Color code decoder with improved scaling for correcting circuit-level noise

Two-dimensional color codes are a promising candidate for fault-tolerant quantum computing, as they have high encoding rates, transversal implementation of logical Clifford gates, and resource-efficient magic state preparation schemes. However, decoding color codes presents a significant challenge due to their structure, where elementary errors violate three checks instead of just two (a key feature in surface code decoding), and the complexity of extracting syndrome is greater. We introduce an efficient color-code decoder that tackles these issues by combining two matching decoders for each color, generalized to handle circuit-level noise by employing detector error models. We provide comprehensive analyses of the decoder, covering its threshold and sub-threshold scaling both for bit-flip noise with ideal measurements and for circuit-level noise. Our simulations reveal that this decoding strategy nearly reaches the best possible scaling of logical failure (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>p</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">f</mml:mi><mml:mi mathvariant="normal">a</mml:mi><mml:mi mathvariant="normal">i</mml:mi><mml:mi mathvariant="normal">l</mml:mi></mml:mrow></mml:msub><mml:mo>&amp;#x223C;</mml:mo><mml:msup><mml:mi>p</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>d</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>) for both noise models, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math> is the noise strength, in the regime of interest for fault-tolerant quantum computing. While its noise thresholds are comparable with other matching-based decoders for color codes (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>8.2</mml:mn><mml:mi mathvariant="normal">&amp;#x0025;</mml:mi></mml:math> for bit-flip noise and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0.46</mml:mn><mml:mi mathvariant="normal">&amp;#x0025;</mml:mi></mml:math> for circuit-level noise), the scaling of logical failure rates below threshold significantly outperforms the best matching-based decoders.