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A quantum simulator is a well controlled quantum system that can simulate thebehavior of another quantum system which may require exponentially largeclassical computing resources to understand otherwise. In the 1980s, Feynmanproposed the use of quantum logic gates on a standard controllable quantumsystem to efficiently simulate the behavior of a model Hamiltonian. Recentexperiments using trapped ions and neutral atoms have realized quantumsimulation of Ising model in presence of external magnetic fields, and showedalmost arbitrary control in generating non-trivial Ising coupling patterns.Here we use laser-cooled trapped 171-Yb+ ions to simulate the emergence ofmagnetism in a system of interacting spins by implementing a fully-connectednon-uniform ferromagnetic Ising model in a transverse magnetic field. To linkthis quantum simulation to condensed matter physics, we measure scalablecorrelation functions and order parameters appropriate for the description oflarger systems, such as various moments of the magnetization. By increasing theIsing coupling strengths compared with the external field, the crossover fromparamagnetism to ferromagnetic order sharpens as the system is scaled up from N= 2 to 9 trapped ion spins. This points toward the onset of a quantum phasetransition that should become infinitely sharp as the system approaches themacroscopic scale. We compare the measured ground state order to theory, whichmay become intractable for non-uniform Ising couplings as the number of spinsgrows beyond 20- 30 and even NP complete for a fully-connected frustrated Isingmodel, making this experiment an important benchmark for large-scale quantumsimulation.

Onset of a Quantum Phase Transition with a Trapped Ion Quantum Simulator