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Quantum processors use the native interactions between effective spins to simulate Hamiltonians or execute quantum gates. In most processors, the native interactions are pairwise, limiting the efficiency of controlling entanglement between many qubits. The capability of manipulating entanglement generated by higher-order interactions is a key challenge for the simulation of many Hamiltonian models appearing in various fields, including high-energy and nuclear physics, as well as quantum chemistry and error correction applications. Here we experimentally demonstrate control over a class of native interactions between trapped-ion qubits, extending conventional pairwise interactions to a higher order. By exploiting state-dependent squeezing operations, we realize and characterize high-fidelity gates and spin Hamiltonians comprising three- and four-body spin interactions. Our results demonstrate the potential of high-order spin interactions as a toolbox for quantum information applications. 

Simulations of nuclear magnetic resonance (NMR) experiments can be an important tool for extracting information about molecular structure and optimizing experimental protocols but are often intractable on classical computers for large molecules such as proteins and for protocols such as zero-field NMR. We demonstrate the first quantum simulation of an NMR spectrum, computing the zero-field spectrum of the methyl group of acetonitrile using four qubits of a trapped-ion quantum computer. We reduce the sampling cost of the quantum simulation by an order of magnitude using compressed sensing techniques. We show how the intrinsic decoherence of NMR systems may enable the zero-field simulation of classically hard molecules on relatively near-term quantum hardware and discuss how the experimentally demonstrated quantum algorithm can be used to efficiently simulate scientifically and technologically relevant solid-state NMR experiments on more mature devices. Our work opens a practical application for quantum computation.