  The simulation complexity of predicting the time evolution of delocalized
many-body quantum systems has attracted much recent interest, and simulations
of such systems in real quantum hardware are promising routes to demonstrating
a quantum advantage over classical machines. In these proposals, random noise
is an obstacle that must be overcome for a faithful simulation, and a single
error event can be enough to drive the system to a classically trivial state.
We argue that this need not always be the case, and consider a modification to
a leading quantum sampling problem-- time evolution in an interacting
Bose-Hubbard chain of transmon qubits [Neill et al, Science 2018] -- where each
site in the chain has a driven coupling to a lossy resonator and particle
number is no longer conserved. The resulting quantum dynamics are complex and
highly nontrivial. We argue that this problem is harder to simulate than the
isolated chain, and that it can achieve volume-law entanglement even in the
strong noise limit, likely persisting up to system sizes beyond the scope of
classical simulation. Further, we show that the metrics which suggest classical
intractability for the isolated chain point to similar conclusions in the noisy
case. These results suggest that quantum sampling problems including nontrivial
noise could be good candidates for demonstrating a quantum advantage in
near-term hardware.
