Machine Learning Decoding of Circuit-Level Noise for Bivariate Bicycle Codes

Fault-tolerant quantum computers will depend crucially on the performance of the classical decoding algorithm which takes in the results of measurements and outputs corrections to the errors inferred to have occurred. Machine learning models have shown great promise as decoders for the surface code; however, this promise has not yet been substantiated for the more challenging task of decoding quantum low-density parity-check (QLDPC) codes. In this paper, we present a recurrent, transformer-based neural network designed to decode circuit-level noise on Bivariate Bicycle (BB) codes. For the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo stretchy="false">[</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>72</mml:mn> <mml:mo>,</mml:mo> <mml:mn>12</mml:mn> <mml:mo>,</mml:mo> <mml:mn>6</mml:mn> <mml:mo stretchy="false">]</mml:mo> <mml:mo stretchy="false">]</mml:mo> </mml:math> BB code, at a physical error rate of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.1</mml:mn> <mml:mi mathvariant="normal">&amp;#x0025;</mml:mi> </mml:math> , our model achieves logical error rates almost <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>5</mml:mn> </mml:math> times lower than belief propagation with ordered statistics decoding (BP-OSD), and roughly <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>5</mml:mn> </mml:math> times larger than a most-likely error decoder. Moreover, while BP-OSD has a wide distribution of runtimes with significant outliers, our model has a consistent runtime and is an order-of-magnitude faster than the worst-case times from a benchmark BP-OSD implementation. On the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo stretchy="false">[</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>144</mml:mn> <mml:mo>,</mml:mo> <mml:mn>12</mml:mn> <mml:mo>,</mml:mo> <mml:mn>12</mml:mn> <mml:mo stretchy="false">]</mml:mo> <mml:mo stretchy="false">]</mml:mo> </mml:math> BB code, our model obtains worse logical error rates but maintains the speed advantage. These results provide initial evidence that machine learning decoders can out-perform conventional decoders on small QLDPC codes, but suggest more complex architectures and/or training procedures are necessary to scale to larger code sizes.