  Digital quantum simulation (DQS) is one of the most promising paths for
achieving first useful real-world applications for quantum processors. Yet even
assuming rapid progress in device engineering and development of fault-tolerant
quantum processors, algorithmic resource optimisation will long remain crucial
to exploit their full power. Currently, Trotterisation provides
state-of-the-art resource scaling. And recent theoretical studies of
Trotterised Ising models suggest that even better performance than expected may
be possible up to a distinct breakdown threshold in empirical performance.
Here, we study multiple paradigmatic DQS models with experimentally realisable
Trotterisations, and evidence the universality of a range of Trotterisation
performance behaviours, including not only the threshold, but also new features
in the pre-threshold regime that is most important for practical applications.
In each model, we observe a distinct Trotterisation threshold shared across
widely varying performance signatures; we further show that an onset of quantum
chaotic dynamics causes the performance breakdown and is directly induced by
digitisation errors. In the important pre-threshold regime, we are able to
identify new distinct regimes displaying qualitatively different quasiperiodic
performance behaviours, and show analytic behaviour for properly defined
operational Trotter errors. Our results rely crucially on diverse new
analytical tools, and provide a previously missing unified picture of
Trotterisation behaviour across local observables, the global quantum state,
and the full Trotterised unitary. This work provides new insights and tools for
addressing important questions about the algorithm performance and underlying
theoretical principles of sufficiently complex Trotterisation-based DQS, that
will help in extracting maximum simulation power from future quantum
processors.
