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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:msub> <a:mi>e</a:mi> <a:mi>c</a:mi> </a:msub> <a:mo>≈</a:mo> <a:mn>0.089</a:mn> </a:mrow> </a:math> for erasure errors and <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mo>≈</b:mo> <b:mn>0.015</b:mn> </b:mrow> </b:math> 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 <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi>T</c:mi> </c:math> -gate injection.