  Quantum hardware has the potential to efficiently solve computationally
difficult problems in physics and chemistry to reap enormous practical rewards.
Analogue quantum simulation accomplishes this by using the dynamics of a
controlled many-body system to mimic those of another system; such a method is
feasible on near-term devices. We show that previous theoretical approaches to
analogue quantum simulation suffer from fundamental barriers which prohibit
scalable experimental implementation. By introducing a new mathematical
framework and going beyond the usual toolbox of Hamiltonian complexity theory
with an additional resource of engineered dissipation, we show that these
barriers can be overcome. This provides a powerful new perspective for the
rigorous study of analogue quantum simulators.
