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

A quantum simulator is a well controlled quantum system that can simulate the behavior of another quantum system which may require exponentially large classical computing resources to understand otherwise. In the 1980s, Feynman proposed the use of quantum logic gates on a standard controllable quantum system to efficiently simulate the behavior of a model Hamiltonian. Recent experiments using trapped ions and neutral atoms have realized quantum simulation of Ising model in presence of external magnetic fields, and showed almost arbitrary control in generating non-trivial Ising coupling patterns. Here we use laser-cooled trapped 171-Yb+ ions to simulate the emergence of magnetism in a system of interacting spins by implementing a fully-connected non-uniform ferromagnetic Ising model in a transverse magnetic field. To link this quantum simulation to condensed matter physics, we measure scalable correlation functions and order parameters appropriate for the description of larger systems, such as various moments of the magnetization. By increasing the Ising coupling strengths compared with the external field, the crossover from paramagnetism 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 phase transition that should become infinitely sharp as the system approaches the macroscopic scale. We compare the measured ground state order to theory, which may become intractable for non-uniform Ising couplings as the number of spins grows beyond 20- 30 and even NP complete for a fully-connected frustrated Ising model, making this experiment an important benchmark for large-scale quantum simulation.