecture 3.
  • The resource that breaks the barrier is exactly what the conjecture excludes: additional planar long-range connectivity beyond nearest-neighbor SWAP-only compilation.
  • The theorem is about circuit design under thin planar connectivity, not about pure 2D mesh routing.

Sources

  • 10.1103/PhysRevLett.129.050504