Neural ensemble decoding for topological quantum error-correcting codes

Abstract

Topological quantum error-correcting codes are a promising candidate for building fault-tolerant quantum computers. Decoding topological codes optimally, however, is known to be a computationally hard problem. Various decoders have been proposed that achieve approximately optimal error thresholds. Due to practical constraints, it is not known if there exists an obvious choice for a decoder. In this paper, we introduce a framework which can combine arbitrary decoders for any given code to significantly reduce the logical error rates. We rely on the crucial observation that two different decoding techniques, while possibly having similar logical error rates, can perform differently on the same error syndrome. We use machine learning techniques to assign a given error syndrome to the decoder which is likely to decode it correctly. We apply our framework to an ensemble of Minimum-Weight Perfect Matching (MWPM) and Hard-Decision Re-normalization Group (HDRG) decoders for the surface code in the depolarizing noise model. Our simulations show an improvement of 38.4%, 14.6%, and 7.1% over the pseudo-threshold of MWPM in the instance of distance 5, 7, and 9 codes, respectively. Lastly, we discuss the advantages and limitations of our framework and applicability to other error-correcting codes. Our framework can provide a significant boost to error correction by combining the strengths of various decoders. In particular, it may allow for combining very fast decoders with moderate error-correcting capability to create a very fast ensemble decoder with high error-correcting capability.

Rectangular surface code under biased noise

“…(13) into Eq. (7,(11)(12), yields an expression for P L X and P L Z in terms of lattice size, physical error rate, and bias.…”

“…3.2, we estimated the logical error rate for the high physical error rate as in Eq. (12,13). This appendix exhibits how these equations are obtained from N = 10 5 data set.…”

“…Recent QEC research focuses on the implementation and construction of the QEC codes [7,8] by considering a biased noise error channel and other channels [9], by designing the efficient decoder using the machine learning [10,11] techniques, and by improving the threshold below which the logical failure rate can be decreased. A framework that applies machine learning techniques for decoding and improves the logical error rate in the depolarizing noise channel is also proposed in [12]. Previous work by Panos et al proposed a concatenated phase flip QEC code [13].…”

Optimizing Quantum Error Correction Codes with Reinforcement Learning

“…For the problem of finding optimized QEC strategies for near-term quantum devices, adaptive machine learning [11] approaches may succeed where brute force searches fail. In fact, machine learning has already been applied to a wide range of decoding problems in QEC [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. Efficient decoding is of central interest in any fault-tolerant scheme.…”

Determination of the semion code threshold using neural decoders

“…Machine learning, and in particular neural networks, has been proposed in recent years as a solution for efficiently decoding stabilizer codes [18][19][20][21][22][23][24][25][26][27][28][29][30][31]. Although there are different approaches to the decoding problem using neural networks, one of the most common consists in applying a very simple decoder to the code.…”