Statistical Mechanics of Quantum Codes*

The many-body dynamics of quantum circuits and quantum error correction are a rapidly developing application of computational statistical mechanics. Phenomena such as the growth of entanglement, robust encoding of quantum information, and successful decoding under noise all lend themselves to statistical mechanics mappings. In this talk, I will highlight some paradigmatic quantum coding phases and phase transitions, along with the computational methods used to study them. I will further discuss a sampling of our ongoing work in this field, which leverages the classical simulatability of Clifford circuits to probe large system sizes. A common thread in this research is “derandomization," evolving a state under a spacetime translation-invariant dynamics. Lack of randomness can reduce the computational complexity of encoding and decoding, while sometimes modifying the universality class of the associated transitions. These transitions include error-correction thresholds in dynamically generated codes; unbinding of entanglement membranes due to measurements and/or dissipation; and fault-tolerant thresholds in codes prepared by multitree circuits. For the latter, tensor network methods for evaluating and updating marginal probabilities underlie the approximate “probability passing” decoding algorithm for realistic noise models.