# Exploiting Anyonic Behavior of Quasicrystals for Topological Quantum Computing

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Correspondence between Anyons and Quasicrystals

#### 2.1. Fibonacci Anyons and Fibonacci ${C}^{*}$-Algebra

#### 2.2. Fibonacci Quasicrystals and the Fibonacci ${C}^{*}$-Algebra

- ${\cup}_{i\in I}{T}_{i}={\mathbb{R}}^{d}$,
- $int\left({T}_{i}\right)\cap int\left({T}_{j}\right)=\xd8$ for all $i\ne j$, and
- ${T}_{i}$ is compact and equal to the closure of its interior ${T}_{i}=\overline{int\left({T}_{i}\right)}$.

## 3. Quasicrystalline Topological Quantum Information Processing

## 4. Implications

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Nielsen, M.A.; Chuang, I.L. Quantum Computation and Quantum Information, 10th Anniversary ed.; Cambridge University Press: Cambridge, UK, 2011. [Google Scholar]
- Barbara, M.; Terhal, B.M. Quantum error correction for quantum memories. Rev. Mod. Phys.
**2015**, 87, 307. [Google Scholar] [CrossRef] - Kelly, J.; Barends, R.; Fowler, A.G.; Megrant, A.; Jeffrey, E.; White, T.C.; Sank, D.; Mutus, J.Y.; Campbell, B.; Chen, Y.; et al. State preservation by repetitive error detection in a superconducting quantum circuit. Nature
**2015**, 519, 66–69. [Google Scholar] [CrossRef] [PubMed] - Djordjevic, I.B. Quantum Information Processing, Quantum Computing, and Quantum Error Correction: An Engineering Approach; Academic Press: Cambridge, MA, USA; Elsevier: Amsterdam, The Netherlands, 2021. [Google Scholar]
- Seedhouse, A.E.; Hansen, I.; Laucht, A.; Yang, C.H.; Dzurak, A.S.; Saraiva, A. Quantum computation protocol for dressed spins in a global field. Phys. Rev. B
**2021**, 104, 235411. [Google Scholar] [CrossRef] - Breuckmann, N.P.; Eberhardt, J.N. Quantum Low-Density Parity-Check Codes. PRX Quantum
**2021**, 2, 040101. [Google Scholar] [CrossRef] - Wang, D.S. A comparative study of universal quantum computing models: Toward a physical unification. Quantum Eng.
**2021**, 3, e85. [Google Scholar] [CrossRef] - Pachos, J.K. Introduction to Topological Quantum Computation; Cambridge University Press: Cambridge, UK, 2012. [Google Scholar]
- Wang, Z. Topological Quantum Computation; Number 112; American Mathematical Society: Providence, RI, USA, 2010. [Google Scholar]
- Bartolomei, H.; Kumar, M.; Bisognin, R.; Marguerite, A.; Berroir, J.M.; Bocquillon, E.; Placais, B.; Cavanna, A.; Dong, Q.; Gennser, U.; et al. Fractional statistics in anyon collisions. Science
**2020**, 368, 173–177. [Google Scholar] [CrossRef] - Ding, L.; Wang, H.; Wang, Y.; Wang, S. Based on Quantum Topological Stabilizer Color Code Morphism Neural Network Decoder. Quantum Eng.
**2022**, 2022, 9638108. [Google Scholar] [CrossRef] - Marcolli, M.; Napp, J. Quantum Computation and Real Multiplication. Math. Comput. Sci.
**2015**, 9, 63–84. [Google Scholar] [CrossRef] - Planat, M.; Aschheim, R.; Amaral, M.M.; Irwin, K. Universal quantum computing and three-manifolds. Symmetry
**2018**, 10, 773. [Google Scholar] [CrossRef] - Planat, M.; Amaral, M.M.; Fang, F.; Chester, D.; Aschheim, R.; Irwin, K. Character varieties and algebraic surfaces for the topology of quantum computing. Symmetry
**2022**, 14, 915. [Google Scholar] [CrossRef] - Baake, M.; Grimm, U. Aperiodic Order; Cambridge University Press: Cambridge, UK, 2013. [Google Scholar]
- Bellissard, J. Gap labelling theorems for Schrödinger’s operators. In From Number Theory to Physics; Luck, J.M., Moussa, P., Waldschmidt, M., Eds.; Les Houches March 89; Springer: Berlin/Heidelberg, Germany, 1992; pp. 538–630. [Google Scholar] [CrossRef]
- Kohmoto, M.; Sutherland, B. Electronic States on a Penrose Lattice. Phys. Rev. Lett.
**1986**, 56, 2740. [Google Scholar] [CrossRef] [PubMed] - Sutherland, B. Self-similar ground-state wave function for electrons on a two-dimensional Penrose lattice. Phys. Rev. B
**1986**, 34, 3904. [Google Scholar] [CrossRef] [PubMed] - Fujiwara, T.; Kohmoto, M.; Tokihiro, T. Multifractal wave functions on a Fibonacci lattice. Phys. Rev. B
**1989**, 40, 7413(R). [Google Scholar] [CrossRef] [PubMed] - Luck, J.M. Cantor spectra and scaling of gap widths in deterministic aperiodic systems. Phys. Rev. B
**1989**, 39, 5834. [Google Scholar] [CrossRef] - Süto, A. Singular continuous spectrum on a cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian. J. Stat. Phys.
**1989**, 56, 525–531. [Google Scholar] [CrossRef] - Benza, V.G. Band spectrum of the octagonal quasicrystal: Finite measure gaps and chaos. Phys. Rev. B Condens. Matter.
**1991**, 44, 10343–10345. [Google Scholar] [CrossRef] - Kaliteevski, M.A.; Br, S.; Abram, R.A.; Krauss, T.F.; Rue, R.D.; Millar, P. Two-dimensional Penrose-tiled photonic quasicrystals: From diffraction pattern to band. Nanotechnology
**2000**, 11, 274. [Google Scholar] [CrossRef] - Florescu, M.; Torquato, S.; Steinhardt, P.J. Complete band gaps in two-dimensional photonic quasicrystals. Phys. Rev. B
**2009**, 80, 155112. [Google Scholar] [CrossRef] - Kalugin, P.; Katz, A. Electrons in deterministic quasicrystalline potentials and hidden conserved quantities. J. Phys. A Math. Theor.
**2014**, 47, 315206. [Google Scholar] [CrossRef][Green Version] - Tanese, D.; Gurevich, E.; Baboux, F.; Jacqmin, T.; Lemaître, A.; Galopin, E.; Sagnes, I.; Amo, A.; Bloch, J.; Akkermans, E. Fractal Energy Spectrum of a Polariton Gas in a Fibonacci Quasiperiodic Potential. Phys. Rev. Lett.
**2014**, 112, 146404. [Google Scholar] [CrossRef] - Gambaudo, J.M.; Vignolo, P. Brillouin zone labelling for quasicrystals. New J. Phys.
**2014**, 16, 043013. [Google Scholar] [CrossRef] - Macé, N.; Jagannathan, A.; Kalugin, P.; Mosseri, R.; Piéchon, F. Critical eigenstates and their properties in one- and two-dimensional quasicrystals. Phys. Rev. B
**2017**, 96, 045138. [Google Scholar] [CrossRef] - Macé, N.; Laflorencie, N.; Alet, F. Many-body localization in a quasiperiodic Fibonacci chain. SciPost Phys.
**2019**, 6, 050. [Google Scholar] [CrossRef] - Sen, A.; Perelman, C.C. A Hamiltonian model of the Fibonacci quasicrystal using non-local interactions: Simulations and spectral analysis. Eur. Phys. J. B
**2020**, 93, 67. [Google Scholar] [CrossRef] - Baggioli, M.; Landry, M. Effective Field Theory for Quasicrystals and Phasons Dynamics. SciPost Phys.
**2020**, 9, 062. [Google Scholar] [CrossRef] - Jagannathan, A. The Fibonacci quasicrystal: Case study of hidden dimensions and multifractality. Rev. Mod. Phys.
**2021**, 93, 045001. [Google Scholar] [CrossRef] - Satija, I.I.; Naumis, G.G. Chern and Majorana modes of quasiperiodic systems. Phys. Rev. B
**2013**, 88, 054204. [Google Scholar] [CrossRef] - Ghadimi, R.; Sugimoto, T.; Tohyama, T. Majorana Zero-Energy Mode and Fractal Structure in Fibonacci-Kitaev Chain. Phys. Soc. Jpn.
**2017**, 86, 114707. [Google Scholar] [CrossRef] - Varjas, D.; Lau, A.; Pöyhönen, K.; Akhmerov, A.R.; Pikulin, D.I.; Fulga, I.C. Topological Phases without Crystalline Counterparts. Phys. Rev. Lett.
**2019**, 123, 196401. [Google Scholar] [CrossRef][Green Version] - Cao, Y.; Zhang, Y.; Liu, Y.B.; Liu, C.C.; Chen, W.Q.; Yang, F. Kohn-Luttinger Mechanism Driven Exotic Topological Superconductivity on the Penrose Lattice. Phys. Rev. Lett.
**2020**, 125, 017002. [Google Scholar] [CrossRef] - Duncan, C.W.; Manna, S.; Nielsen, A.E.B. Topological models in rotationally symmetric quasicrystals. Phys. Rev. B
**2020**, 101, 115413. [Google Scholar] [CrossRef] - Liu, T.; Cheng, S.; Guo, H.; Xianlong, G. Fate of Majorana zero modes, exact location of critical states, and unconventional real-complex transition in non-Hermitian quasiperiodic lattices. Phys. Rev. B
**2021**, 103, 104203. [Google Scholar] [CrossRef] - Hua, C.B.; Liu, Z.R.; Peng, T.; Chen, R.; Xu, D.H.; Zhou, B. Disorder-induced chiral and helical Majorana edge modes in a two-dimensional Ammann-Beenker quasicrystal. Phys. Rev. B
**2021**, 104, 155304. [Google Scholar] [CrossRef] - Fraxanet, J.; Bhattacharya, U.; Grass, T.; Rakshit, D.; Lewenstein, M.; Dauphin, A. Topological properties of the longrange Kitaev chain with Aubry-Andre-Harper modulation. Phys. Rev. Res.
**2021**, 3, 013148. [Google Scholar] [CrossRef] - Rosa, M.I.N.; Ruzzene, M.; Prodan, E. Topological gaps by twisting. Commun. Phys.
**2021**, 4, 130. [Google Scholar] [CrossRef] - Sarangi, S.; Nielsen, A.E.B. Effect of coordination on topological phases on self-similar structures. Phys. Rev. B
**2021**, 104, 045147. [Google Scholar] [CrossRef] - Fan, J.; Huang, H. Topological states in quasicrystals. Front. Phys.
**2022**, 17, 13203. [Google Scholar] [CrossRef] - Zhang, Y.; Liu, X.; Belić, M.R.; Zhong, W.; Zhang, Y.; Xiao, M. Propagation Dynamics of a Light Beam in a Fractional Schrödinger Equation. Phys. Rev. Lett.
**2015**, 115, 180403. [Google Scholar] [CrossRef] - Elitzur, S.; Moore, G.W.; Schwimmer, A.; Seiberg, N. Remarks on the Canonical Quantization of the Chern–Simons-Witten Theory. Nucl. Phys. B
**1989**, 326, 108–134. [Google Scholar] [CrossRef] - Trebst, S.; Troyer, M.; Wang, Z.; Ludwig, A.W.W. A Short Introduction to Fibonacci Anyon Models. Prog. Theor. Phys. Suppl.
**2008**, 176, 384–407. [Google Scholar] [CrossRef] - Bratteli, O. Inductive limits of finite-dimensional C
^{*}-algebras. Trans. Am. Math. Soc.**1972**, 171, 195–234. [Google Scholar] [CrossRef] - Davidson, K.R. C
^{*}-Algebras by Example; Fields Institute Monographs; Fields Institute for Research in Mathematical Sciences: Toronto, ON, Canada, 1996; ISSN 1069-5273. [Google Scholar] - Hannaford, P.; Sacha, K. Condensed matter physics in big discrete time crystals. AAPPS Bull.
**2022**, 32, 12. [Google Scholar] [CrossRef] - Connes, A. Non-Commutative Geometry; Academic Press: Boston, MA, USA, 1994. [Google Scholar]
- Sadun, L. Tilings, tiling spaces and topology. Philos. Mag.
**2006**, 86, 875–881. [Google Scholar] [CrossRef] - Tasnadi, T. Penrose Tilings, Chaotic Dynamical Systems and Algebraic K-Theory. arXiv
**2002**, arXiv:math-ph/0204022. [Google Scholar] [CrossRef] - Jones, V.F.R. Index for Subfactors. Invent. Math.
**1983**, 72, 1–26. Available online: http://eudml.org/doc/143011 (accessed on 1 January 2022). [CrossRef] - Kauffman, L.H.; Lomonaco, S.J. Braiding, Majorana fermions, Fibonacci particles and topological quantum computing. Quantum Inf. Process.
**2018**, 17, 201. [Google Scholar] [CrossRef] - Goodman, F.M.; Wenzl, H. The Temperley-Lieb algebra at roots of unity. Pac. J. Math.
**1993**, 161, 307–334. [Google Scholar] [CrossRef] - Feiguin, A.; Trebst, S.; Ludwig, A.W.W.; Troyer, M.; Kitaev, A.; Wang, A.; Freedman, M.H. Interacting Anyons in Topological Quantum Liquids: The Golden Chain. Phys. Rev. Lett.
**2007**, 98, 160409. [Google Scholar] [CrossRef] - Zhang, H.; Liu, C.X.; Gazibegovic, S.; Xu, D.; Logan, J.A.; Wang, G.; van Loo, N.; Bommer, J.D.; de Moor, M.W.; Car, D.; et al. Retraction Note: Quantized Majorana conductance. Nature
**2021**, 591, E30. [Google Scholar] [CrossRef] - Gazibegovic, S.; Car, D.; Zhang, H.; Balk, S.C.; Logan, J.A.; De Moor, M.W.; Cassidy, M.C.; Schmits, R.; Xu, D.; Wang, G.; et al. RETRACTED ARTICLE: Epitaxy of advanced nanowire quantum devices. Nature
**2017**, 548, 434–438. [Google Scholar] [CrossRef] - Zhang, Y.; Wu, Z.; Belić, M.R.; Zheng, H.; Wang, Z.; Xiao, M.; Zhang, Y. Photonic Floquet topological insulators in atomic ensembles. Laser Photonics Rev.
**2015**, 9, 331–338. [Google Scholar] [CrossRef] - Flouris, K.; Jimenez, M.M.; Debus, J.D.; Herrmann, H.J. Confining massless Dirac particles in two-dimensional curved space. Phys. Rev. B
**2018**, 98, 155419. [Google Scholar] [CrossRef] - Zhang, Z.; Wang, R.; Zhang, Y.; Kartashov, Y.V.; Li, F.; Zhong, H.; Guan, H.; Gao, K.; Li, F.; Zhang, Y.; et al. Observation of edge solitons in photonic graphene. Nat. Commun.
**2020**, 11, 1902. [Google Scholar] [CrossRef] [PubMed] - Saraswat, V.; Jacobberger, R.M.; Arnold, M.S. Materials Science Challenges to Graphene Nanoribbon Electronics. ACS Nano
**2021**, 15, 3674–3708. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**The segment of the window in perpendicular space for the Fibonacci chain is shown at each inflation/deflation level. The L tiles are in red and S tiles in blue. On the horizontal axis, we show specific Fibonacci-chain configurations, where the number of tiles grows with the Fibonacci sequence. The sequences ${\left({x}_{i}\right)}_{n}$ are given by vertical lines. For example, we show two possible sequences at ${x}_{1}$ and ${x}_{2}$.

**Figure 2.**In (

**a**), we show three inflations tracking two positions ${x}_{1}=110$ and ${x}_{2}=111$ over the inflation levels with the fat rhombus in red and the thin in blue. In (

**b**), we introduce the ribbon description. The ribbons are constructed by straight lines (smooth for illustration purposes on the image) going from the center of one tile to the center of an adjacent tile following the Fibonacci rules on the same level as the inflations. For example, the ribbon ${R}_{b}$ (the blue in the nth level) goes over the following tiles in the three levels shown: TFFT, FTFFTF and FTFTFFTFTF. Note that a ribbon going over an F in one level will go over an F and T in the next inflation level, and a ribbon going over an S will always go to an F.

**Figure 3.**A tile flip that sends ribbons ${R}_{b}$ from FTFTF

**FT**FTF to FTFTF

**TF**FTF given a factor of ${\varphi}^{-2}$ on the associated states. The Ribbon ${R}_{a}$ has a change in orientation on the flip position.

**Figure 4.**A Bratteli diagram for the Fibonacci chain (similar for the Penrose tiling with fat (F) and thin (T) rhombus), where each path, i, to a node gives a ${x}_{i}$, and the different inflation levels n are shown. The number in parentheses is the number of paths to that node at level N, $n=1,\dots ,N$, which gives the Hilbert-space dimension for the associated subspace with sequences ${\left({x}_{i}\right)}_{N}=$L or S.

**Table 1.**A dictionary comparing concepts related to Fibonacci anyons and TQC with a quantum–mechanical Fibonacci chain is provided.

Fibonacci Anyons | Quantum Fibonacci Chain |
---|---|

Anyon | Tile |

0, 1 | S, L |

d-fold degeneracy | # of tiles |

Fusion with 1 (anyon destruction) | Deflation (tiles merging) |

Braid $B=FR{F}^{-1}$ | ${\rho}_{A}\left({B}_{n}\right)=A\varphi {E}_{n}+{A}^{-1}\mathbb{I}$ |

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**MDPI and ACS Style**

Amaral, M.; Chester, D.; Fang, F.; Irwin, K.
Exploiting Anyonic Behavior of Quasicrystals for Topological Quantum Computing. *Symmetry* **2022**, *14*, 1780.
https://doi.org/10.3390/sym14091780

**AMA Style**

Amaral M, Chester D, Fang F, Irwin K.
Exploiting Anyonic Behavior of Quasicrystals for Topological Quantum Computing. *Symmetry*. 2022; 14(9):1780.
https://doi.org/10.3390/sym14091780

**Chicago/Turabian Style**

Amaral, Marcelo, David Chester, Fang Fang, and Klee Irwin.
2022. "Exploiting Anyonic Behavior of Quasicrystals for Topological Quantum Computing" *Symmetry* 14, no. 9: 1780.
https://doi.org/10.3390/sym14091780