基于离散外微积分的用于表面码解码的物理信息图神经网络
Physics-Informed Graph-Neural Decoding of the Surface Code: the Logical Signal as an Exact Topological Pairing
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中文总结 AI 辅助
研究针对表面码解码问题，提出基于离散外微积分的物理信息图神经网络解码器，通过离散泊松方程学习边权重，利用可微求解器获取电流，证明特定电流差值为拓扑配对，该方法在旋转表面码解码中表现良好，能精确读出逻辑信号。
中文摘要 AI 辅助
我们介绍了一种用于表面码的物理信息图神经网络解码器。在综合症图上的离散泊松方程作为一种硬归纳偏差，图编码器据此学习适应综合症的边权重。一个可微求解器返回节点势和相关的边电流。我们的核心观察是逻辑错误信号不是由电流的任何部分携带，而是由综合症相对于两个码边界的位置携带。我们在由逻辑算子连接的两个边界上各放置一个汇，并读取它们消耗的电流之差。我们证明这个单一数字是一种拓扑配对：它用一个从一个边界上的 +1 到另一个边界上的 -1 的平滑坐标对每个激发探测器加权，并对投票进行求和。该坐标由码拓扑和学习到的度量共同确定，生成一个精确且无自由参数的读出。在电路级去极化噪声下的旋转表面码上，这个单一拓扑标量与最佳全场读出匹配。对于一个逻辑量子比特，逻辑信号是一维的，所以投影到它上面不会丢弃任何东西。在更大的码距下，情况更明显，表明随着场变得更大且更稀疏，分离这种配对更有帮助。
英文摘要
We develop a physics-informed graph neural network (GNN) decoder for the surface code that solves a discrete Poisson equation on the syndrome graph, with the syndrome as the charge source. We compare four readout architectures for extracting the logical-error probability: a potential-based readout that maps the Poisson field through a multilayer perceptron, two current-based readouts under single- and two-sink Dirichlet boundary conditions, and a diffusion-based variant. Comparing these, we show that the solver's edge current is a pure gradient flow whose harmonic (circulating) part vanishes identically. The logical signal therefore cannot be read as a component of the current itself; it is instead a topological pairing between the syndrome and a boundary-fixed harmonic coordinate that distinguishes the two code boundaries linked by the logical operator. We prove that this pairing is evaluated exactly and in closed form, with no learned readout parameters, as the net current drained between the two boundary sinks. On the rotated surface code under circuit-level depolarising noise, this single closed-form scalar matches the best full-field readout and, at larger code distance, significantly exceeds the single-sink current pool, so that isolating the pairing helps more, not less, as the field grows larger and sparser. The decoder is not intended to surpass minimum-weight perfect matching, near-optimal for this noise model; its contribution is an interpretable characterisation of the logical signal itself.
