  Recently, Hastings & Haah introduced a quantum memory defined on the
honeycomb lattice. Remarkably, this honeycomb code assembles weight-six parity
checks using only two-local measurements. The sparse connectivity and two-local
measurements are desirable features for certain hardware, while the weight-six
parity checks enable robust performance in the circuit model.
  In this work, we quantify the robustness of logical qubits preserved by the
honeycomb code using a correlated minimum-weight perfect-matching decoder.
Using Monte Carlo sampling, we estimate the honeycomb code's threshold in
different error models, and project how efficiently it can reach the "teraquop
regime" where trillions of quantum logical operations can be executed reliably.
We perform the same estimates for the rotated surface code, and find a
threshold of $0.2\%-0.3\%$ for the honeycomb code compared to a threshold of
$0.5\%-0.7\%$ for the surface code in a controlled-not circuit model. In a
circuit model with native two-body measurements, the honeycomb code achieves a
threshold of $1.5\% < p <2.0\%$, where $p$ is the collective error rate of the
two-body measurement gate - including both measurement and correlated data
depolarization error processes. With such gates at a physical error rate of
$10^{-3}$, we project that the honeycomb code can reach the teraquop regime
with only $600$ physical qubits.
