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The Penrose Tiling is a Quantum Error-Correcting Code

Zhi Li, Latham Boyle

Abstract

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

The Penrose Tiling is a Quantum Error-Correcting Code

Abstract

The Penrose tiling (PT) is an intrinsically non-periodic way of tiling the plane, with many remarkable properties. A quantum error-correcting code (QECC) is a clever way of protecting quantum information from noise, by encoding the information with a sophisticated type of redundancy. Although PTs and QECCs might seem completely unrelated, in this paper we point out that PTs give rise to (or, in a sense, are) a remarkable new type of QECC. In this code, quantum information is encoded through quantum geometry, and any local errors or erasures in any finite region, no matter how large, may be diagnosed and corrected. We also construct variants of this code (based on the Ammann-Beenker and Fibonacci tilings) that can live on finite spatial tori, in discrete spin systems, or in an arbitrary number of spatial dimensions. We discuss connections to quantum computing, condensed matter physics, and quantum gravity.
Paper Structure (27 sections, 8 theorems, 31 equations, 7 figures, 1 table)

This paper contains 27 sections, 8 theorems, 31 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Penrose Tilings
  3. Defining the Tilings
  4. Inflation and Deflation
  5. Indistinguishability and Recoverability
  6. The Penrose Tiling as a QECC
  7. QECCs: A Brief Introduction
  8. Constructing the PT QECC
  9. Discrete Realization via the Fibonacci Quasicrystals
  10. 1D Fibonacci Quasicrystals
  11. Discrete QECC from Symbolic Substitutions
  12. Finite Realization via Ammann–Beenker tilings
  13. Discussion
  14. Quantum Error-Correcting Code Space
  15. Local Recoverability
  16. ...and 12 more sections

Key Result

Lemma 1

If the set $\{\tau n_i ~(\text{mod} 1)\}$ is dense in the unit circle $\mathbb{R}/\mathbb{Z}$ for a subset of integers $\{n_i\}$, then $\{a_{n_i}\}$ determines $\gamma~(\text{mod} 1)$.

Figures (7)

  • Figure 1: Parallel between PTs and QECCs. (a) In a PT: examining a finite region $K$ tells you nothing about which PT you are in, but examining the complementary region $K^c$ allows you to reconstruct full PT. Here, purple lines are the edges of tiles and blue lines are Ammann lines. (b) In a QECC: examining a logical state (code state) in a finite region $K$ tells you nothing about which logical state you're in, but examining the complementary region $K^c$ allows you to reconstruct the logical state on the full space.
  • Figure 2: (a) Two fundamental rhombi for Penrose tilings, thin (green) and thick (blue), decorated by arrows or Ammann bars to define the matching rule. (b) The inflation rule for Penrose tilings. Note that inflation of a parent rhombus produces several half offspring rhombi, which are to be combined with other half offspring rhombi (indicated by lighter color), coming from inflation of neighboring parent rhombi.
  • Figure 3: An illustration for the wavefunction Eq. (\ref{['eq:codegs']}). Given an (infinite) Penrose tiling $T$, each term on the right-hand side represents a Euclidean transformed version of the original tiling, denoted by $gT$. Here, four patches are drawn from the same tiling $T$, serving to illustrate the relative translations and rotations among the $gT$'s.
  • Figure 4: (a) A finite piece of a 1D Fibonacci quasicrystal (scaled to fit), also represented as $LSLLSLSLLSLL$ or 101101011011. (b) Inflation rule $(L,S)\to(LS,L)$.
  • Figure 5: (a) The two fundamental tiles in the Ammann–Beenker tiling, decorated by Ammann bars indicating a matching rule. The arrow in each square shows its orientation (so it is only symmetric under reflection across the corresponding diagonal). (b) The corresponding inflation rule.
  • ...and 2 more figures

Theorems & Definitions (16)

  • proof
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Lemma 2
  • proof
  • Proposition 2
  • proof
  • Lemma 3
  • ...and 6 more