A precisely controlled quantum system may reveal a fundamental understanding of another, less accessible system of interest. A universal quantum computer is currently out of reach, but an analogue quantum simulator that makes relevant observables, interactions and states of a quantum model accessible could permit insight into complex dynamics. Several platforms have been suggested and proof-of-principle experiments have been conducted. Here, we operate two-dimensional arrays of three trapped ions in individually controlled harmonic wells forming equilateral triangles with side lengths 40 and 80 μm. In our approach, which is scalable to arbitrary two-dimensional lattices, we demonstrate individual control of the electronic and motional degrees of freedom, preparation of a fiducial initial state with ion motion close to the ground state, as well as a tuning of couplings between ions within experimental sequences. Our work paves the way towards a quantum simulator of two-dimensional systems designed at will.

The cluster state represents a highly entangled state which is one central object for measurementbased quantum computing. Here we study the robustness of the cluster state on the two-dimensional square lattice at zero temperature in the presence of external magnetic fields by means of different types of high-order series expansions and variational techniques using infinite Projected Entangled Pair States (iPEPS). The phase diagram displays a first-order phase transition line ending in two critical end points. Furthermore, it contains a characteristic self-dual line in parameter space allowing many precise statements. The self-duality is shown to exist on any lattice topology. I. MOTIVATIONThe exploitation of quantum mechanics to build a quantum computer is a very active area in current research, because it is expected to be capable of solving classically hard problems in a polynomial amount of time 1 yielding a deeper understanding of the quantum world. To this end it has been shown that a universal quantum computer can be built by only a small set of elementary operations, namely arbitrary single-qubit rotations plus certain two-qubit gates like CZ or cNOT 2,3 . Especially the two-qubit operations turn out to be complicated to implement in experiment.Measurement-based quantum computing is a fascinating alternative approach for a quantum computer 4 . The essential idea is to prepare a highly-entangled initial quantum state on which only single-qubit measurement are sufficient to run a quantum algorithm. Meaurements with respect to an arbitrary basis are easy to perform in experiment. This feature comes with the price, that the initial state is hard to prepare in nature. One class of such highly-entangled states useful for measurementbased quantum computation are cluster states.One natural way of realizing a cluster state would be to cool down appropriate Hamiltonians having the cluster state as a ground state. Indeed, so-called cluster Hamiltonians exist but contain typically multi-site interactions which are very rare in nature. As a consequence, simpler models containing solely two-spin interactions have been proposed in the literature having the cluster Hamiltonian as an effective low-energy model. But it has been shown recently that it is very challenging to protect approximative cluster states against additional perturbations 5 . Another approach to study such systems efficiently, could be to prepare the cluster Hamiltonian with a quantum simulator 6-8 . However simulating multi-spin interactions with respect to the desired topology will probably be a challenge.In any case it is important to check whether the cluster state is stable and protected against additional perturbations. This has been the subject of several works in recent years which mostly concentrate on additional magnetic fields as a perturbation 9,10 . The latter studies either investigated the change of entanglement of the perturbed cluster state or explored the complete breakdown of the cluster state due to a phase transition which serves a...

Motivated by the possibility of universal quantum computation under noise perturbations, we compute the phase diagram of the 2d cluster state Hamiltonian in the presence of Ising terms and magnetic fields. Unlike in previous analysis of perturbed 2d cluster states, we find strong evidence of a very well defined cluster phase, separated from a polarized phase by a line of 1st and 2nd order transitions compatible with the 3d Ising universality class and a tricritical end point. The phase boundary sets an upper bound for the amount of perturbation in the system so that its ground state is still useful for measurement-based quantum computation purposes. Moreover, we also compute the local fidelity with the unperturbed 2d cluster state. Besides a classical approximation, we determine the phase diagram by combining series expansions and variational infinite Projected entangled-Pair States (iPEPS) methods. Our work constitutes the first analysis of the non-trivial effect of few-body perturbations in the 2d cluster state, which is of relevance for experimental proposals. PACS numbers: 03.67.-a, 03.65.Ud, 02.70.-c Introduction.-What defines universal computationability of a quantum computer? Or in other words, when can a device be used to perform universal quantum compuation under practical assumptions? Certainly, this question is of paramount importance if we aim to develop large-scale quantum algorithms eventually. A way to address this problem is through the exciting approach of measurement-based quantum computation (MBQC) introduced by Raussendorf and Briegel [1]. In this setting, a quantum computation is performed by implementing local measurements on a highly-entangled quantum state, known as cluster state [2]. As such, MBQC is a universal model of quantum computation. Therefore, it is relevant to know how much can one perturb the cluster state before it can no longer be used as a resource for quantum computation. If we see the cluster state as the ground state of a many-body system, then the relevant question is whether the ground state of the perturbed system is still qualified for quantum computation or not.Unlike in 1d, the ground state of such a "cluster" Hamiltonian in 2d (the 2d cluster state) is a universal resource for MBQC. The robustness of this state can be naturally analyzed at zero temperature by considering the effect of different Hamiltonian perturbations [3][4][5][6]. The case of a magnetic field along the x direction was first considered in Ref. [3], where the model was shown to be self-dual [7], and where the whole energy spectrum was itself also dual (up to degeneracies) to that of the quantum compass model [8,9], to the Xu-Moore model [10,11], and to the toric code in a transverse field [12]. The phase transition in all these models takes place at the self-dual point and it is found to be first order [12,13]. The more intricate case of combined magnetic fields along x and z directions was later considered in Ref. [5]. Here a line of 1st-order transitions is detected in the x − z plane

We present two methods for characterization of motional-mode configurations that are generally applicable to the weak and strong-binding limit of single or multiple trapped atomic ions. Our methods are essential to realize control of the individual as well as the common motional degrees of freedom. In particular, when implementing scalable radio-frequency trap architectures with decreasing ion-electrode distances, local curvatures of electric potentials need to be measured and adjusted precisely, e.g., to tune phonon tunneling and control effective spin-spin interaction. We demonstrate both methods using single 25 Mg + ions that are individually confined 40 µm above a surface-electrode trap array and prepared close to the ground state of motion in three dimensions. [14], yielding increasing interaction strengths by decreasing system length scales. Correspondingly, higher-order terms need to be considered, in order to enable precise control of interaction potentials. For example, in microfabricated surface-electrode ion trap arrays [15-17] local potentials, dominated by applied electric trapping potentials, define motional modes. For envisioned quantum simulations, motional degrees of freedom can be exploited either within individual sites or between different sites. This, in turn, requires adjustment of motional-mode configurations, i.e., individual orientation of the normal-mode vectors and related motional frequencies, to enable individual, tunable inter-site interactions [17]. In this letter, we introduce and experimentally demonstrate two distinct methods for the analysis of motional-mode configurations that are generally applicable to the weak and strong-binding limit. For the latter, we cool single ions close to the ground state of motion in three dimensions.To introduce our system, we consider a single ion with charge Q and mass m, harmonically bound in three di-