  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.
