Source-linked AI summary
Comparing neural network based decoders for the surface code
Savvas Varsamopoulos, Koen Bertels, Carmen G. Almudever
TL;DR
Decoding surface-code errors must balance performance against the limited time available for error correction. The paper implements and compares low- and high-level neural-network decoders, then evaluates a recurrent high-level decoder across code distances and noise settings. The authors report a favorable performance–execution-time balance and comparable-to-Blossom performance within the tested distances and models.
Problem
Surface-code decoders must balance decoding performance with execution time because error correction allows only a limited time for decoding.
Source: §I · p. 5Method
The paper compares low- and high-level neural-network decoders and studies how training data and network choices affect performance and execution time.
Source: A. Choosing the training dataset · p. 9, B. Structure of the neural network · p. 11, §V · p. 16Results
The high-level RNN offers a balance of decoding performance and execution time; performance equivalent to Blossom was achieved through d ≤9 for depolarizing noise and d ≤7 for circuit noise.
Source: §V · p. 16Takeaways & Limitations
For the tested settings, the authors identify the high-level RNN as a decoder that is easier to train and can achieve better performance than the FFNN while decoding higher code distances.
Source: §4 · p. 13, §V · p. 16Training-dataset growth constrained the tested range to d=9 for depolarizing noise and d=7 for circuit noise.
Source: A. Choosing the training dataset · p. 9
Abstract
from arXiv · showhide
Matching algorithms can be used for identifying errors in quantum systems, being the most famous the Blossom algorithm. Recent works have shown that small distance quantum error correction codes can be efficiently decoded by employing machine learning techniques based on neural networks (NN). Various NN-based decoders have been proposed to enhance the decoding performance and the decoding time. Their implementation differs in how the decoding is performed, at logical or physical level, as well as in several neural network related parameters. In this work, we implement and compare two NN-based decoders, a low level decoder and a high level decoder, and study how different NN parameters affect their decoding performance and execution time. Crucial parameters such as the size of the training dataset, the structure and the type of the neural network, and the learning rate used during training are discussed. After performing this comparison, we conclude that the high level decoder based on a Recurrent NN shows a better balance between decoding performance and execution time and it is much easier to train. We then test its decoding performance for different code distances, probability datasets and under the depolarizing and circuit error models.
I. Quantum error correction
The surface code detects errors through ancilla parity checks and decodes their syndrome into data-qubit corrections. Decoder choice balances performance against the limited time available for error correction.
I. Quantum error correction
- Surface-code parity checks use ancilla measurements to detect changes without directly measuring and collapsing the data qubits.Changes between consecutive parity-check outcomes are detection events; collecting outcomes forms the error syndrome.Source: §I · pp. 4–5
- Data-qubit errors form chains whose endpoints produce detection events, while measurement errors appear as repeated-check patterns across cycles.Multiple cycles help distinguish these error types before corrections are proposed.Source: §I · p. 5, Figure 5 · p. 5
- Maximum-likelihood decoding offers higher performance than Blossom but scales exponentially with code distance, whereas optimized Blossom scales linearly with qubit count.This performance–execution-time trade-off matters because decoding has a limited time budget.Source: §I · p. 5
- The paper compares neural-network decoders with each other and un-optimized Blossom as an alternative approach to decoding.The comparison examines decoding performance and execution time.Source: §I · p. 6
II. Neural network based decoders
Neural-network decoders can target physical-qubit corrections or logical-state corrections. This paper implements both types and compares their decoding performance and execution time with un-optimized Blossom.
II. Neural network based decoders
- The neural networks use supervised learning, mean squared error as the cost function, and ReLU activation; the recurrent networks use LSTM cells.Source: §II · p. 6
- Low-level decoders seek exact physical-level corrections, while high-level decoders seek corrections that restore the logical state [14] [11] [12] [16] [17].Source: §II · p. 6
- The paper implements low-level and high-level neural-network decoders and compares them with un-optimized Blossom on performance and execution time.Source: §II · p. 6
A. Inputs/Outputs
The decoder types differ in what their networks predict from an error syndrome: physical-qubit error probabilities or logical-state error probabilities.
A. Inputs/Outputs
- A low-level decoder maps the syndrome to an error-probability distribution for each data qubit, aiming to correct the physical errors.Source: A. Inputs/Outputs · p. 6
- A high-level decoder maps the syndrome to probabilities over logical-state errors rather than predicting corrections for every data qubit.Its four possible logical errors are Ī, X̄, Z̄, and Ȳ.Source: A. Inputs/Outputs · p. 6
- The high-level decoder’s logical-state prediction is simpler because it has four possible outputs, unlike the low-level decoder’s output for each data qubit.Source: A. Inputs/Outputs · p. 6
B. Sampling and training process
Training data are sampled into syndrome-to-output mappings, with frequencies retained for degenerate syndromes. The implementation then differs in how each decoder validates or corrects its proposed output.
B. Sampling and training process
- Sampling records syndrome inputs and outputs while tracking frequencies because different error sets can produce the same syndrome.Those frequencies define per-data-qubit probability distributions for low-level decoding and per-logical-state distributions for high-level decoding.Source: B. Sampling and training process · p. 6
- Training maps stored inputs to outputs and stops when at least 99% of training inputs are predicted correctly.Source: B. Sampling and training process · p. 7
C. Implementation details
- Low-level decoding checks whether sampled physical-qubit corrections reproduce the observed syndrome and resamples if they do not, making execution time non-constant.The need for re-evaluation depends on physical error rate, code distance, and resampling strategy.Source: C. Implementation details · p. 7
- High-level decoding combines a fast simple decoder’s valid corrections with neural-network logical corrections, allowing both modules to operate in parallel.The neural network uses syndromes from all cycles to identify whether the simple decoder’s corrections cause a logical error.Source: C. Implementation details · pp. 7–8
III. Implementation parameters
The authors compare high- and low-level neural-network decoders under the same conditions, evaluating decoding performance and execution time. For training data, they seek a small but diverse dataset that reaches Blossom’s performance while balancing decoding quality against cost.
III. Implementation parameters
- The comparison evaluates decoding performance and execution time for high- and low-level decoders under the same conditions.Decoding performance is assessed through lifetime simulations; execution time is measured from the first syndrome in a decoding window to decoder output.Source: §III · p. 8
A. Choosing the training dataset
- As code distance increases, the syndrome space grows exponentially; the tested dataset approach reached d=7 for circuit noise and d=9 for depolarizing noise.For circuit noise at d=7, the full space was estimated at 3.1x10^144; distributed decoding is mentioned as a potential way to scale further [36] and [37].Source: A. Choosing the training dataset · p. 9
- A single-probability dataset is sampled at one physical error rate and tested across many, but may generalize poorly when test probabilities differ.At low sampling rates, the samples are less diverse; the approach nevertheless uses one trained network across tested error rates.Source: A. Choosing the training dataset · p. 10
- Multiple-probability datasets train a separate network for each sampled error rate and test it at that rate and nearby rates, providing more relevant training information.The paper selects frequent syndromes and grows dataset size until performance reaches the Blossom baseline; it does not claim to find the optimal dataset.Source: A. Choosing the training dataset · p. 10
B. Structure of the neural network
The tested network configurations varied in layer and node counts, and training accuracy helped select a configuration for continued training. Results favor the high-level decoder and RNNs over the respective alternatives, while network structure depends on the data as well as the network type.
B. Structure of the neural network
- The configuration sweep tested layer and node choices for d=3 and d=5, both decoder levels, RNNs and FFNNs, and the depolarizing model.Training stopped at 5000 epochs for initial comparison, and the configuration with the highest training accuracy was selected.Source: B. Structure of the neural network · p. 11, Figure 8 · p. 11
- Training-sample similarity, dataset size, and network type affect configuration, with layer count more affected by sample similarity than dataset size.The last hidden layer was set to the output-node count and earlier hidden layers had decreasing node counts; the selected configuration was continued until it reached at least Blossom’s decoding performance.Source: B. Structure of the neural network · p. 11
- The high-level decoder outperformed the low-level decoder, and the RNN slightly outperformed the FFNN at d=3; the RNN–FFNN gap was more evident at larger distances and under circuit noise.The authors report that RNN training is easier as dataset size increases, whereas FFNN training becomes harder; they attribute the RNN’s circuit-noise fit to its temporal dynamics.Source: B. Structure of the neural network · pp. 11–12, Figure 9 · p. 12
1. Batch size
The section describes training choices and timing considerations for the neural-network decoders. It concludes that high-level decoders can outperform low-level decoders with error-rate-constant execution time, while RNNs trade higher performance and easier training for longer execution time.
1. Batch size
- Batches of 1000 or 10000 samples were used depending on dataset size; batch updates can improve training accuracy and time by updating weights multiple times per iteration.The authors say this typically leads to faster convergence.Source: §1 · p. 12
2. Learning rate
- The learning rate sets the size of weight updates, and the authors decreased it stepwise when training accuracy did not improve after a specified number of iterations.The stated learning-rate range was 0.01 to 0.0001, with 50 iterations given as an example trigger.Source: §2 · p. 12
3. Generalization
- Training continued until predicted and desired values were close to the third decimal digit for more than 95% of training samples.The authors frame this criterion around generalization to inputs unknown during training, rather than only correct predictions on known inputs.Source: §3 · p. 12
4. Training time and execution time
- The reported Figure 10 timing illustrates behavior rather than representative execution-time values; a hardware implementation is needed for realistic time estimates.The paper reports that training times ranged from half a day to three days on a 28-hyperthread-core 2GHz CPU with 384GB memory, and notes that training platform affects timing.Source: §4 · pp. 12–13
- High-level decoder execution time appears to increase linearly with code distance, although it remains constant with physical error rate.The authors provide calculated average decoding time per surface-code cycle under depolarizing noise for tested distances in Table III.Source: §4 · p. 13, Table III · p. 13
- High-level decoders can achieve higher decoding performance than low-level decoders while keeping execution time constant with physical error rate.The low-level decoder repeats predictions to obtain valid corrections, so its execution time remains non-constant even with a carefully designed repetition step.Source: §4 · p. 13
- RNNs achieved higher decoding performance than FFNNs, mainly due to using more weights, but required more execution time and were easier to train.The authors report that FFNN training and evaluation become extremely difficult as code distance increases; they chose high-level RNN decoders to balance execution time and performance.Source: §4 · p. 13
IV. Results
The results section evaluates the high-level RNN decoder across code distances, error models, and training-dataset probability strategies. It describes how the decoding flow differs between depolarizing and circuit noise, and how decoding performance is measured.
IV. Results
- The high-level RNN decoder is evaluated across code distances, depolarizing and circuit noise, and single- and multiple-probability datasets.Performance is assessed through lifetime simulations, logical-error rates, pseudo-thresholds, and thresholds across code distances.Source: §IV · pp. 13–14
- For depolarizing noise, the syndrome is sent directly to the simple decoder because perfect syndrome measurements are assumed.The paper describes this flow alongside the circuit-noise decoding process.Source: §IV · p. 14, Figure 11 · p. 14
- For circuit noise, one neural network separates data-qubit from measurement-error detection events, and another predicts whether simple-decoder corrections cause a logical error.The second network’s corrections are combined with the simple decoder’s corrections to cancel any logical error created by the latter.Source: §IV · p. 14
A. Depolarizing error model
With perfect syndrome measurements under depolarizing noise, the high-level decoder performs better when trained on multiple probability datasets. Its multiple-dataset threshold exceeds Blossom’s reported threshold.
A. Depolarizing error model
- Five training datasets were sampled at physical error rates 0.2, 0.15, 0.1, 0.08, and 0.05.Perfect syndrome measurements allow the logical error rate to be calculated per error-correction cycle.Source: A. Depolarizing error model · p. 14
- Multiple probability datasets yield lower logical error rates and higher pseudo-thresholds than a single probability dataset.The paper attributes the improved performance to the multiple-dataset decoder having more relevant information.Source: A. Depolarizing error model · p. 14, Table IV · p. 14
- The multiple-probability decoder reaches a 0.146 threshold, versus 0.140 for the single-probability decoder and 0.142 for Blossom.The comparison covers code distances d=3, 5, 7, and 9 under depolarizing noise with perfect syndrome measurements.Source: A. Depolarizing error model · pp. 14–15, Figure 12 · p. 15
B. Circuit noise model
Under circuit noise with imperfect syndrome measurements, multiple-probability training improves decoding performance, especially at larger code distances. Its threshold is 3.2x10−3, compared with 2.25x10−3 for single-probability training.
B. Circuit noise model
- Multiple-probability training performs significantly better, especially as code distance increases; at d=3, both approaches perform equally well.For d=7, performance is closer to d=5 above the pseudo-threshold, but resembles d=7 below it, where training samples become relevant again.Source: B. Circuit noise model · p. 15
- Blossom was not included for circuit-noise code distances above 3 because it could not be used with imperfect measurements at those distances.The authors note that their pseudo-threshold results are close to circuit-noise results reported in the literature.Source: B. Circuit noise model · p. 15
- The circuit-noise threshold is 3.2x10−3 for multiple-probability training, versus 2.25x10−3 for single-probability training.Blossom’s reported threshold is approximately 6x10−3.Source: B. Circuit noise model · p. 15, Table V · p. 15
V. Conclusions
The paper compares high- and low-level neural-network decoders and finds that the high-level decoder offers better performance at equal dataset size. Its RNN variant is easier to train than an FFNN, and the tested performance matches Blossom through d≤9 for noiseless depolarizing noise and d≤7 for circuit noise.
V. Conclusions
- The high-level decoder achieves better performance at the same dataset size, with execution time constant in physical error rate and linear in code distance.FFNNs can run faster, but RNNs are easier to train and can decode higher code distances with better performance.Source: §V · p. 16
- Multiple datasets sampled at different physical error rates can improve performance because their samples are more relevant.The authors report Blossom-equivalent performance through d≤9 for noiseless depolarizing noise and successful decoding through d≤7 for circuit noise with noisy syndrome measurements.Source: §V · p. 16
- The study could not test higher code distances because dataset size grows exponentially as code distance increases linearly.The authors identify distributed decoding as a possible solution.Source: §V · p. 16