Quantum Memory and Autonomous Computation in Two Dimensions

Standard approaches to quantum error correction (QEC) require active maintenance using measurements and classical processing. The possibility of passive QEC has so far only been established in an unphysical number of spatial dimensions. In this work, we present a simple method for autonomous QEC in two spatial dimensions, formulated as a quantum cellular automaton with a fixed, local and translation-invariant update rule. The construction uses hierarchical, self-simulating control elements based on the classical schemes from the seminal results of Gács (1986, 1989) together with a measurement-free concatenated code. We analyze the system under a local noise model and prove a noise threshold below which the logical errors are suppressed arbitrarily with increasing system size and the memory lifetime diverges in the thermodynamic limit. The scheme admits a continuous-time implementation as a time-independent, translation-invariant local Lindbladian with engineered dissipative jump operators. Further, the recursive nature of our protocol allows for the fault-tolerant encoding of arbitrary quantum circuits and thus constitutes a self-correcting universal quantum computer.