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 ...

Spontaneous symmetry breaking is a fundamental concept in many areas of physics, ranging from cosmology and particle physics to condensed matter 1 . A prime example is the breaking of spatial translation symmetry, which underlies the formation of crystals and the phase transition from liquid to solid. Analogous to crystals in space, the breaking of translation symmetry in time and the emergence of a "time crystal" was recently proposed 2,3 , but later shown to be forbidden in thermal equilibrium [4][5][6] . However, nonequilibrium Floquet systems subject to a periodic drive can exhibit persistent time-correlations at an emergent sub-harmonic frequency [7][8][9][10] . This new phase of matter has been dubbed a "discrete time crystal" (DTC) 10,11 . Here, we present the first experimental observation of a discrete time crystal, in an interacting spin chain of trapped atomic ions. We apply a periodic Hamiltonian to the system under many-body localization (MBL) conditions, and observe a sub-harmonic temporal response that is robust to external perturbations. Such a time crystal opens the door for studying systems with long-range spatial-temporal correlations and novel phases of matter that emerge under intrinsically non-equilibrium conditions 7 .For any symmetry in a Hamiltonian system, its spontaneous breaking in the ground state leads to a phase transition 12 . The broken symmetry itself can assume many different forms. For example, the breaking of spinrotational symmetry leads to a phase transition from paramagnetism to ferromagnetism when the temperature is brought below the Curie point. The breaking of spatial symmetry leads to the formation of crystals, where the continuous translation symmetry of space is replaced by a discrete one.We now pose an analogous question: can the translation symmetry of time be broken? The proposal of such a "time crystal" 2 for time-independent Hamiltonians has led to much discussion, with the conclusion that such structures cannot exist in the ground state or any thermal equilibrium state of a quantum mechanical system 4-6 . A simple intuitive explanation is that quantum equilibrium states have time-independent observables by construction; thus, time translation symmetry can only be spontaneously broken in non-equilibrium systems 7-10 . In particular, the dynamics of periodically-driven Floquet systems possesses a discrete time translation symmetry governed by the drive period. This symmetry can be further broken into "super-lattice" structures where physical observables exhibit a period larger than that of the drive. Such a response is analogous to commensurate charge density waves that break the discrete translation symmetry of their underlying lattice 1 . The robust subharmonic synchronization of the many-body Floquet system is the essence of the discrete time crystal phase 7-10 . In a DTC, the underlying Floquet drive should generally be accompanied by strong disorder, leading to manybody localization 13 and thereby preventing the quantum system from absorbing the drive energy...