Measuring Errors and Signatures of Chaos in Trotterized Quantum Simulation Using Atomic Qudits

Noisy, intermediate-scale quantum (NISQ) processors are improving rapidly but remain well short of requirements for fault tolerant computation. In the meantime, much effort has focused on the development of quantum simulators that operate without error correction. So-called “digital” processors can simulate non-native Hamiltonians through Trotterization, wherein the evolution is broken into discrete steps using a Trotter-Suzuki expansion. When simulating the evolution over a total time T, this introduces Trotter errors that scale inversely with the number of time steps. For optimal performance, this must be weighed against the native errors inherent to the processor hardware implementation, which scale roughly in proportion with the number of time steps. Notably, the optimal step size can be affected by the appearance of chaos in the Trotterized dynamics, which leads to hypersensitivity to both Trotter and native errors. We investigate each of these error regimes in quantum simulations running on a small, highly-accurate quantum (SHAQ) processor based on the combined electron-nuclear spins of a Cs-133 atom. As a concrete example, we focus on the Lipkin-Meshkov-Glick Hamiltonian, which when Trotterized becomes the Quantum-Kicked-Top - a well-studied system that exhibits chaos and dynamical instability. We use our SHAQ processor to study the sensitivity of observables to errors, and distinguish between Trotter and native errors in quantum simulation. Finally, we show that fidelity-OTOC measurements can be implemented and used to identify the presence of chaos and instabilities as they appear and disappear with changing Trotter step size