Observation of a fault tolerance threshold with concatenated codes

We introduce a fault-tolerant protocol for code concatenation of a generalized Shor code using a butterfly network architecture with high-noise thresholds and low ancilla overhead to allow implementation on current devices. We develop a probability passing decoder using tensor networks that applies Bayesian updates to the marginal error probabilities after each layer of checks, achieving a state-preparation threshold of ec≈0.089 for erasure errors and ≈0.015 for unheralded noise. We implement our state-preparation protocol on ion-trap hardware with added noise to demonstrate the threshold behavior in a real quantum device. We further theoretically test the performance of our scheme as a quantum memory and for universal quantum computation through the preparation of low-noise magic states for state distillation and T-gate injection.