In this publication, we theoretically and numerically demonstrate a powerful analog-digital hybrid quantum simulation protocol that can be used to simulate spin-dynamics on a two dimensional lattice using a linear chain of ions. The simulation protocol makes use of native long range interactions between trapped ion spins, that are then digitally modified to achieve the target spin-spin interaction graph. On one hand, the hybrid simulation scheme is more flexible and requires less experimental control over individual ions than an analog quantum simulation, where the Hamiltonian is simulated by fine-tuning the control parameters. On the other hand, the hybrid simulation scheme does not build the entire time evolution out of quantum logic gates, unlike a fully digital quantum simulator, and therefore is more robust against Trotterization errors. This work is performed in collaboration with researchers from University of California, Berkeley and Army Research Laboratory, Maryland.

Abstract: Controlling the interaction graph between spins or qubits in a quantum simulator allows user-controlled tailoring of native interactions to achieve a target Hamiltonian. The flexibility of engineering long-ranged phonon-mediated spin-spin interactions in a trapped ion quantum simulator offers such a possibility. Trapped ions, a leading candidate for simulating computationally hard quantum many-body dynamics, are most readily trapped in a linear 1D chain, limiting their utility for readily simulating higher dimensional spin models. In this work, we introduce a hybrid method of analog-digital simulation for simulating 2D spin models and dynamically changing interactions to achieve a new graph using a linear 1D chain. The method relies on time domain Hamiltonian engineering through a successive application of Stark shift gradient pulses, and wherein the pulse sequence can simply be obtained from a Fourier series decomposition of the target Hamiltonian over the space of lattice couplings. We focus on engineering 2D rectangular nearest-neighbor spin lattices, demonstrating that the required control parameters scale linearly with ion number. This hybrid approach offers compelling possibilities for the use of 1D chains in the study of Hamiltonian quenches, dynamical phase transitions, and quantum transport in 2D and 3D. We discuss a possible experimental implementation of this approach using real experimental parameters.