Quantum error-correcting codes from aperiodic monotiles: the Hat and the Spectre

arXiv:2607.15326v1 Announce Type: new Abstract: Li and Boyle showed that the Penrose tiling defines a quantum error-correcting code: superpositions of tilings over isometry orbits protect quantum information against erasure of any bounded region. We extend the construction to the aperiodic monotiles discovered by Smith, Myers, Kaplan and Goodman-Strauss. For the Hat, we prove strong local indistinguishability for all Hat tilings, and we prove local recoverability unconditionally for all nonsingular Hat tilings via the torus parametrization of the underlying cut-and-project scheme. The remaining singular case reduces to one sharply posed geometric question -- can a region that is a union of hats be retiled a second way? -- which we verify computationally has no counterexample up to a substantial scale: a certified $2490$-tile patch admits precisely one tiling by hats, so all $2^{2490}$ of its tile-subregions retile uniquely. Unlike the Penrose, Ammann-Beenker and Fibonacci tilings, both monotiles form two local-indistinguishability classes, so their code spaces split into two erasure-correcting sectors carrying a superselected classical label. Whether the label survives depends on which isometries are gauged: the Spectre's classes are exchanged by a $30^{\circ}$ rotation and merge once all proper isometries are gauged, whereas the Hat's are exchanged only by reflections. Under the physically natural gauge group $SE(2)$, the Hat code therefore stores one robust classical bit -- the handedness of its long-range order, readable in any bounded window with separation $\Delta = |K|\sqrt{5}/3$ -- alongside its protected quantum sectors. It is the reflexible monotile, not the chiral one, that carries the chirality bit. We give the exact Perron-Frobenius data of both codes, including the per-class reflected-Hat frequencies $(3\mp\sqrt{5})/6$ and the Spectre orientation-class frequencies $(5\pm\sqrt{15})/10$.

Abstract:Li and Boyle showed that the Penrose tiling defines a quantum error-correcting code: superpositions of tilings over isometry orbits protect quantum information against erasure of any bounded region. We extend the construction to the aperiodic monotiles discovered by Smith, Myers, Kaplan and Goodman-Strauss. For the Hat, we prove strong local indistinguishability for all Hat tilings, and we prove local recoverability unconditionally for all nonsingular Hat tilings via the torus parametrization of the underlying cut-and-project scheme. The remaining singular case reduces to one sharply posed geometric question -- can a region that is a union of hats be retiled a second way? -- which we verify computationally has no counterexample up to a substantial scale: a certified $2490$-tile patch admits precisely one tiling by hats, so all $2^{2490}$ of its tile-subregions retile uniquely. Unlike the Penrose, Ammann-Beenker and Fibonacci tilings, both monotiles form two local-indistinguishability classes, so their code spaces split into two erasure-correcting sectors carrying a superselected classical label. Whether the label survives depends on which isometries are gauged: the Spectre's classes are exchanged by a $30^{\circ}$ rotation and merge once all proper isometries are gauged, whereas the Hat's are exchanged only by reflections. Under the physically natural gauge group $SE(2)$, the Hat code therefore stores one robust classical bit -- the handedness of its long-range order, readable in any bounded window with separation $\Delta = |K|\sqrt{5}/3$ -- alongside its protected quantum sectors. It is the reflexible monotile, not the chiral one, that carries the chirality bit. We give the exact Perron-Frobenius data of both codes, including the per-class reflected-Hat frequencies $(3\mp\sqrt{5})/6$ and the Spectre orientation-class frequencies $(5\pm\sqrt{15})/10$.

Quantum Physics (quant-ph); Metric Geometry (math.MG)

arXiv:2607.15326 [quant-ph]

(or arXiv:2607.15326v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2607.15326 Focus to learn more arXiv-issued DOI via DataCite

flower_classes.json

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pickles/bigpatch.pkl

pickles/flower_results.pkl

pickles/hat_kites.pkl

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pickles/radius2_results.pkl

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prototile_geometry.json

radius2_classes.json

spectre_census.json

verify_hat_retiling.py

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Quantum error-correcting codes from aperiodic monotiles: the Hat and the Spectre

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