A Code-Agnostic Graph Neural Network Decoder from the Detection Error Model

The gist The POLYMECHANON, a graph neural network (GNN) decoder for quantum error correction whose only input is the detection error model (DEM) of a quantum code under a given noise model, represents a code-agnostic approach that decodes any stabiliser code without redesigning the decoder on page 1.

How it works

The core innovation is representing the detection error model (DEM) of a quantum code under a given noise model as a tripartite graph built out of detectors, error mechanisms and logical observables on page 1. Every input feature is computed by using the quantum code as data rather than design choice on page 1. This structure allows the same architecture to decode in principle any stabiliser code, under any noise model that can be expressed as a detection error model, once trained on the DEM on page 1.

The tripartite graph G is explicitly represented with vertices V = D ∪ EM ∪ L and edges E = ED,EM ∪ EEM,L on page 7. The nodes are defined as:

**- D-nodes (detectors): one per detector, carrying the syndrome bit 0/1; EM-nodes (error mechanisms): one per fault mechanism; L-nodes (logical observables): one per logical qubit on page 7. The edges ED,EM and EEM,L encode the matrices H and L previously defined on page 7. This representation makes the belief-propagation structure [72] of the problem explicit on page 7. Decoding consists in determining the probabilities of EM-nodes and L-nodes in G, having as inputs solely the syndrome bits for the D-nodes and the prior probabilities in the EM-nodes and in the EM-nodes and the L-nodes on page 7. The probabilistic information depending only on the noise model and on the specific faults on page 7. The DEM treats error mechanisms as independent between each others, so that the prior probability over error configurations e = (e1,..., eNEM) ∈ F NEM 2 factorises as P(e) = N YEM j=1 p ej j (1 − pj) 1−ej, (3) where pj is the prior probability of the single j-th mechanism on page 7. Each mechanism is equivalently described by the logit of its prior probability, λj = logpj/(1 − pj), (4) the quantity carried by the corresponding EM-node, which we define later in this paragraph on page 7. Given a syndrome σ, a maximum-likelihood decoder returns the most probable of two scenarios, that the error e triggered a logical error or that it did not on page 7. The decoded outcome should be ˆl = arg maxl P(l σ), i.e. the logical operator ˆl that for a given syndrome σ is most likely to have happened on page 7. However, the sum runs over an exponentially large number of error configurations, making the exact maximization intractable in general on page 7. This unfeasible regime gives the motivation to consider approximate decoders such as belief propagation [71, 72] and machine-learned decoders as the one presented in this work on page 7. The DEM has a natural tripartite structure that we make explicit by representing it as a tripartite graph G = (V, E) — V and E being the vertices and the edges of G respectively on page 7. Specifically, V = D ∪ EM ∪ L and E = ED,EM ∪ EEM,L) (Fig. 3) as described in what follows on page 7. As for the error mechanisms, they are initially endowed with the logits of their prior probabilities on page 7. Finally, edges ED,EM and EEM,L encode respectively the matrices H and L previously defined on page 7. This representation makes the belief-propagation structure [72] of the problem explicit on page 7. Decoding consists in determining the probabilities of EM-nodes and L-nodes in G, having as inputs solely the syndrome bits for the D-nodes and the prior probabilities in the EM-nodes and in the EM-nodes and the L-nodes on page 7. The DEM has a natural tripartite structure that we make explicit by representing it as a tripartite graph G = (V, E) — V and E being the vertices and the edges of G respectively on page 7. Specifically, V = D ∪ EM ∪ L and E = ED,EM ∪ EEM,L) (Fig. 3) as described in what follows on page 7. As for the error mechanisms, they are initially endowed with the logits of their prior probabilities on page 7. Finally, edges ED,EM and EEM,L encode respectively the matrices H and L previously defined on page 7. This representation makes the belief-propagation structure [72] of the problem explicit on page 7. Decoding consists in determining the probabilities of EM-nodes and L-nodes in G, having as inputs solely the syndrome bits for the D-nodes and the prior probabilities in the EM-nodes and in the EM-nodes and the L-nodes on page 7. The DEM has a natural tripartite structure that we make explicit by representing it as a tripartite graph G = (V, E) — V and E being the vertices and the edges of G respectively on page 7. Specifically, V = D ∪ EM ∪ L and E = ED,EM ∪ EEM,L) (Fig. 3) as described in what follows on page 7. As for the error mechanisms, they are initially endowed with the logits of their prior probabilities on page 7. Finally, edges ED,EM and EEM,L encode respectively the matrices H and L previously defined on page 7. This representation makes the belief-propagation structure [72] of the problem explicit on page 7. Decoding consists in determining the probabilities of EM-nodes and L-nodes in G, having as inputs solely the syndrome bits for the D-nodes and the prior probabilities in the EM-nodes and in the EM-nodes and the L-nodes on page 7. The DEM has a natural tripartite structure that we make explicit by representing it as a tripartite graph G = (V, E) — V and E being the vertices and the edges of G respectively on page 7. Specifically, V = D ∪ EM ∪ L and E = ED,EM ∪ EEM,L) (Fig. 3) as described in what follows on page 7. As for the error mechanisms, they are initially endowed with the logits of their prior probabilities on page 7. Finally, edges ED,EM and EEM,L encode respectively the matrices H and L previously defined on page 7. This representation makes the belief-propagation structure [72] of the problem explicit on page 7. Decoding consists in determining the probabilities of EM-nodes and L-nodes in G, having as inputs solely the syndrome bits for the D-nodes and the prior probabilities in the EM-nodes and in the EM-nodes and the L-nodes on page 7.

1. Improve decoder generalizability by training a single model across multiple code families, as demonstrated by a single model trained across codes of different families outperforms uncorrelated MWPM on the graph-like codes seen during training. This enables the system to decode unseen codes whose size is comparable to those seen in training, although accuracy drops as the target graph grows beyond the largest one seen during training.

2. Enhance real-time decoding capability by leveraging a fixed computational graph structure where the cost of our decoder does not grow with the noise strength, making it suitable for real-time decoding on neutral-atom processors. This allows for a deterministic latency, which is crucial for keeping pace with the syndrome stream.

3. Implement confidence-based post-selection to lower logical error rates by more than an order of magnitude on the surface code while keeping more than 90% of the shots, using a confidence metric defined as c = min k=1,..., NL max(ˆpk, 1 − pˆk). This strategy allows for rejecting high-weight error events, which are precisely those where the decoder can fail.

4. Optimize decoding cost by utilizing the fact that the GNN's per-shot latency is flat in physical error rate, as its cost is set by the size of graph and of the neural architecture, not by the noise. This makes it cheaper per shot on every code of the family, by a factor between 2.5 and 5.1 at the lowest noise.

5. Develop a framework for adapting to diverse error models beyond simple detection error models, as the decoder only requires a DEM, which is all that an optimal decoder should see in order to operate its correction. This allows the system to address atom loss, where the DEM changes from shot to shot, and other noise types without requiring redesigning the architecture.

We present POLYMECHANON, a graph neural network (GNN) decoder for quantum error correction whose only input is the detection error model (DEM) of a quantum code under a given noise model. We represent the DEM as a tripartite graph of detectors, error mechanisms and logical observables, where every input feature is computed by using the quantum code as data rather than design choice. In this way, the same architecture decodes in principle any stabiliser code, under any noise model that can be expressed as a detection error model, once trained on the DEM. We test this approach along four directions. Firstly, on the rotated surface code the decoder outperforms correlated MWPM under both phenomenological and circuit-level noise, with up to 25% fewer logical failures. Secondly, on a family of high-rate qLDPC codes it matches BP+OSD on the smaller codes and surpasses it on the larger ones, with up to 16% fewer logical failures on [![130,4,6]!]. Thirdly, its decoding time does not depend on the physical error rate, whereas that of BP+OSD grows with it: on [![130,4,6]!] its effective cost per shot is of the order of a few ms on a single GPU against a few tens of milliseconds for BP+OSD on a single CPU core, and its fixed computational graph makes it a candidate for real-time decoding on neutral-atom processors. Finally, a single model trained across codes of different families outperforms uncorrelated MWPM on the graph-like codes seen during training and stays within 10% of BP+OSD on the others, while it tends to fail to generalize to unseen codes, particularly larger ones. Its probabilistic output enables confidence-based post-selection, lowering the logical error rate by more than an order of magnitude on the surface code while keeping more than 90% of the shots. Since only the DEM changes, new codes and noise models can be decoded without redesigning the decoder.