  The requirements for fault-tolerant quantum error correction can be
simplified by leveraging structure in the noise of the underlying hardware. In
this work, we identify a new type of structured noise motivated by neutral atom
qubits, biased erasure errors, which arises when qubit errors are dominated by
detectable leakage from only one of the computational states of the qubit. We
study the performance of this model using gate-level simulations of the XZZX
surface code. Using the predicted erasure fraction and bias of metastable
$^{171}$Yb qubits, we find a threshold of 8.2% for two-qubit gate errors, which
is 1.9 times higher than the threshold for unbiased erasures, and 7.5 times
higher than the threshold for depolarizing errors. Surprisingly, the improved
threshold is achieved without bias-preserving controlled-not gates, and instead
results from the lower noise entropy in this model. We also introduce an XZZX
cluster state construction for measurement-based error correction,
hybrid-fusion, that is optimized for this noise model. By combining fusion
operations and deterministic entangling gates, this construction preserves the
intrinsic symmetry of the XZZX code, leading to a higher threshold of 10.3% and
enabling the use of rectangular codes with fewer qubits. We discuss a potential
physical implementation using a single plane of atoms and moveable tweezers.
