Hamiltonian Engineering on a hyper-sphere with a trapped ion quantum simulator

Trapped ion systems utilize the collective normal modes to mediate effective spin-spin interactions, thereby realizing quantum computers and simulators. We demonstrate ways to engineer Hamiltonians with a trapped ion chain by addressing the axial modes, in which the mode spectrum is sparse and has near equal mode spacings. This allows us to generate flexible spin-spin interactions with mode addressing while eliminating all residual spin-phonon interactions.  We demonstrate quantum simulations of spin Hamiltonians in high-dimensional geometries by applying frequency-modulated two-tone laser fields to sequentially address different modes. By choosing the gate time and interaction strength of each layer, we encode different connectivity configurations of qubits and apply a digital-analog model to implement Ising Hamiltonians on a square-lattice (4 qubits), a sphere (6 qubits), and a four-dimensional hyper-sphere (8 qubits). Our results can be applied to fast parallel gates and stimulate a new direction of digital-analog hybrid quantum simulation.