来自非周期单瓦片的量子纠错码：帽子瓦片和幽灵瓦片
Quantum error-correcting codes from aperiodic monotiles: the Hat and the Spectre
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中文总结 AI 辅助
研究将量子纠错码构造从彭罗斯镶嵌扩展到非周期单瓦片，如帽子瓦片和幽灵瓦片。通过证明相关性质、解决几何问题及计算验证等，得到两种瓦片的局部不可区分性类别等，还给出码的佩龙 - 弗罗贝尼乌斯数据，帽子码能存储经典比特及保护量子扇区。
中文摘要 AI 辅助
李和博伊尔表明彭罗斯镶嵌定义了一种量子纠错码：等距轨道上镶嵌的叠加可保护量子信息免受任何有界区域的擦除。我们将此构造扩展到史密斯、迈尔斯、卡普兰和古德曼 - 施特劳斯发现的非周期单瓦片。对于帽子瓦片，我们证明了所有帽子镶嵌的强局部不可区分性，并且通过底层切割投影方案的环面参数化，无条件地证明了所有非奇异帽子镶嵌的局部可恢复性。其余奇异情况归结为一个尖锐的几何问题——一个由帽子组成的区域能否以第二种方式重新镶嵌？——我们通过计算验证，在相当大的规模内没有反例：一个经过认证的2490瓦片补丁恰好有一种由帽子组成的镶嵌方式，所以其所有2的2490次方个瓦片子区域都唯一地重新镶嵌。与彭罗斯、阿曼 - 本克和斐波那契镶嵌不同，这两种单瓦片都形成两个局部不可区分性类别，所以它们的码空间分裂为两个携带超选经典标签的擦除校正扇区。标签是否留存取决于测量哪些等距变换：幽灵瓦片的类别通过30度旋转交换，并且一旦测量所有适当的等距变换就会合并，而帽子瓦片的类别仅通过反射交换。在物理上自然的规范群SE(2)下，帽子码因此存储一个稳健的经典比特——其长程有序的手性，可在任何间距为Δ = |K|√5/3的有界窗口中读取——以及其受保护的量子扇区。携带手性比特的是可反射的单瓦片，而不是手性单瓦片。我们给出了两种码的精确佩龙 - 弗罗贝尼乌斯数据，包括每个类别的反射帽子频率(3±√5)/6和幽灵瓦片方向类频率(5±√15)/10。
英文摘要
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 $Δ= |K|\sqrt{5}/3$ -- 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 $(5\pm\sqrt{15})/10$.
