Conservation laws and quantum error correction

Quantum error-correcting codes are essential to the realisation of scalable quantum computation. They defend encoded quantum information by making stabilizer measurements to identify the occurrence of errors. 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 three-dimensional fracton codes, as well as classical fractal 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.