  From the perspective of quantum many-body physics, the Floquet code of
Hastings and Haah can be thought of as a measurement-only version of the Kitaev
honeycomb model where a periodic sequence of two-qubit XX, YY, and ZZ
measurements dynamically stabilizes a toric code state with two logical qubits.
However, the most striking feature of the Kitaev model is its intrinsic
fractionalization of quantum spins into an emergent gauge field and itinerant
Majorana fermions that form a Dirac liquid, which is absent in the Floquet
code. Here we demonstrate that by varying the measurement strength of the
honeycomb Floquet code one can observe features akin to the fractionalization
physics of the Kitaev model at finite temperature. Introducing coherent errors
to weaken the measurements we observe three consecutive stages that reveal
qubit fractionalization (for weak measurements), the formation of a Majorana
liquid (for intermediate measurement strength), and Majorana pairing together
with gauge ordering (for strong measurements). Our analysis is based on a
mapping of the imperfect Floquet code to random Gaussian fermionic circuits
(networks) that can be Monte Carlo sampled, exposing two crossover peaks. With
an eye on circuit implementations, our analysis demonstrates that the Floquet
code, in contrast to the toric code, does not immediately break down to a
trivial state under weak measurements, but instead gives way to a long-range
entangled Majorana liquid state.
