Constant-overhead quantum error correction with thin planar connectivity

Quantum LDPC codes may provide a path to build low-overhead fault-tolerant quantum computers. However, as general LDPC codes lack geometric constraints, na\"ive layouts couple many distant qubits with crossing connections which could be hard to build in hardware and could result in performance-degrading crosstalk. We propose a 2D layout for quantum LDPC codes by decomposing their Tanner graphs into a small number of planar layers. Each layer contains long-range connections which do not cross. For any CSS code with a degree-$\delta$ Tanner graph, we design stabilizer measurement circuits with depth at most $(2\delta +2)$ using at most $\lceil \delta/2 \rceil$ layers. We observe a circuit-noise threshold of 0.28\% for a positive-rate code family using 49 physical qubits per logical qubit. For a physical error rate of $10^{-4}$, this family reaches a logical error rate of $10^{-15}$ using fourteen times fewer physical qubits than the surface code.