Local Probabilistic Decoding of a Quantum Code

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="monospace">f</mml:mi><mml:mi mathvariant="monospace">l</mml:mi><mml:mi mathvariant="monospace">i</mml:mi><mml:mi mathvariant="monospace">p</mml:mi></mml:mrow></mml:math> is an extremely simple and maximally local classical decoder which has been used to great effect in certain classes of classical codes. When applied to quantum codes there exist constant-weight errors (such as half of a stabiliser) which are uncorrectable for this decoder, so previous studies have considered modified versions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="monospace">f</mml:mi><mml:mi mathvariant="monospace">l</mml:mi><mml:mi mathvariant="monospace">i</mml:mi><mml:mi mathvariant="monospace">p</mml:mi></mml:mrow></mml:math>, sometimes in conjunction with other decoders. We argue that this may not always be necessary, and present numerical evidence for the existence of a threshold for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="monospace">f</mml:mi><mml:mi mathvariant="monospace">l</mml:mi><mml:mi mathvariant="monospace">i</mml:mi><mml:mi mathvariant="monospace">p</mml:mi></mml:mrow></mml:math> when applied to the looplike syndromes of a three-dimensional toric code on a cubic lattice. This result can be attributed to the fact that the lowest-weight uncorrectable errors for this decoder are closer (in terms of Hamming distance) to correctable errors than to other uncorrectable errors, and so they are likely to become correctable in future code cycles after transformation by additional noise. Introducing randomness into the decoder can allow it to correct these "uncorrectable" errors with finite probability, and for a decoding strategy that uses a combination of belief propagation and probabilistic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="monospace">f</mml:mi><mml:mi mathvariant="monospace">l</mml:mi><mml:mi mathvariant="monospace">i</mml:mi><mml:mi mathvariant="monospace">p</mml:mi></mml:mrow></mml:math> we observe a threshold of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>&amp;#x223C;</mml:mo><mml:mn>5.5</mml:mn><mml:mi mathvariant="normal">&amp;#x0025;</mml:mi></mml:math> under phenomenological noise. This is comparable to the best known threshold for this code (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>&amp;#x223C;</mml:mo><mml:mn>7.1</mml:mn><mml:mi mathvariant="normal">&amp;#x0025;</mml:mi></mml:math>) which was achieved using belief propagation and ordered statistics decoding [Higgott and Breuckmann, 2022], a strategy with a runtime of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math> as opposed to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math> when parallelised) runtime of our local decoder. We expect that this strategy could be generalised to work well in other low-density parity check codes, and hope that these results will prompt investigation of other previously overlooked decoders.