Machine-learning-assisted correction of correlated qubit errors in a topological code

P. Baireuther; T. E. O'Brien; B. Tarasinski; C. W. J. Beenakker

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Machine-learning-assisted correction of correlated qubit errors in a topological code

P. Baireuther, T. E. O'Brien, B. Tarasinski, C. W. J. Beenakker

arXiv:1705.07855v3May 22, 2017quant-phcond-mat.dis-nn

TL;DR

Fault-tolerant quantum computation needs efficient correction of errors in encoded qubits. This paper trains a recurrent neural-network decoder from experimentally accessible data, achieving better performance than blossom by detecting X–Z correlations while retaining performance over many cycles.

Problem
Efficient decoding is needed to detect and correct accumulated errors in encoded quantum information, but existing surface-code neural decoders lack practical memory or experimental-data training.
Method
The paper uses an LSTM recurrent neural network trained on measured syndromes and final readouts to decode surface-code error-correction cycles.
Results
ηd = 0.81, a 26% improvement over the blossom decoder, in a density-matrix simulation of the 17-qubit Surface-17 experiment.
Takeaways & Limitations
The decoder provides a realistic option for forthcoming experimental surface-code implementations by learning correlations from physical-system data without a priori noise-model assumptions.
Takeaways & Limitations
At low error rates, training may lack enough two-qubit errors, and scaling to lattice-surgery or braiding-deformed surface codes remains necessary.

Abstract

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A fault-tolerant quantum computation requires an efficient means to detect and correct errors that accumulate in encoded quantum information. In the context of machine learning, neural networks are a promising new approach to quantum error correction. Here we show that a recurrent neural network can be trained, using only experimentally accessible data, to detect errors in a widely used topological code, the surface code, with a performance above that of the established minimum-weight perfect matching (or blossom) decoder. The performance gain is achieved because the neural network decoder can detect correlations between bit-flip (X) and phase-flip (Z) errors. The machine learning algorithm adapts to the physical system, hence no noise model is needed. The long short-term memory layers of the recurrent neural network maintain their performance over a large number of quantum error correction cycles, making it a practical decoder for forthcoming experimental realizations of the surface code.

1 Introduction

Efficient decoding is essential for fault-tolerant quantum computation, and surface codes are attractive scalable candidates. This work develops a recurrent neural-network decoder that learns from measured data and improves on blossom decoding by recognizing correlated errors.

1 Introduction: Surface codes encode a logical qubit nonlocally and combine favorable small-circuit performance with scalability to larger circuits.
1 Introduction: Machine learning promises flexibility across topological codes and error types beyond classic decoding algorithms.
1 Introduction: Existing neural-network decoders lack either memory for repeated stabilizer cycles or the ability to learn directly from measured data.
1 Introduction: A recurrent neural-network decoder improves on blossom decoding in a realistic simulation of a forthcoming surface-code experiment.The decoder separately detects X and Z errors and their Y correlations.

2 Overview of the surface code

The surface code spreads one logical qubit across noisy physical qubits and uses repeated commuting parity checks to reveal error syndromes. Decoding uses these measurement histories to choose corrections that minimize logical errors, with blossom as an efficient but imperfect decoder.

2 Overview of the surface code: A QEC code encodes one logical qubit across N noisy physical qubits using N −1 binary parity checks.A complete measurement cycle projects the system into a two-dimensional logical subspace.
2 Overview of the surface code: Repeated syndrome measurements track discrete transitions between logical subspaces, allowing a decoder to infer corrections from the system’s history.
2 Overview of the surface code: Maximum-likelihood decoding is NP-hard in general, so polynomial-time decoders trade optimality for sufficient error-mitigation performance.
2 Overview of the surface code: The surface code arranges data qubits on a d×d lattice with checkerboard stabilizer measurements and boundary checks totaling N −1 measurements.

3 Neural network detection of correlated errors

The decoder uses a recurrent network with internal memory to process arbitrary-duration syndrome histories and learn from experimentally accessible measurements. Its target is to identify correlations across error types and across repeated cycles.

3 Neural network detection of correlated errors: Pauli-Y errors combine X and Z errors, creating correlations that blossom decoding does not optimally detect.These correlations motivate neural-network pattern recognition.
3 Neural network detection of correlated errors: An LSTM recurrent decoder maintains performance over an unspecified number of cycles with computational cost per cycle independent of total cycle count.Its short- and long-term memory states retain information across cycles.
3 Neural network detection of correlated errors: The network is trained using measured syndromes and final data-qubit readouts rather than a listing of the errors that occurred.
3 Neural network detection of correlated errors: Two separate binary-classification decoders are trained because the final logical qubit may be measured in either the x or z basis.

4 Related Work

Related work improves on blossom by modeling X–Z correlations or using machine learning, while this paper combines correlation detection with recurrent memory and experimental-data training. Existing approaches differ in optimality, runtime, noise assumptions, and cycle-length flexibility.

4.1 Approaches going beyond blossom decoding: Correlations between Pauli-X and Pauli-Z errors improve decoding performance and can also be incorporated into adapted blossom algorithms.
4.1 Approaches going beyond blossom decoding: Correlated corrections improved distance-3 decoding by approximately 24% at comparable depolarization probability, similar to the neural-network improvement.
4.1 Approaches going beyond blossom decoding: DMRG and Monte Carlo methods can outperform blossom, but they involve restricted noise models or increased runtime costs.The Monte Carlo approach adds O(N^2) computational cost.
4.2 Approaches based on machine learning: The recurrent network differs from earlier machine-learning decoders by detecting correlations across arbitrary stabilizer-measurement cycles.
4.2 Approaches based on machine learning: Earlier machine-learning studies without circuit-level noise cannot be directly compared quantitatively with this decoder.

5 Design of the neural network decoder

The decoder uses recurrent neural networks with LSTM memory to process syndrome histories while separating repeated-cycle decoding from final-syndrome processing. Two task-specific networks combine recurrent outputs with syndrome information to estimate final parity corrections.

5 Design of the neural network decoder: The recurrent network uses LSTM layers and fully connected evaluation layers to transform syndrome inputs into parity predictions.The LSTM layers receive syndrome increments from both x- and z-stabilizer measurements.
5 Design of the neural network decoder: Figure 2 depicts two neural networks with boxes for inputs and outputs, ovals for LSTM and evaluation layers, and dashed arrows for internal memory flow.Solid arrows represent data flow through the decoder.
5 Design of the neural network decoder: The final-syndrome input is supplied only to the decoder operating in its own basis, distinguishing it from syndrome histories available to both decoders.This design addresses the final syndrome’s break from time-translational invariance.
5 Design of the neural network decoder: The decoder splits time-translational decoding into two tasks: estimating accumulated parity from syndrome increments and estimating final-syndrome adjustments.The final parity probability is formed from the two task probabilities using a probabilistic sum.
5 Design of the neural network decoder: Two LSTM-based networks process different syndrome windows: network 1 receives T increments, while network 2 receives the last T0 increments and the final syndrome increment.For Surface-17, optimal task-2 performance uses T0 = 3 cycles, supporting decoding for experiments of three or more cycles.

6 Neural network performance

The neural decoder is evaluated on a 17-qubit distance-3 surface code under a Pauli error model and compared with blossom decoding. It achieves a lower logical error rate, with the advantage emerging as correlated Y errors increase.

6 Neural network performance: The performance comparison uses the 17-qubit distance-3 Surface-17 code with independent X, Y, and Z error opportunities and measurement errors.The model includes px, py, and pz qubit error probabilities and pm measurement-error probability.
6 Neural network performance: Figure 3 evaluates the probability of correctly decoding the logical qubit over stabilizer-measurement cycles using 5 · 10^4 datasets and bootstrapped 3σ error bars.The plotted curves are two-parameter fits to Eq. (8).
6 Neural network performance: As py increases at fixed px = pz = 0.048% and pm = 0.14%, blossom produces a larger logical error rate than the neural decoder.The rates are extracted from fitted decay curves as in Figure 3.
6 Neural network performance: 0.209% versus 0.274% logical error rate per cycle: the neural network outperforms the blossom decoder under the tested Pauli error model.The rates are obtained by fitting logical-fidelity decay curves with a two-parameter fit.
6 Neural network performance: When py = 0, the decoders perform similarly because X and Z errors are independent, whereas the neural decoder gains an advantage once py is approximately px.The comparison is intended to test whether detecting error correlations explains the improvement.

7 Conclusion and outlook

The recurrent neural-network decoder outperforms standard MWPM decoding for correlated bit- and phase-flip errors and is designed for experimental QEC. A Surface-17 density-matrix simulation supports its practical viability, while several scaling and low-error-rate challenges remain.

7 Conclusion and outlook: The recurrent neural network outperforms MWPM in the presence of correlated bit-flip and phase-flip errors.Its long short-term memory layer supports operation across multiple quantum-error-correction cycles.
7 Conclusion and outlook: Training solely on experimental data without a priori theoretical noise assumptions makes the decoder a realistic option for forthcoming QEC implementations.The authors report that the training data require roughly 60 s to generate on experimental hardware.
7 Conclusion and outlook: ηd = 0.81, a 26% improvement over the blossom decoder, was achieved in a Surface-17 density-matrix simulation with superconducting transmon parameters.The simulation covered fidelity decay over 100 cycles and used 4 · 10^6 training datasets.
7 Conclusion and outlook: Future work must address low-error-rate training, other topological codes, and surface-code variants used for lattice surgery or braiding.The decoder may need modification to avoid dependence on a single code distance.

A.1 Architecture

The decoder uses two recurrent networks whose probabilistic outputs are combined into a bit-flip logical-error probability. Each network uses LSTM layers and a rectified-linear evaluation layer, with the second network also receiving the final syndrome increment.

A.1 Architecture: The decoder combines two network outputs as p = p1(1 − p2) + p2(1 − p1), yielding the probability that the logical qubit incurred a bit-flip error.The networks process syndrome-increment sequences, with the second also using the final syndrome increment.
A.1 Architecture: The individual probability outputs have a complement symmetry, but this ambiguity does not affect the networks’ joint operation.It can be removed by testing on an all-zero-syndrome example.
A.1 Architecture: Both networks contain two LSTM layers followed by a fully connected evaluation layer with 64 rectified linear units.The first layer receives syndrome increments, while subsequent evaluation inputs differ between the two networks.
A.1 Architecture: The first network evaluates the final second-layer LSTM output, whereas the second concatenates that output with the final syndrome increment.Both outputs are passed through a rectified-linear activation before evaluation.

A.2 Training and evaluation

The networks are trained on millions of variable-length syndrome sequences and selected using long-sequence validation. Evaluation uses separate test data, parity predictions, and a fitted logical-error rate per cycle.

A.2 Training and evaluation: The networks are trained simultaneously on 4 · 10^6 sequences of lengths T = 11–20 cycles using mini-batches of 64 and the Adam optimizer.Training uses a learning rate of 10^-3 and one epoch contains 10^4 mini-batches.
A.2 Training and evaluation: Validation uses 10^4 sequences of lengths T = 81–100 cycles, storing the best decoder and stopping after 100 epochs without improvement.Three decoders are trained, and the instance with the lowest validation logical error rate is selected.
A.2 Training and evaluation: The test set contains 5·10^4 sequences with T = 300 for the Pauli error model and T = 100 for the density-matrix simulation.Unlike training and validation, test sequences provide final syndrome increments and final error parity after each cycle.
A.2 Training and evaluation: Predictions classify parity as even when p < 0.5 and odd otherwise, with logical fidelity averaged over the test set to obtain the logical error rate per cycle.A two-parameter fit to Eq. (8) is used to obtain that rate.

B Parity-bit error versus Pauli-frame-update error

The paper shows that parity-bit error and Pauli-frame-update error rates are equivalent. In experiments, parity output is also more appropriate because the true Pauli frame depends on measuring the data qubits.

B Parity-bit error versus Pauli-frame-update error: The Pauli-frame-update error rate ϵP equals the parity-bit error rate ϵ used to evaluate the decoder.The equivalence follows because differing logical commutation parities indicate that the calculated and true frame updates differ by a logical operator.
B Parity-bit error versus Pauli-frame-update error: A Pauli frame is a list of Pauli X, Y, or Z errors on data qubits, with stabilizer-separated frames treated as equivalent.The logical-operator choice has gauge freedom because multiplying by a stabilizer leaves the logical subspace unchanged.
B Parity-bit error versus Pauli-frame-update error: The decoder forms a Pauli-frame update by multiplying a simple frame by XL or ZL according to its predicted parity bit.An error occurs when this calculated update differs from the true Pauli-frame update.
B Parity-bit error versus Pauli-frame-update error: In physical experiments, the true Pauli frame is undetermined until data qubits are measured, so predicting a parity bit avoids predicting quantum-random measurement outcomes.This makes parity output more convenient and appropriate for the neural network.