Tensor-network decoders for process tensor descriptions of non-Markovian noise

Quantum error correction (QEC) is essential for fault-tolerant quantum computation. Often in QEC errors are assumed to be independent and identically distributed and can be discretised to a random Pauli error during the execution of a quantum circuit. In real devices, however, the noise profile is much more complex and contains non-trivial spatiotemporal correlations, such as cross-talk, non-Markovianity, and their mixtures. Here, we examine the performance of two paradigmatic QEC codes in the presence of complex noise by using process tensors to represent spatiotemporal correlations beyond iid errors. This integration is an instance of the recently proposed $\mathit{\text{strategic}}\mathit{\text{code}}$, which combines QEC with process tensors. In particular, we construct the maximum likelihood (ML) decoder for a quantum error correction code with a process tensor. To understand the computational overhead and implications of this approach, we implement our framework numerically for small code instances and evaluate its performance. We also propose a method to evaluate the performance of strategic codes and construct the ML decoder with an efficient tensor network approximation. Our results highlight the possible detrimental effects of correlated noise and potential pathways for designing decoders that account for such effects.