  Recently, a lot of effort has been devoted towards designing erasure qubits
in which dominant physical noise excites leakage states whose population can be
detected and returned to the qubit subspace. Interest in these erasure qubits
has been driven by studies showing that the requirements for fault-tolerant
quantum error correction are significantly relaxed when noise in every gate
operation is dominated by erasures. However, these studies assume perfectly
accurate erasure checks after every gate operation which generally come with
undesirable time and hardware overhead costs. In this work, we investigate the
consequences of using an imperfect but overhead-efficient erasure check for
fault-tolerant quantum error correction with the surface code. We show that,
under physically reasonable assumptions on the imperfect erasure checks, the
threshold error rate is still at least over twice that for Pauli noise. We also
study the impact of imperfect erasure checks on the effective error distance
and find that it degrades the effective distance under a general error model in
which a qubit suffers from depolarizing noise when interacting with a leaked
qubit. We then identify a more restrictive but realistic noise model for a
qubit that interacts with a leaked qubit, under which the effective error
distance is twice that for Pauli noise. We apply our analysis to recently
proposed superconducting dual-rail erasure qubits and show that achieving good
performance surface code quantum memories with relaxed system requirements is
possible.
