Non-Markovianity of Logical Errors in Early Fault-tolerant Memory Experiments

It has been previously shown that logical qubits encoded in quantum error correcting codes can exhibit non-Markovian dynamical evolution, even when the underlying physical noise is Markovian. It is not fully understood how inhibitory this phenomenon will be to application of gate-based characterization techniques like gate set tomography (which assume Markovianity) to early fault-tolerant quantum systems. In previous work, we studied QEC circuits where each round is decoded independently, showed that the effect occurs only if there are syndrome readout errors, and that the effect is strong particularly for short depth and non-fault-tolerant circuits. To better understand how this non-Markovian phenomenon would affect the characterization of early fault-tolerant quantum computers over many rounds of QEC, we simulate fault-tolerant QEC circuits designed to tolerate a small number of syndrome errors, and where the number of syndrome extraction rounds accessible to the decoder can vary. We include examples of small distance codes, and analyze the interplay between syndrome errors, time correlations, and logical error dynamics.