Conservation laws and quantum error correction
A quantum error-correcting code depends on a classical decoding algorithm that uses the outcomes of stabilizer measurements to determine the error that needs to be repaired. Likewise, the design of a decoding algorithm depends on the underlying physics of the quantum error-correcting code that it needs to decode. The surface code, for instance, can make use of the minimum-weight perfect-matching decoding algorithm to pair the defects that are measured by its stabilizers due to its underlying charge parity conservation symmetry. In this talk I will argue that this perspective on decoding gives us a unifying principle to design decoding algorithms for exotic codes, as well as new decoding algorithms that are specialised to the noise that a code will experience. I will describe new decoders for exotic fracton codes we have designed using these principles. I will also discuss how the symmetries of a code change if we focus on restricted noise models, and how we have leveraged this observation to design high-threshold decoders for biased noise models. In addition to these examples, this talk will focus on recent work on decoding the color code, where we found a high-performance decoder by investigating the defect conservation laws at the boundaries of the color code. Remarkably, our results show that we obtain an advantage by decoding this planar quantum error-correcting code by matching defects on a manifold that has the topology of a Moebius strip.