Errors from Dynamical Structural Instabilities of Floquet Maps in Quantum Simulation

We study the behavior of errors in the quantum simulation of  spin systems with long-range multibody interactions resulting from the  Trotter-Suzuki decomposition of the time evolution operator. We  identify a regime where the Floquet operator underlying the Trotter  decomposition undergoes sharp changes even for small variations in the  simulation step size. This results in a time evolution operator that  is very different from the dynamics generated by the targeted  Hamiltonian, which leads to a proliferation of errors in the quantum  simulation. These regions of sharp change in the Floquet operator,  referred to as structural instability regions, appear typically at  intermediate Trotter step sizes and in the weakly interacting regime,  and are thus complementary to recently revealed quantum chaotic  regimes of the Trotterized evolution [L. M. Sieberer et al. npj  Quantum Inf. 5, 78 (2019); M. Heyl, P. Hauke, and P. Zoller, Sci. Adv.  5, eaau8342 (2019)]. We characterize these structural instability  regimes in p-spin models, transverse-field Ising models with  all-to-all p-body interactions, and analytically predict their  occurrence based on unitary perturbation theory. We further show that  the effective Hamiltonian associated with the Trotter decomposition of  the unitary time-evolution operator, when the Trotter step size is  chosen to be in the structural instability region, is very different  from the target Hamiltonian, which explains the large errors that can  occur in the simulation in the regions of instability. These results  have implications for the reliability of near-term gate based quantum  simulators, and reveal an important interplay between errors and the  physical properties of the system being simulated.

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