Mixed-state Phases: A New Window into Error Correction and Machine Learning

Advances in controllably open quantum systems have enabled both a vast landscape of non-equilibrium mixed state phases of matter and fault tolerant quantum memories. I will first review how Markov length, the length scale with which the conditional mutual information (CMI) decays exponentially, is an essential quantity characterizing mixed-state phases. Then, I will establish a correspondence between the fault tolerance of local stabilizer codes experiencing measurement and physical errors and the mixed state phases of decohered resource states in one higher dimension. This motivates a diagnostic of fault-tolerance--the spacetime Markov length-- which is determined by the decay of the CMI of repeated syndrome measurement outcomes in spacetime. The diagnostic is independent of the decoder, and its divergence signals the intrinsic breakdown of fault tolerance. Finally, I will apply these physical insights to machine learning, identifying and characterizing classes of distributions that are hard to learn.