  Quantum circuits make it possible to simulate the continuous-time dynamics of
a many-body Hamiltonian by implementing discrete Trotter steps of duration
$\tau$. However, when $\tau$ is sufficiently large, the discrete dynamics
exhibit qualitative differences compared to the original evolution, potentially
displaying novel features and many-body effects. We study an interesting
example of this phenomenon, by considering the integrable Trotterization of a
prototypical integrable model, the XXZ Heisenberg spin chain. We focus on the
well-known bipartition protocol, where two halves of a large system are
prepared in different macrostates and suddenly joined together, yielding
non-trivial nonequilibrium dynamics. Building upon recent results and adapting
the generalized hydrodynamics (GHD) of integrable models, we develop an exact
large-scale description of an explicit one-dimensional quantum-circuit setting,
where the input left and right qubits are initialized in two distinct product
states. We explore the phenomenology predicted by the GHD equations, which
depend on the Trotter step and the gate parameters. In some phases of the
parameter space, we show that the quantum-circuit large-scale dynamics is
qualitatively different compared to the continuous-time evolution. In
particular, we find that a single microscopic defect at the junction, such as
the addition of a single qubit, may change the nonequilibrium macrostate
appearing at late time.
