Physics-Informed Graph Neural Networks for Surface Code Decoding via Discrete Exterior Calculus

Abstract

We introduce a physics-informed graph neural network decoder for the surface code. A discrete Poisson equation on the syndrome graph acts as a hard inductive bias on which a graph encoder learns syndrome-adaptive edge weights. A differentiable solver returns the node potentials and the associated edge current. Our central observation is that the logical-error signal is not carried by any part of that current, but by how the syndrome sits relative to the two code boundaries. We place a sink on each of the two boundaries linked by the logical operator and read the difference of the currents they drain. We prove that this single number is a topological pairing: it weighs each excited detector by a smooth coordinate that runs from +1 on one boundary to -1 on the other, and sums the votes. The coordinate is fixed by both the code topology and the learned metric, generating an exact and free-parameter readout. On the rotated surface code under circuit-level depolarising noise, this one topological scalar matches the best full-field readout. That parity is itself the result: for one logical qubit the logical signal is one-dimensional, so projecting onto it discards nothing. At larger code distance the picture strengthens indicating that isolating the pairing helps more as the field grows larger and sparser.