Quantum memory and autonomous computation in two dimen- sions

Standard approaches to quantum error correction (QEC) require active maintenance using measurements and classical processing. Passive QEC, by contrast, has so far been established only in unphysical spatial dimensions. Here, we give an explicit scheme for autonomous quantum error correction and computation in two dimensions, formulated as a dissipative quantum cellular automaton with a fixed, local and translation-invariant update rule. The construction uses hierarchical, self-simulating control elements based on ideas from the seminal classical results of G\'acs (1986, 1989) together with a measurement-free concatenated quantum code. We prove the existence of a nonzero noise threshold under a local noise model. Below this threshold, logical errors on encoded initial states are suppressed exponentially with increasing system size and the memory lifetime diverges in the thermodynamic limit. We also describe an implementation in continuous time as a time-independent, translation-invariant local Lindbladian using engineered dissipative jump operators. The recursive nature of our protocol allows for the fault-tolerant execution of quantum circuits specified by the initial state, and thus constitutes a self-correcting quantum computer capable of universal computation.