  Hypergraph product codes are a promising avenue to achieving fault-tolerant
quantum computation with constant overhead. When embedding these and other
constant-rate qLDPC codes into 2D, a significant number of nonlocal connections
are required, posing difficulties for some quantum computing architectures. In
this work, we introduce a fault-tolerance scheme that aims to alleviate the
effects of implementing this nonlocality by measuring generators acting on
spatially distant qubits less frequently than those which do not. We
investigate the performance of a simplified version of this scheme, where the
measured generators are randomly selected. When applied to hypergraph product
codes and a modified small-set-flip decoding algorithm, we prove that for a
sufficiently high percentage of generators being measured, a threshold still
exists. We also find numerical evidence that the logical error rate is
exponentially suppressed even when a large constant fraction of generators are
not measured.
