We show that a large number of ions forming a 2D Coulomb crystal provides an almost ideal system for scalable quantum computation and quantum simulation. In particular, the coupling of the internal states to the motion of the ions transverse to the crystal plane allows one to implement two-qubit quantum gates. We analyze in detail the decoherence induced by anharmonic couplings, and show that very high gate fidelities can be achieved with current experimental setups. DOI: 10.1103/PhysRevLett.96.250501 PACS numbers: 03.67.Lx, 03.67.Pp, 42.50.Vk The search for a physical system where quantum computation is feasible is at the focus of an intense theoretical and experimental activity [1]. Ion traps are by now among the most promising candidates for a many-qubit quantum processor. In this system, qubits are stored in internal electronic states, and collective vibrational modes of the ions allow us to induce quantum gates between them [2]. Following this idea, the building blocks for quantum computation have already been demonstrated in experiments with a few qubits [3]. Most of the current efforts to scale up the size of current ion quantum processors rely on the fabrication of arrays of microtraps [4], in which a large number of ions can be stored and shuttled. Even though an astonishing progress has been achieved in this direction in the last years, the scalability of this system still demands technical advances in microfabrication and trap design [5].Penning traps provide us with an alternative trapping scheme, where a large number of ions (10 4 -10 6 ) can be confined by a potential with approximate cylindrical symmetry [6]. Axial confinement is induced by a static electric field, whereas radial confinement is a result of the rotation of the ions under an axial magnetic field. If the axial confinement is strong enough, ions arrange themselves in a triangular lattice on a single plane, which corresponds to a classical two-dimensional (2D) Wigner crystal. The appeal of this system lies on the fact that ions are naturally ordered in a 2D regular array, without the need of individual micropotentials. Furthermore, ions are separated by distances of the order of tens of microns, such that they are individually addressable by optical means [7]. Thus, ions in Penning traps may appear as ideally suited for quantum computation and quantum simulation. However, this system has never been considered for this task [8]. First, because the complicated vibrational level structure of the crystal makes it difficult to apply here schemes that require resolution of single vibrational modes. In addition to that, typical schemes usually rely on the coupling of qubits to modes in directions parallel to the crystal. In current experiments with Penning traps, Doppler cooling of ions has reached temperatures of at most 1 mK, which implies occupation numbers of 10 2 -10 3 in the in-plane vibrational modes, so that it seems not to be possible to use them for quantum operations.In this Letter we show how to circumvent these problems by ex...