Concatenation Schemes for Topological Fault-tolerant Quantum Error Correction

We investigate a family of fault-tolerant quantum error correction schemes based on the concatenation of small error detection or error correction codes with the three-dimensional cluster state. We propose fault-tolerant state preparation and decoding schemes that effectively convert every circuit-level error into an erasure error, leveraging the cluster state's high threshold against such errors. We find a set of codes for which such a conversion is possible, and study their performance against the standard circuit-level depolarizing model. Our best performing scheme, which is based on a concatenation with a classical code, improves the threshold by $16.5\%$ and decreases the spacetime overhead by $32\%$ compared to the scheme without concatenation, with each scheme subject to a physical error rate of $10^{-3}$ and achieving a logical error rate of $10^{-6}$.