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.
翻译:Qantum LDPC 代码可能为建设低管防故障量计算机提供一条路径。 但是, 由于普通 LDPC 代码缺乏几何限制, NA\ “ 动态布局” 将许多具有交叉连接的远方 ⁇ 点数组合成硬硬件, 并可能导致性能降解交叉话。 我们建议对量子LDPC 代码进行二维布局, 将其坦纳图解成少量的平面层。 每个层都包含不相交的长距离连接。 对于具有度- $\ delta$ Tanner 图形的 CSS 代码, 我们设计深度为$( 2\ delta +2) 的稳定器测量线路, 最多使用$\ lceil\ delta/2\ rceil\ = rceil$ 平面值。 我们用比地表代码少14倍的物理平面代码, 我们观察到正率代码家族的电路段- 0. 28 阈值阈值阈值阈值阈值为 028 。 对于实际误率 10 4 美元, 这个家族来说, 这个家庭逻辑误率为 10 15 。