  Unlike the surface code, quantum low-density parity-check (QLDPC) codes can
have a finite encoding rate, potentially lowering the error correction
overhead. However, finite-rate QLDPC codes have nonlocal stabilizers, making it
difficult to design stabilizer measurement circuits that are low-depth and do
not decrease the effective distance. Here, we demonstrate that a popular family
of finite-rate QLDPC codes, hypergraph product codes, has the convenient
property of distance-robustness: any stabilizer measurement circuit preserves
the effective distance. In particular, we prove the depth-optimal circuit in
[Tremblay et al, PRL 129, 050504 (2022)] is also optimal in terms of effective
distance.
