A network is frustrated when competing interactions between nodes prevent each bond from being satisfied. This compromise is central to the behaviour of many complex systems, from social and neural networks to protein folding and magnetism. Frustrated networks have highly degenerate ground states, with excess entropy and disorder even at zero temperature. In the case of quantum networks, frustration can lead to massively entangled ground states, underpinning exotic materials such as quantum spin liquids and spin glasses. Here we realize a quantum simulation of frustrated Ising spins in a system of three trapped atomic ions, whose interactions are precisely controlled using optical forces. We study the ground state of this system as it adiabatically evolves from a transverse polarized state, and observe that frustration induces extra degeneracy. We also measure the entanglement in the system, finding a link between frustration and ground-state entanglement. This experimental system can be scaled to simulate larger numbers of spins, the ground states of which (for frustrated interactions) cannot be simulated on a classical computer.

Frustration, or the competition between interacting components of a network, is often responsible for the complexity of many body systems, from social [1] and neural [2] networks to protein folding [3] and magnetism [4][5][6]. In quantum magnetic systems, frustration arises naturally from competing spin-spin interactions given by the geometry of the spin lattice or by the presence of long-range antiferromagnetic couplings. Frustrated magnetism is a hallmark of poorly understood systems such as quantum spin liquids, spin glasses [7,8] and spin ices [9], whose ground states are massively degenerate and can carry high degrees of quantum entanglement [10,11]. The controlled study of frustrated magnetism in materials is hampered by short dynamical time scales and the presence of impurities, while numerical modeling is generally intractable when dealing with dynamics beyond N ∼ 30 particles [12]. Alternatively, a quantum simulator [13] can be exploited to directly engineer prescribed frustrated interactions between controlled quantum systems, and several small-scale experiments have moved in this direction [11,[14][15][16][17][18][19]. In this article, we perform a quantum simulation of a long-range antiferromagnetic quantum Ising model with a transverse field, on a crystal of up to N = 16 trapped 171 Yb + atoms. We directly control the amount of frustration by continuously tuning the range of interaction and directly measure spin correlation functions and their dynamics through spatially-resolved spin detection. We find a pronounced dependence of the magnetic order on the amount of frustration, and extract signatures of quantum coherence in the resulting phases.Cold atoms are ideal platforms for the simulation of frustrated spin models, with the ability to tailor interactions with external fields and perform projective measurements of the individual spins [14]. Neutral atomic systems are typically limited to nearest neighbor interactions [17], although geometrically-frustrated interactions can be realized in certain optical lattice geometries [20]. The natural long-range interaction between cold atomic ions [21] has led to the engineering of Ising couplings between individual trapped ion spins [15,19,22,23] and the observation of spin frustration and quantum entanglement in the smallest system of three spins [11]. In this article we implement variable-range antiferromagnetic (AFM) Ising interactions and transverse magnetic fields with up to N = 16 atomic ion spins, using optical dipole forces. We directly measure the emergence and frustration of magnetic order through spatially-resolved imaging of the spins, in a system that approaches a complexity level where it becomes difficult or impossible to calculate the ground state order or the spin dynamics.We simulate the Ising model with long range antiferromagnetic interactions, given by the Hamiltonian (h = 1)where σ where 0 < α < 3 [21]. For B J ij on all pairs, the spins are polarized along the effective transverse magnetic field in the ground state |↑ y ↑ y ↑ y · · ·...

A quantum simulator is a well-controlled quantum system that can follow the evolution of a prescribed model whose behaviour may be difficult to determine. A good example is the simulation of a set of interacting spins, where phase transitions between various spin orders can underlie poorly understood concepts such as spin liquids. Here we simulate the emergence of magnetism by implementing a fully connected non-uniform ferromagnetic quantum Ising model using up to 9 trapped 171 Yb + ions. By increasing the Ising coupling strengths compared with the transverse field, the crossover from paramagnetism to ferromagnetic order sharpens as the system is scaled up, prefacing the expected quantum phase transition in the thermodynamic limit. We measure scalable order parameters appropriate for large systems, such as various moments of the magnetization. As the results are theoretically tractable, this work provides a critical benchmark for the simulation of intractable arbitrary fully connected Ising models in larger systems.