Constant-Overhead Quantum Error Correction with Thin Planar Connectivity

TLDR

Quantum low‑density parity‑check codes promise low‑overhead fault‑tolerant quantum computing, but their lack of geometric constraints leads to long‑range, crossing connections that are difficult to implement and may cause crosstalk. The authors propose a two‑dimensional layout for quantum LDPC codes that decomposes Tanner graphs into a few planar layers. They design stabilizer‑measurement circuits for any CSS code with degree‑δ Tanner graphs that fit into at most ⌈δ/2⌉ planar layers and achieve depth no greater than (2δ+2), ensuring non‑crossing long‑range connections. The proposed layout achieves a 0.28 % circuit‑noise threshold with 49 physical qubits per logical qubit, and at a physical error rate of 10⁻⁴ it attains a logical error rate of 10⁻¹⁵ while using fourteen times fewer qubits than the surface code.

Abstract

Quantum low density parity check (LDPC) codes may provide a path to build low-overhead fault-tolerant quantum computers. However, as general LDPC codes lack geometric constraints, naïve 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 Calderbank-Shor-Steane code with a degree-δ Tanner graph, we design stabilizer measurement circuits with depth at most (2δ+2) using at most ⌈δ/2⌉ 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.