A quantum simulator is a restricted class of quantum computer that controls the interactions between quantum bits in a way that can be mapped to certain difficult quantum many-body problems. As more control is exerted over larger numbers of qubits, the simulator can tackle a wider range of problems, with the ultimate limit being a universal quantum computer that can solve general classes of hard problems. We use a quantum simulator composed of up to 53 qubits to study a non-equilibrium phase transition in the transverse field Ising model of magnetism, in a regime where conventional statistical mechanics does not apply. The qubits are represented by trapped ion spins that can be prepared in a variety of initial pure states. We apply a global long-range Ising interaction with controllable strength and range, and measure each individual qubit with near 99% efficiency. This allows the single-shot measurement of arbitrary many-body correlations for the direct probing of the dynamical phase transition and the uncovering of computationally intractable features that rely on the long-range interactions and high connectivity between the qubits.There have been many recent demonstrations of quantum simulators with varying numbers of qubits and degrees of individual qubit control [1]. For instance, small numbers of qubits stored in trapped atomic ions [2,3] and superconducting circuits [4] have been used to simulate various magnetic spin or Hubbard models with individual qubit state preparation and measurement. Large numbers of atoms have simulated similar models, but with global control and measurements [5] or with correlations that only appear over a few atom sites [6]. An outstanding challenge is to increase qubit number while maintaining individual qubit control and measurement, with the goal of performing simulations or algorithms that cannot be efficiently solved classically. Atomic systems are excellent candidates for this scaling, because their qubits can be made virtually identical, with flexible and reconfigurable control through external optical fields and high initialization and detection efficiency for individual qubits. Recent work with neutral atoms [7,8] has demonstrated many-body quantum dynamics with up to 51 atoms coupled through van der Waals Rydberg interactions, and the current work presents the optical control and measurement of a similar number of atomic ions interacting through their long-range Coulomb-coupled motion.We perform a quantum simulation of a dynamical phase transition (DPT) with up to 53 trapped ion qubits. The understanding of such nonequilibrium behavior is of great interest to a wide range of subjects, from social science [9] and cellular biology [10] to astrophysics [11] and quantum condensed matter physics [12]. Recent theoretical studies of DPT [13][14][15][16][17][18][19][20] involve the transverse field Ising model (TFIM), the quintessential model of quantum phase transitions [21]. A recent experiment investigated a DPT with up to 10 trapped ion qubits, where the transverse field ...

We have reexamined type-I intermittency, i.e., alternating bursts of regular and chaotic behavior just preceding a tangent bifurcation of the quadratic map family. By studying period-3 intermittency for the Henon map family numerically we establish the strong one-dimensional character of intermittency in the phase space. The disappearance of intermittency as the one-dimensional character of the map is lost (i.e., the dissipation is decreased) is reflected in a qualitative change in the behavior of the Lyapunov exponent in the period-1 basin of attraction. PACS numbers: 05.45.+bIn the brief euphoria attendent upon the discovery of the universal period-doubling route to chaos by Feigenbaum and others [1], attempts were made to link irregular behaviors of natural systems to properties of onehump maps of the interval, and other simple maps. One of these attempts was the discussion of intermittency in low-dimensional dynamical systems by Pomeau and Manneville (PM) [2], as a toy model of the complex, spatiotemporal intermittency of turbulent hydrodynamics. For a low-dimensional dynamical system intermittency is the occurrence of alternating bursts of periodic and chaotic behavior in its long-time behavior, for a small range of a control parameter. By citation consensus the most vivid example pointed out by PM is type-I intermittency which precedes the tangent bifurcation to a periodic attractor (of a practically low period) for a one-hump map of the interval. Two factors make this example particularly simple. First, the ease of visualizing the dynamics of 1D maps, so that a plot of the third iterate, y n +3, of the quadratic map, y" + \ =/i ->> w 2 , as a function of y n , for H slightly below its period-3 tangent bifurcation value, 53 = 1.75, clearly reveals the nature of the intermittent process (see Fig. 1): The period-3 orbit gets trapped in the narrow channels between y n +i(y n ) and the period-3 fixed-point line, >> W +3 =SB .V//, and the fraction of the time spent in regular behavior increases as the map parameter approaches its period-3 tangent bifurcation value, varying as e~l , where e^s^-pi. Concomitantly, the Lyapunov exponent, X(e) oc e ,/2 . It is of equal importance for inter-mittency that the quadratic map only has one attractor for each value of its parameter, thus crucially avoiding the homoclinic effects which both enrich and plague the more realistic Henon family, FIG. I. y n +)(yn) in the trapping region of the logistic map for /i at an intermittent parameter value preceding the tangent bifurcation to period-3 stability. Inset: The narrow channel giving rise to regular period-3 bursts.

Quantum computers and simulators may offer significant advantages over their classical counterparts, providing insights into quantum many-body systems and possibly improving performance for solving exponentially hard problems, such as optimization and satisfiability. Here, we report the implementation of a low-depth Quantum Approximate Optimization Algorithm (QAOA) using an analog quantum simulator. We estimate the ground-state energy of the Transverse Field Ising Model with long-range interactions with tunable range, and we optimize the corresponding combinatorial classical problem by sampling the QAOA output with high-fidelity, single-shot, individual qubit measurements. We execute the algorithm with both an exhaustive search and closed-loop optimization of the variational parameters, approximating the ground-state energy with up to 40 trapped-ion qubits. We benchmark the experiment with bootstrapping heuristic methods scaling polynomially with the system size. We observe, in agreement with numerics, that the QAOA performance does not degrade significantly as we scale up the system size and that the runtime is approximately independent from the number of qubits. We finally give a comprehensive analysis of the errors occurring in our system, a crucial step in the path forward toward the application of the QAOA to more general problem instances.