Fault-tolerant measurement-free quantum error correction with multi-qubit gates

Measurement-free quantum error correction (MFQEC) offers an alternative to standard measurement-based QEC in platforms with an unconditional qubit reset gate. We revisit the question of fault tolerance (FT) for a measurement-free variant of the Steane code that leverages multi-qubit gates and redundant syndrome extraction, finding previously overlooked phase-flip errors that undermine FT. We then construct a revised MFQEC circuit that is resistant to all single-qubit errors, but which nonetheless cannot tolerate certain correlated errors. In order to investigate FT systematically, we introduce an efficient method to classically simulate MFQEC circuits with (i) Clifford gates for syndrome extraction, (ii) syndrome-controlled Pauli operations for decoding, and (iii) a Pauli noise model. We thereby find a pseudothreshold of $\sim0.7\%$ for our revised MFQEC Steane code under a restricted noise model previously considered in the literature. We then relax noise model assumptions to identify general requirements for FT with multi-qubit gates, finding that existing multi-qubit neutral atom gates are incompatible with fault-tolerant syndrome extraction in a straightforward implementation of both measurement-based and measurement-free variants of the Steane code. Decomposing multi-qubit gates into two-qubit gates similarly spoils FT. Finally, we discuss the theoretical ingredients that are necessary to recover FT for MFQEC codes, including single-shot FT and a recent proposal by Heußen \textit{et al.} [arXiv:2307.13296] to achieve FT by ``copying'' errors onto an ancilla register. By combining multi-qubit gates, redundant syndrome extraction, and copy-assisted FT, we construct a measurement-free and fault-tolerant variant of the Steane code with a pseudothreshold of $\sim0.1\%$.