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. 2017 Sep 8;7(1):11003.
doi: 10.1038/s41598-017-11266-1.

Deep Neural Network Probabilistic Decoder for Stabilizer Codes

Stefan Krastanov  1   2 , Liang Jiang  3   4
Affiliations

Deep Neural Network Probabilistic Decoder for Stabilizer Codes

Stefan Krastanov et al. Sci Rep. .

Abstract

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.

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Conflict of interest statement

The authors declare that they have no competing interests.

Figures

Figure 1
Figure 1
Quantum Error Correcting Codes. A very general class of QEC codes is the class stabilizer codes, defined by the stabilizer subgroup of the physical qubits that leaves the state of the logical qubits unperturbed. Our neural architecture can be readily applied to such codes, however many codes of practical interest (like the one we are testing against) have additional structure that would be interesting to consider. The example in (a) shows a small patch of a toric code, which is a CSS code (the stabilizer operators are products of only Z or only X operators, permitting us to talk of Z and X syndromes separately). Moreover, the toric code possesses a lattice structure that provides for a variety of decoders designed to exploit that structure. Our decoder, depicted in (b), does not have built-in knowledge of that structure, rather it learns it through training. Due to size constraints, the depictions present only a small subset of all qubit or syndrome nodes.
Figure 2
Figure 2
Decoder performance for toric codes of distances 5 and 7. The x axis is the depolarization rate of the physical qubits (the probability that an X, Y, or Z error has occurred), while the y axis is the fraction of properly decoded code iterations (the conjugate of the logical error rate). The neural network decoder (rectangular markers) significantly outperforms the minimal weight perfect matching decoder (triangular markers), both in terms of threshold and logical error rate. For the above plots, neural networks with 18 hidden layers were used.
Figure 3
Figure 3
Correlations learned by the neural network. The neural network and MWPM decoder performances for a distance 5 code from Fig. 2 are repeated in this plot. To visualize the importance of taking into account correlations between errors, we also plot the square of the “corrected fraction” for a neural and a MWPM decoder decoding only the Z stabilizer (this neural decoder was trained only on Z stabilizer data). The neural decoders outperforms MWPM both when decoding only the Z stabilizer and when decoding Z and X together. If there were no correlations between errors then the squared value for decoding Z would be the same as the value for decoding both Z and X for each decoder. However, the difference is much more substantial between the two neural decoders, demonstrating the limitations of MWPM and similar decoders that do not account for correlations.
Figure 4
Figure 4
Sampling the neural network. (Arrays in the diagram are in bold font, H is the parity check matrix of the code) After the neural network is trained its output can efficiently be evaluated for any given syndrome s. The output array E is interpreted as a list of probabilities for each qubit for an error to have happened. An array e (whether an error occurred at each qubit) is sampled from E. In the loop we check whether the guess e actually produces the same syndrome as the initially given one. If not, we resample only the qubits taking part in the stabilizer measurement corresponding to the incorrect elements of the syndrome. If the loop runs for more than a set number of iterations we declare failure to decode (a detected, however not corrected, error). As for any other decoding algorithm the final result may be wrong if the total error that occurred is of particularly low probability (i.e. of high weight). In the case of a general stabilizer code H stands for the list of stabilizer operators.
Figure 5
Figure 5
Sampling overhead versus decoder performance. Sampling possible errors from the output of the neural network is an iterative process not guaranteed to reach an acceptable solution, therefore we need to set an upper bound on how many iterations are permitted before giving up (which would result in a detected but not corrected error). The plot gives the performance of our decoder trained on toric codes of different distances with respect to the maximal permitted number of iterations. The dashed lines give the MWPM decoder performances for the same codes as a reference. Codes up to distance 9 (containing 162 physical qubits) are practical, but extending using our decoder for codes with more than 242 physical qubits would be prohibitive due to the sampling overhead. The evaluations were done for 10% depolarization rate on the physical qubits.
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