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The Penrose tiling (PT) is an intrinsically non-periodic way of tiling theplane, with many remarkable properties. A quantum error-correcting code (QECC)is a clever way of protecting quantum information from noise, by encoding theinformation with a sophisticated type of redundancy. Although PTs and QECCsmight seem completely unrelated, in this paper we point out that PTs give riseto (or, in a sense, are) a remarkable new type of QECC. In this code, quantuminformation is encoded through quantum geometry, and any local errors orerasures in any finite region, no matter how large, may be diagnosed andcorrected. We also construct variants of this code (based on the Ammann-Beenkerand Fibonacci tilings) that can live on finite spatial tori, in discrete spinsystems, or in an arbitrary number of spatial dimensions. We discussconnections to quantum computing, condensed matter physics, and quantumgravity.

The Penrose Tiling is a Quantum Error-Correcting Code