Source-linked AI summary
Deep Neural Network Probabilistic Decoder for Stabilizer Codes
Stefan Krastanov, Liang Jiang
TL;DR
Existing decoders require significant modifications to use correlations between X and Z errors. This paper models syndrome-conditioned error probabilities with a neural network and samples candidate errors, reporting higher toric-code thresholds than several decoders.
Problem
Other decoders need significant modifications to employ correlations between Z and X errors, which occur together in Y errors.
Source: Results · p. 4Method
A neural network represents the error distribution conditioned on a measured syndrome, and sampling from it produces candidate errors.
Source: Results · p. 4, Methods · p. 6Results
Nearly 2 percentage points higher threshold (around 16.4%) and roughly 10 percentage points higher corrected fraction than MWPM under depolarization.
Source: Results · p. 4Takeaways & Limitations
The neural decoder can be applied to any stabilizer code, although its reported testing uses the toric code.
Source: Discussion · p. 7Sampling makes the decoder impractically inefficient for codes larger than roughly 200 qubits; it learns only per-qubit marginal error probabilities, not their correlations.
Source: Discussion · p. 7
Abstract
from arXiv · showhide
Neural networks can efficiently encode the probability distribution of errors in an error correcting code. Moreover, these distributions can be conditioned on the syndromes of the corresponding errors. This paves a path forward for a decoder that employs a neural network to calculate the conditional distribution, then sample from the distribution - the sample will be the predicted error for the given syndrome. We present an implementation of such an algorithm that can be applied to any stabilizer code. Testing it on the toric code, it has higher threshold than a number of known decoders thanks to naturally finding the most probable error and accounting for correlations between errors.
Results
On toric codes under depolarization, the neural decoder outperforms MWPM in threshold and corrected fraction. The reported gains are associated with direct optimization for probable errors and learning correlations between X and Z stabilizers.
Results
- Nearly 2 percentage points higher threshold (around 16.4%) and roughly 10 percentage points higher corrected fraction than MWPM under depolarization.The threshold compares favorably with renormalization-group decoders at 15.2% and correlation-tuned decoders at 13.3%; a renormalization-group decoder enhanced by a sparse code decoder also reaches 16.4%.Source: Figure 2 · p. 3, Results · p. 4
- Sampling makes decoding impractically slow above 200 physical qubits, while the architecture can be applied to any stabilizer code.The evaluation uses the toric code under the depolarization error model.Source: Results · pp. 3–4
- For a distance-5 code, performance improves with network size until gains diminish beyond about 15 hidden layers.Figure 2 uses 18-hidden-layer networks.Source: Figure 2 · p. 3, Results · p. 4
- Direct training for the most probable error and joint learning of Z and X stabilizers are identified as two characteristics behind the threshold gain.Under depolarization, one third of errors are Y errors, equivalent to simultaneous X and Z errors.Source: Results · p. 4
Methods
The decoder trains a neural network to map measured syndromes to error probabilities, then samples and checks candidate errors. The methods also specify its network setup, training data, and sampling procedure.
Methods
- Figure 3 compares joint Z-and-X decoding with the squared corrected fraction from Z-only decoding to illustrate the role of error correlations.The Z-only neural decoder was trained only on Z-stabilizer data.Source: Figure 3 · p. 5
- The network maps syndrome inputs to physical-error outputs, using tanh hidden-layer activations, sigmoid outputs, and hidden layers four times the input-layer size.The architecture was selected after hyperparameter searches on distance-3-to-6 toric codes; random search is described as a current best practice for hyperparameter selection.Source: Methods · p. 5
- Training used one billion syndrome-error pairs in batches of 512, generated from a depolarization model with equally likely X, Y, and Z errors conditional on an error.A new network was trained for each fidelity, and testing showed some robustness to a different error rate.Source: Methods · pp. 5–6
- Input normalization to mean zero and standard deviation one improved network performance.Under depolarization rate 1−p, the Z- and X-eigenvalue flip rates are each Pe = 2/3(1−p).Source: Methods · p. 6
- The network output is interpreted as per-qubit error probabilities conditioned on the syndrome; candidate errors are sampled and checked against that syndrome, with resampling for mismatches.A hard-decision message-passing procedure resamples only qubits involved in incorrect syndrome elements.Source: Methods · p. 6
- The code for building, training, and evaluating the decoder is publicly available on the authors’ web page.Scripts for the presented figures are available upon request, and pretrained networks can also be provided.Source: Methods · p. 6
Discussion
The discussion presents the neural decoder as a compressed alternative to lookup tables, applicable to stabilizer codes, while noting its sampling-efficiency limit. It also points to architecture extensions for toric-code structure and renormalization-group decoding.
Discussion
- The decoder compresses error-distribution information into a neural network, whereas complete lookup tables exceed the stated size bounds for tested toric codes.The training dataset for tested codes did not exceed 10 billion examples and could be made orders of magnitude smaller by reusing samples in stochastic gradient descent.Source: Discussion · p. 6
- The sampling loop checks whether a sampled error reproduces the syndrome, resampling only qubits in violated stabilizers and declaring failure after a preset iteration limit.The final result may still be wrong when the actual error has particularly low probability.Source: Figure 4 · p. 7
- The design is presented as applicable to any stabilizer code and as one of the best choices for codes with fewer than 200 qubits.The authors tested it on the toric code but state that the design does not depend on that code's specific structure.Source: Discussion · p. 7
- For codes larger than roughly 200 qubits, probabilistic sampling makes the decoder impractically inefficient; the authors attribute this partly to learning only per-qubit error marginals.They suggest that outputting correlation information and using more advanced belief propagation could address the problem.Source: Discussion · p. 7
- The authors propose toric-code-specific architectures exploiting lattice structure and translational symmetry, and suggest using the decoder in a renormalization-group decoder.They note that a network trained on a fixed patch could potentially be used for toric codes of any size.Source: Discussion · p. 7
Additional Information
SK contributed the project code, while SK and LJ jointly contributed design, analysis, and manuscript preparation. The authors report no competing financial interests.
Additional Information
- SK contributed the code for the project.The passage identifies SK as the code contributor.Source: Additional Information · p. 8
- SK and LJ jointly contributed to design, analysis, and manuscript preparation.These contributions are reported jointly.Source: Additional Information · p. 8
- The authors report no competing financial interests.This is the paper's competing-interests statement.Source: Additional Information · p. 8
Supplementary Materials
The supplementary material describes selective resampling, diminishing returns from adding network depth, and software utilities for designing and evaluating decoders.
Performance of sampling through “hard-decision message passing”
- Hard-decision message passing resamples only qubits connected to violated syndrome checks, preserving already properly decoded components.The supplementary figure compares this sampling with naive resampling and also shows message passing on an untrained network, which performs poorly.Source: Performance of sampling through… · p. 10
Performance vs depth of the neural network
- Increasing network depth improves performance up to a point, after which added layers bring minor gains that do not outweigh their expense.The example is for a 5x5 toric code.Source: Performance vs depth of the neural… · p. 10
Software package
- The software provides command-line utilities to design and evaluate deep neural network decoders with arbitrary hyperparameters.It is based on Keras and TensorFlow and can run on GPU accelerators.Source: Software package · p. 10