Transversal logical gates offer the opportunity for fast and low-noise logic, particularly when interspersed by a single round of parity check measurements of the underlying code. Using such circuits for the surface code requires decoding across logical gates, complicating the decoding task. We show how one can decode across an arbitrary sequence of transversal gates for the unrotated surface code, using a fast “logical observable” minimum-weight perfect matching-based decoder, and benchmark its performance in Clifford circuits under circuit-level noise. We propose windowed logical observable matching decoders to address the problem of fully efficient decoding: our basic windowed decoder is computationally efficient under the restriction of quiescent (slow) resets. Our “advanced” two-step windowed decoder can be computationally inefficient but allows fast resets. For both windowed decoders we identify errors which scale sublinearly in d—depending on the structure of the circuit—which can lead to logical failure, and we propose methods to adapt the decoding to remove such failures. Our work highlights the complexity and interest in efficient decoding of fast logic for the surface code.

Neural network decoders can achieve a lower logical error rate compared to conventional decoders, like minimum-weight perfect matching, when decoding the surface code. Furthermore, these decoders require no prior information about the physical error rates, making them highly adaptable. In this study, we investigate the performance of such a decoder using both simulated and experimental data obtained from a transmon-qubit processor, focusing on small-distance surface codes. We first show that the neural network typically outperforms the matching decoder due to better handling of errors leading to multiple correlated syndrome defects, such as Y errors. When applied to the experimental data of Google Quantum AI [R. Acharya, Nature (London) 614, 676 (2023)10.1038/s41586-022-05434-1], the neural network decoder achieves logical error rates approximately 25% lower than minimum-weight perfect matching, approaching the performance of a maximum-likelihood decoder. To demonstrate the flexibility of this decoder, we incorporate the soft information available in the analog readout of transmon qubits and evaluate the performance of this decoder in simulation using a symmetric Gaussian-noise model. Considering the soft information leads to an approximately 10% lower logical error rate, depending on the probability of a measurement error. The good logical performance, flexibility, and computational efficiency make neural network decoders well-suited for near-term demonstrations of quantum memories.