Optimizing error correction intervals for qubits in noisy memories

Quantum error correction protects qubits stored in noisy memories. One can perform repeated rounds of error correction to recover from errors accumulated between correction intervals. Unfortunately, there are overheads such as gate and measurement errors associated with performing error correction operations that require fine tuning of the timing between error correction rounds. We investigate the optimization of the error correction interval in discrete quantum error correction codes when qubit storage times are random. Specifically, we focus on scenarios where logical errors can be modeled as independent flips per correction round (memoryless) like CSS codes and other error correction schemes under generalized Pauli error channels. We find that under certain noise models (dephasing and depolarizing) and error correction schemes, the optimization does not depend upon the storage time patterns yielding an invariancy for the optimization condition. We further explore the conditions for this invariancy and study noise models where it breaks and the performance cost of operating under the memoryless assumption. Lastly we explore this problem in the spatial dimension and find that the results translate to optimization of repeater link lengths in multipath quantum networks.