Correlated decoding of logical qubit algorithms with transversal gates

Quantum error correction is essential to perform reliable quantum computation at scale. Recent experiments have realized error-corrected quantum algorithms on a multi-qubit logical processor, crucially relying on the use of transversal gates. Here we observe that the performance of such algorithms can be substantially improved by accounting for physical error propagation during transversal gates and decoding the logical qubits jointly. We find that this correlated decoding significantly improves the thresholds of both Clifford and non-Clifford transversal entangling gates, and we explore several decoders offering different computational runtimes and accuracies. We then apply correlated decoding to deep logical circuits with noisy syndrome extraction and find that significantly higher fidelities can be reached by utilizing this technique to reduce the number of rounds of noisy syndrome extraction per gate. This correlated decoding technique offers key advantages in early fault-tolerant computation, as well as the possibility for reduction in the spacetime cost of deep circuit logical algorithms.