The Anti-de Sitter/Conformal Field Theory correspondence has given us a precise description of quantum gravity in asymptotically Anti-de Sitter space, as a quantum field theory (without gravity) with one fewer spacetime dimension. This proposal has passed many tests, but the emergence of the extra dimension on the gravitational side remains rather mysterious. I will present recent work explaining how this emergence can be naturally understood as a manifestation of quantum error correction, an idea originally introduced to solve the seemingly unrelated problem of protecting a quantum computer from decoherence. We will see that this gives precise meaning to speculations over the past decade relating boundary entanglement to the emergence of bulk geometry; to emphasize this precision I will illustrate the proposal using an exactly soluble model of the correspondence constructed via tensor networks. This talk should be quite accessible to beginning graduate students!