Exploiting Anyonic Behavior of Quasicrystals for Topological Quantum Computing

symmetry SS

Article

ExploitingAnyonicBehaviorofQuasicrystalsforTopological QuantumComputing

MarceloAmaral* ,DavidChester ,FangFang andKleeIrwin

QuantumGravityResearch,LosAngeles,CA90290,USA * Correspondence:marcelo@quantumgravityresearch.org

Abstract: Theconcreterealizationoftopologicalquantumcomputingusinglow-dimensionalquasiparticles,knownasanyons,remainsoneoftheimportantchallengesofquantumcomputing.A topologicalquantumcomputingplatformpromisestodelivermorerobustqubitswithadditional hardware-levelprotectionagainsterrorsthatcouldleadtothedesiredlarge-scalequantumcomputation.Weproposequasicrystalmaterialsassuchanaturalplatformandshowthattheyexhibitanyonic behaviorthatcanbeusedfortopologicalquantumcomputing.Differentfromanyons,quasicrystals arealreadyimplementedinlaboratories.Inparticular,westudythecorrespondencebetweenthe fusionHilbertspacesofthesimplestnon-abeliananyon,theFibonaccianyons,andthetilingspacesof theone-dimensionalFibonaccichainandthetwo-dimensionalPenrosetilingquasicrystals.Aconcrete encodingonthesetilingspacesoftopologicalquantuminformationprocessingisalsopresentedby makinguseofinflationanddeflationofsuchtilingspaces.Whileweoutlinethetheoreticalbasisfor suchaplatform,detailsonthephysicalimplementationremainopen.

Keywords: topologicalquantumcomputing;anyons;quasicrystals;quasicrystallinecodes;tiling spaces

Citation: Amaral,M.;Chester,D.; Fang,F.;Irwin,K.Exploiting AnyonicBehaviorofQuasicrystals forTopologicalQuantumComputing. Symmetry 2022, 14,1780. https:// doi.org/10.3390/sym14091780

AcademicEditor:Ignatios Antoniadis

Received:15July2022 Accepted:23August2022 Published:26August2022

Publisher’sNote: MDPIstaysneutral withregardtojurisdictionalclaimsin publishedmapsandinstitutionalaffiliations.

1.Introduction

Whilequantumcomputershavebeenexperimentallyrealized,obtaininglarge-scale fault-tolerantquantumcomputationstillremainsachallenge.Sincequbitsareverysensitive totheenvironment,itisnecessarytosolvetheproblemofdecoherence[1].Software algorithmshavebeenproposedbyresearchersinthefield[2–6].Acomparativestudywith theprosandconsofvariousquantumcomputingmodelsisreviewedin[7].Thereviews mentionedhighlightthedifficultywithscalablequantumerrorcorrectionsandpointoutthe needfordifferentapproaches.Adifferentseminalsolutionistoaddhardware-levelerror correctionviatopologicalquantumcomputation(TQC)[8,9].Inparticular,non-abelian anyonscanprovideuniversalquantumcomputation[8].Theoretically,low-dimensional anyonicsystemsareahallmarktopologicalphaseofmatter,whichcouldbeusedfor TQCifaconcreteimplementationcouldbeachieved.Whileabeliananyonshavebeen experimentallyrealized[10],concreteevidenceofnon-abeliananyonsstillremainselusive. Interestingly,iftopologicalquantumcomputerhardwarecanbeimplemented,additional software-levelerrorcorrectioncanbeadded[11].

Copyright: ©2022bytheauthors. LicenseeMDPI,Basel,Switzerland. Thisarticleisanopenaccessarticle distributedunderthetermsand conditionsoftheCreativeCommons Attribution(CCBY)license(https:// creativecommons.org/licenses/by/ 4.0/).

TheChern–Simonstheory,whenappliedtothefractionalquantumHalleffectand latticemodelssuchasthetoriccode,constitutestheoreticalframeworksforusinganyons forTQC[8,9].Thesesystemssupportemergentquasiparticleexcitationsthatshowanyonic orfractionalstatistics.Thefusionrulesandbraidpropertiesofanyonsareusefulfor implementingTQC.Thequasiparticlesthatencodethetopologicalinformationdefinethe structureofthefusionHilbertspace.IntheChern–Simonstheory,anyonsareclassifiedby anintegerparametercalledthelevel k,whichappearsintheactionofthetheory.There areinfinitelevels; k = 2 definesAbeliananyons,whilegreaterlevelsdefinenon-Abelian anyons.TheFibonaccianyonisthequintessentialandsimplestnon-abeliananyonat

thelevel k = 3[8,9].Forourpurposes,thefusionHilbertspaceforFibonaccianyonsis describedbytheFibonacci C∗-algebra[12].

DuetothepotentialofTQCandtheexperimentaldifficultyofimplementingnonAbeliananyons,itisworthunderstandingwhatformsofTQCarepossibleingeneral. Previously,weco-authoredanon-anyonicproposalofTQCfromthree-dimensionaltopology[13]anddiscussedtheirassociatedcharactervarieties[14].Here,westudyquasicrystals describedbythegeometriccut-and-projectmethod[15].Theaimistoshowthattiling spacesassociatedwithquasicrystalsexhibitanyonicbehavior,whichcanleadtoTQC implementations.Morespecifically,weaimtoestablishlower-dimensionalquasicrystalsas anewcandidatetoimplementTQC.

Althoughcrystallographicmaterialshavewell-developedtheories,mainlyBlochand Floquet’stheories,thesetheoriesdonotworkproperlyforthetopologicalaspectsof quasicrystalsduetothelackoftranslationalsymmetry[16].Nevertheless,theconnection betweenlower-dimensionalquasicrystalswithhigher-dimensionallatticesallowsusto adaptandtouseaspectsoftheknowncrystallographictheoriesconsideringthesubspaces ofthehigher-dimensionalHilbertspaces.Thephysicsofaperiodicorderisagrowingand activefieldofresearch[16–32].Topologicalsuperconductorshavebeeninvestigatedin quasicrystals,suggestingthattheycanexhibittopologicalphasesofmatter[33–43].

Wepresentaconnectionbetweenanyonsandone-andtwo-dimensionalquasicrystals, suchasthe5-foldPenrosetiling,bytheisomorphismbetweentheanyonicfusionHilbert spaceandthesubspacesoflatticesHilbertspacesdescribingquasicrystaltilingspaces. BothspaceshavedimensionsthatgrowwiththeFibonaccisequence.Atheoremfrom functionalanalysissaysthattwoHilbertspacesareisomorphicif,andonlyif,theyhave thesamedimensions.WeproposethatthesesubspacesarefusionHilbertspacesand showanisomorphismbetweentheFibonacci C∗-algebraofFibonaccianyonsanda C∗algebraassociatedwiththetilingspacesofquasicrystals.The C∗-algebraofinterestallows fortheimplementationofrepresentationsofthebraidgroupnecessaryfortopological quantumcomputing.Itisworthmentioningthat,withintheBlochtheoryforperiodic atomicstructures,theenergylevelquantizationmapstotheperiodicpointgroupsymmetry. Aswithsimilarapproachesthatgobeyondtheperiodicstructures,e.g.,[44],quasicrystal approachesmakeuseofthisbyrestrictingtosubspacesofthecrystallinestructures.

Thispaperisorganizedasfollows:inSection 2,wereviewanddiscusselementsof anyonicfusionHilbertspacesandtheFibonacci C∗-algebrastoestablishthecorrespondence withthetilingspacesofquasicrystals.InSection 3,wediscussaspectsofinformation processingintilingspaces.WepresentdiscussionsandimplicationsinSection 4

2.CorrespondencebetweenAnyonsandQuasicrystals

Thequintessentialandsimplestnon-abeliananyonistheFibonaccianyon[8,9].We willshowtheisomorphismbetweenanyonicfusionHilbertspacesandquasicrystalline HilbertspacesattheleveloftheFibonaccianyonsandFibonacciquasicrystals,namely theone-dimensionalFibonaccichainandthe5-foldtwo-dimensionalPenrosetiling.The nameFibonacciinFibonaccianyonsisduethedimensionsoftheirHilbertspacesbeing awell-knownFibonaccinumber,and,inthecaseofthementionedquasicrystals,wewill showthattheyhavethesamebehavior,justifyingthenameFibonacci.

2.1.FibonacciAnyonsandFibonacciC∗-Algebra

Therearedifferentwaystodescribeanyons,includingtheChern–Simons(CS)theory andlatticeHamiltonianapproach[8,9].ForCStheory,itiswellknownthatthereisan additionalgauge-invarianttermthatcanbeaddedtotheMaxwellorYang–MillsLagrangian in(2+1)dimensions.ThisCStermistopological,asitdoesnotdependonthemetric[8,45]. Atlowtemperatures,thistermdominates.Inthenon-abeliancase,theactionisinvariant underSU(2) ∼ = Spin(3)andcanbewrittenasaGaussconstraintonawavefunctionofthe gaugefields.

Inthepresenceofsources(representationsofaLiealgebra),anyonicbehavior,suchas fusionandbraiddynamics,canbefoundwithsufficientcontrolofthelow-temperature Hamiltonian,Lagrangian,orGaussianconstraints.Thedegenerategroundstateofthe effectivetheoryisassociatedwiththeCSsourcesformtheso-calledfusionHilbertspace, whichisproposedasafault-toleranttopologicalquantumcomputingsubstrate.Inthecase ofFibonaccianyons,thesourcescanonlybeinthetwolower-dimensionalrepresentations ofSO(3),thespin-0andspin-1representations,withthefusionrules 1 ⊗ 1 = 0 ⊕ 1 0 ⊗ 1 = 1 1 ⊗ 0 = 1.(1)

Ifwehave N spin-1representationsassourcesandstarttofusethem,theycan builddifferentfusionpathsthatcanleadtoeitherspin-1orspin-0representationswith certainprobabilities.Thedifferentpathstofusethe N spin-1sourcestoonlyonespin-1 orspin-0sourcecanbeseenasstatesinafusionHilbertspace HN ,whereitsdimension growswiththenumberoforiginalspin-1sourcesandisgivenbytheFibonaccisequence, ((0,1, )1,2,3,5,8,13, ... , Fib(N + 1))[46],i.e., HN = CFib(N+1),where Fib(N + 1) isthe N + 1thFibonaccinumber.

Rotatingonephysicalsourcearoundtheotherisequivalenttoanoperationinthe fusionHilbertspacedescribedbytheso-calledbraidoperators(higher-dimensionalrepresentationsofthebraidgroup),whichleadstonon-trivialstatisticsgiventhenecessary quantumevolutionfortopologicalquantumcomputation.Theexplicitconstructionof braidoperators, B,isgivenasexamplesin([46],Sections2.4and2.5)throughtheso-called F-matricesand R-matricesoperatinginthefusionHilbertspace.Forthecaseoffusingtwo anyonsintoathirdone,thisprocessisafive-dimensionalspace,andtheexplicitmatrices inasuitablebasecanbegivenby

MoredetailsonFibonaccianyonsarewellknownandcanbefoundinRef.[46]and referencestherein.LessknownistheisomorphismofthefusionHilbertspaceswith representationsofcertain C∗-algebras,inparticular,theso-calledFibonacci C∗-algebra[12]. In[12],itisshownthatthefusionrulesdeterminethedataofaBrattelidiagram[47],which specifiesanapproximatelyfinite-dimensional(AF) C∗-algebrawitharepresentationona Hilbertspace,whichisisomorphictotheanyonicfusionHilbertspace.AnAF C∗-algebra A isgivenbyadirectlimit A = lim −→An ofafinite-dimensional C∗-algebra An,where An isadirectsumofmatrixalgebrasover C, An = ⊕Nn k=1Mrk (C).Similarly,aHilbert-space representationof A, HA,isobtainedasadirectlimitofasystemoffinite-dimensional Hilbertspaces HA n , HA n = ⊕Nn k=1Crk .ABrattelidiagramyieldsaunique C∗-algebraand allowsforasimplercomputationofthedimensionoftheHilbert-spacerepresentationsof thisalgebrabycountingthenumberofpathstoacertainnode.FortheFibonacci C∗-algebra, see([48],ExampleIII.2.6)and([12],Section3.2),fortheBrattelidiagramillustrationandthe dimensionoftheHilbert-spacecomputation.Theisomorphismbetweentherepresentations ofHilbertspacesandtheanyonic-fusionHilbertspacesisgivenin([12],Lemma3.3),where thedimensionsofFibonaccianyonsandtheFibonacci C∗-algebrabothgrowwiththe Fibonaccisequence.

2.2.FibonacciQuasicrystalsandtheFibonacciC∗-Algebra

Inanalogywiththeanyoniccase,wewillprovideaphysicaldescriptionoftheanyonic behaviorofquasicrystalstoallowforconcretephysicalimplementationandthentheassociatedeffectivefusionHilbertspacetodealwithtopologicalquantuminformationprocessing. ItismorecommontodealwithquasicrystalsfromthepointofviewofBlochtheoryfor periodicmany-bodyatomicquantumsystems,butevenwithinthispointofviewthereare differentimplementations.Whilethequasicrystalliteratureisfastgrowing,wementionthe quasicrystallineextensionoftheBlochtheoryincontextofthegap-labellingtheorem[16] andthediscoveryofafewexactsolutionsforquasicrystalHamiltonians [17–19,25,28,32]. Wealsohighlightmoredevelopmentsintermsofcomputationsofthespectrumandband structure[20–24,26,27]andthestudyoftopologicalproperties[33–39].Finally,quasicrystalshavebeenactivelystudiedinrecentyears[29–31,40–43,49].Fromourunderstanding, thedifferentapproacheshaveconvergentresults,includingtheself-similarstructureof theenergyspectrum,bandstructure,andtopologicalprotectedphases.Thegeometric cut-and-projectmethod,oritsmoregeneralform,calledmodelsets,describesthisstructure. ThestartingpointistheperiodicBlochtheoryconsideringtheSchrodingerequation foraparticleovertheatomicstructurewithaperiodicpotential V(r + R)= V(r) forall latticevectors R ofagivenlattice L.Withthissetup,theHamiltoniancommuteswiththe translationoperators,andtheBlochtheorydiagonalizesbothsimultaneously.Forthis,one introducesthereciprocallattice L∗ withprimitivetranslationvectors K,wherethescalar product R K isanintegermultipleof2π.Theeigenfunctionsaresuchthat k existsas

ψk+K (r + R)= eik R ψk (r),(3)

inwhich ψk (r) theBlochwavefunctionson Rn × Rn (r intheVoronoicell V and k inits dual V∗,alsocalledBrillouinzone).Thecurvesofthespectrumareperiodicinadual reciprocalspace,andtheentirebandstructureisdefinedbythebandstructureinsidethe firstBrillouinzone.

OurideaistostudytheHilbertspaceof ψ’ssatisfyingBloch’stheorem,suchthat ||ψ||2 < ∞.Wethenintroduce,foreach k ∈ V∗,theHilbertspace Hk ofthefunctions u on Rn,suchthat u(r + R)= eik R u(r),(4) and ||u||2 < ∞,with HL = ⊕Hk,andthedimensiongrowswiththenumberofpointson thelattice.TheHilbertspacesforaparticleoveranaperiodicpotentialfromaquasicrystal willbeseenasasubspaceofthelatticeHilbertspace HL,andwewillneedtoreviewthe cut-and-projectmethodtoobtainthequasicrystalfromthelattice L

Weconsideracut-and-projectscheme(CPS)tobea3-tuplet G = Rd , Rd , L ,where theparallelspace Rd andtheperpendicularspace Rd arerealeuclideanspaces, L is thelatticein E = Rd × Rd ,andistheembeddingspacewithtwonaturalprojections π: Rd

T = {Ti | i ∈ I} Rd , where I isacountableindexset,andthefragments Ti of T are non-emptyclosedsetsin Rd subjecttotheconditions

1. ∪i∈I Ti = Rd ,

2. int(Ti ) ∩ int(Tj )= Øforall i = j,and

3. Ti iscompactandequaltotheclosureofitsinterior Ti = int(Ti )

Whilethisistrivialforlatticeswithuniqueunitcells,quasicrystalshavemorethan oneunitcell.Multiplequasicrystalswiththesamenumberofpoints N from L projectedto theparallelspacecanleadtodifferenttilingsdependingontheshiftparameter γ Theconstructionaboveidentifiesthequasicrystalpointsetasasubsetoftheoriginal latticeintheembeddingspaceanditsHilbertspace H asasubspaceofthelatticeHilbert space HL.Anexplicitexampleisgivenin([16],Section3.2)fortheone-dimensionalFibonaccichainderivedfromthe Z2 lattice.Thisprovidesaccesstothephysicalproperties ofquasicrystals,suchastheirelectronicstructure.However,thefulltilingstructureisnot properlycapturedbythesedescriptions.Toaddressthedifferenttilingconfigurationsof quasicrystals,itisstandardtoconsidertheassociated C∗-algebrastructures([50],Sections II.3andV.10)andthenotionoftilingspaces[51].AsimplewaytolookatthisistodecomposethequasicrystallineHilbertspace H furtheraccordingtotileconfigurations.The one-dimensionalFibonaccichainandthetwo-dimensionalPenrosetilingcanbedescribed byonlytwotiles.FortheFibonaccichain,theyarecalledlong(L)andshort(S)edges.For thePenrosetiling,theycanbegiveneitherbyafatrhombus(F)andathinrhombus(T)or twoquadrilateralscalledkitesanddarts.

WecanthenconsidertheHilbertspaces HL,F and HS,T associatedwiththetwodifferent tiles.Thefrequencyoftheappearanceofthesetilesinsometilingisconstantandgrows withtheFibonaccisequence,given,atsomestep,as F(N) forLorFto F(N 1) forSorT. FromtheBlochtheory,thenumberofstatesdependsonthenumberofpointsinthelattice, whichtranslatestothenumberoftiles.Alatticetriviallyhasonlyonetile.Forquasicrystals, thenumbergrowsdifferentlydependingonthetilingconsidered.BoththeFibonacci chainandthePenrosetilingcontaintwofundamentaltilesthatgrowwiththeFibonacci sequence.Assuch,theHilbertspaces HL,F and HS,T subspacesofaquasicrystallineHilbert space(whicharesubspacesoflatticesHilbertspaces)havedimensionsthatgrowwith thenumberoftilesaddedtothequasicrystalinthesamewaythatthedimensionsofthe anyonicfusionHilbertspacesgrowwiththeadditionofanyons.Followingthediscussion fromtheprevioussection,weconcludethatthesequasicrystallinesubspacesarecandidates fortheimplementationofrepresentationsoftheFibonacci C∗-algebraassociatedwith Fibonaccianyons.WeseethetilesemergingfromtheBlochtheoryplayingthesameroleof thenon-abelian SO(3) sourcesintheChern–Simonstheory.

Anotherperspectiveistoconsiderthetilingspace,whichleadstoHilbertspacesthat areisomorphictotheonesconsideredabovewithdimensionsgrowingwiththeFibonacci sequence.Basically,westartwithaquasicrystalpointset γ andassociatesatilingwith it.Then,wecanshiftthepointsetbyshiftingthewindowinperpendicularspaceusing γ⊥.Eachshiftgeneratesanewtilingwiththesametilesbutwithadifferentconfiguration, wherethesetilescanbeseeninbothparallelandperpendicularspacesduetothestarmap. Thedifferenceisthat,inparallelspace,thereisagrowthofthequasicrystalwithtilesof fixedlength,while,intheperpendicularspace,eachpointaddedrescalesthetilesand reorganizestheconfigurationleadingtoarescalingofthespace,whichisusuallycalled inflation or deflation fortheinverseprocess.Eachtilingisapointintheso-calledtiling space,whichencodesallpossibletilingsthatcanbemadewithafixedCPSandwindow. Toencodethisinformation,wecanfixapoint x insidethewindowintheperpendicular space.Asthepointsareprojected,with π⊥(L),wecantrackthetiletypearound x after anewpointisprojected.Then,wecangeneratedifferenttilingsfromdifferentshiftsand trackthesequenceoftilesaroundthatpoint x overthedifferentsequenceofprojections. Equivalently,onecanuseonlyoneprojectionandtracktheevolutionofdifferent positionsinsidethewindow.Eachtilingisdescribedbyasequencethatencodesthe

evolutionoftilesaround x intheperpendicularspaceasthequasicrystalsgrowinparallel space.BylabellingtheFibonacci-chainandPenrose-tilinglettersLorFasthesymbols 1,andSorTas0wecanassociatedifferentsequences (xi )n of0sand1swith x,where i indexesthedifferentsequencesofprojections,and n ∈ N isthelevelinonesequenceof projections.Theonlyconstraintonthesesequences,whicharisesfromthegeometryofthe CPSwithfixedwindow,isthat,if (xi )n = 0,then (xi )n+1 = 1.Weillustratethisforthe FibonaccichaininFigure 1,where x1 = 1111101...,and x2 = 011011...,forexample.

Figure1. ThesegmentofthewindowinperpendicularspacefortheFibonaccichainisshownateach inflation/deflationlevel.The L tilesareinredand S tilesinblue.Onthehorizontalaxis,weshow specificFibonacci-chainconfigurations,wherethenumberoftilesgrowswiththeFibonaccisequence. Thesequences (xi )n aregivenbyverticallines.Forexample,weshowtwopossiblesequencesat x1 and x2.

ThePenrosetilingisshowninFigure 2,where x1 = 110...,and x2 = 111... Additionally,anequivalencerelationisdefinedonthisspaceofsequences.Tilings Ti and Tj withsome m,suchthat (xi )n =(xj )n for n ≥ m,areequivalent.Thisispresented indetailin([50],SectionsII.3andV.10)forthetilingspaceofthePenrosetilingwiththe constructionofa C∗-algebra A associatedwiththisspace.Remarkably,thisalgebraisthe sameFibonacci C∗-algebra;theHilbert-spacerepresentationsareisomorphictotheanyonic fusionHilbertspaces[12].Inthenextsection,wepresentdetailedaspectsofthisalgebra, quasicrystalphysicsinterpretations,andtopologicalquantumcomputation.

LetusconsideraconcretesolutionofaHamiltonianforaquasicrystal.Despite thedifficultieswiththegeneralizationoftheBloch’sandFloquet’stheories,therearea fewknownexactsolutionsforquasicrystalHamiltonians.Someofthestatesolutions oftheso-calledtight-bindingmodelfortheFibonaccichainandthePenrosetilingare known [17–19,25,28,32].Thesestatesincludezero-energydegeneratestatesandhavea similarformtotheBlochwavefunction,Equation(4),givenby

ψ(i)= C(i)eκh(i) (7) where κ ∈ R isaconstant, C(i) arelocalsite-dependentperiodicfunctionsgiventhe localamplitudesand h(i) isanon-localheightfielddependentonthegeometryofthe specifictiling.FortheFibonaccichaininEquation(7),thezero-energystatetakestheform ψ(2i)=( 1)i eκh(2i) with κ = ln φ,andthefield h(2i) givenby h(i)= ∑ 0≤j≤i B(2j → 2(j + 1)),(8) where B(LS)= 1, B(SL)= 1,and B(LL)= 0.ForthePenrosetiling,both κ and C(i) are computednumerically[28],buttheribbondescriptiondiscussedaboveallowsustoaccess

theFibonaccichainsubspacesdirectly.Notethataflip LS → SL,suchasthetheoneforthe ribbon Rb inFigure 3,changesthestatebyafactorof φ 2 , ψLS (i)= φ 2ψSL (i)

Figure2. In(a),weshowthreeinflationstrackingtwopositions x1 = 110 and x2 = 111 overthe inflationlevelswiththefatrhombusinredandthethininblue.In(b),weintroducetheribbon description.Theribbonsareconstructedbystraightlines(smoothforillustrationpurposesonthe image)goingfromthecenterofonetiletothecenterofanadjacenttilefollowingtheFibonaccirules onthesamelevelastheinflations.Forexample,theribbon Rb (theblueinthe nthlevel)goesoverthe followingtilesinthethreelevelsshown:TFFT,FTFFTFandFTFTFFTFTF.Notethataribbongoing overanFinonelevelwillgooveranFandTinthenextinflationlevel,andaribbongoingoveranS willalwaysgotoanF.

3.QuasicrystallineTopologicalQuantumInformationProcessing

FollowingtheBlochtheory,aquantum–mechanicalquasicrystalisdescribedbya Hilbertspace,whichisasubspaceofaHilbertspacedescribingahigher-dimensional crystal(thelattice L fromtheprevioussection).Inprinciple,thisgivesusamechanism togrowaquasicrystalwhilemaintainingthequantumsuperpositionoftilingsinatiling space.Thisgrowthisdescribedbythesequencesof0sand1s(encodingthedifferenttwo tilesintheFibonaccichainorPenrosetiling) (xi )n,suchthat,if (xi )n = 0,then (xi )n+1 = 1 andissubjecttosomeequivalencerelation,suchastheonedescribedintheprevious

Figure3. Atileflipthatsendsribbons Rb fromFTFTFFTFTFtoFTFTFTFFTFgivenafactorof φ 2 on theassociatedstates.TheRibbon Ra hasachangeinorientationontheflipposition.

sectionwithoneassociatedalgebra A.Aslightlydifferent,butequivalentwaytoaddress thetilingspaceistoconsiderfinitesequences (xi )n, n = 1, , N subjecttothesamerule and,withaequivalencerelationgivenby (xi )N =(xj )N ,constructthealgebra A asthe inductivelimitoffinite-dimensionalalgebras AN with AN asadirectsumofthematrix algebras[52].FortheFibonaccichainandPenrosetilingdescribedbyjusttwotiles,theset ofequivalenceclasseshasonlytwoelements,withthenumberofbothtilesgrowingwith theFibonaccisequence(forexample L growswith F(N + 1) and S with F(N)),whichgives AN = Mdn L ⊕ Mdn S with dn l = F(N + 1) and dn S = F(N).Theembeddingof AN in AN+1 is givenby dn+1 L = dn L + dn S and dn+1 S = dn L.Toconducttheinverseprocessandmergetiles, onecandefineaprojectionatthestep N bymeansoftheoperationtoforgetthatstep, remainingwithsequenceswith n = 1,..., N 1.

Onecanthenconsiderprojections En actingontheassociatedHilbertspacesdefinedby AN ,suchthat En mapstheHilbertspace Hdn L to Hdn 1 L orsubspacesof Hdn L associatedwith AN withthesubspacesof Hdn 1 L associatedwith AN 1 [53].Following([50],Lemma5in sectionV.10),weconsiderasequenceof En orthogonalprojections,knownasJones–Wenzl projections,suchthatthefollowingrelationshold E2 n = En (9) En Em En = φ 2 En,if |n m| = 1 (10) En Em = Em En,if |n m| > 1, (11) where,formoregeneralquasicrystals,onecouldconsiderEquation(10)tobe En Em En = [2] 2 q En,withtheso-calledquantumnumbers [n]q givenby [n]q = qn q n q q 1 (12) with q = e πi r .InthecaseofEquations(9)–(11),wehave q,afifthrootofunity, r = 5,and wecallthealgebra AN (q)

InthestudyofFibonaccianyons,theTemperley–Liebalgebrawithgenerators Fn is typicallyused,suchthat En = φ 1 Fn,see([8],Section8.2.2)and[54].Thealgebradefined bytheprojections En,Equations(9)–(11),isisomorphictotheFibonacci C∗-algebraofthe FibonaccianyonsandFibonacciquasicrystals,theproofcanbeseenbyexplicitlyderiving itsBrattelidiagram[53].Thequasicrystalprojectionscanbeusedtoimplementthebraid operationsnecessaryforquantumevolutiontoimplementtopologicalquantumcomputing. Inthecaseofanyons,movingoneanyonaroundtheotherisanon-trivialoperationencoded inthebraidgroupoperationsonthefusionHilbertspace.Fornon-abeliananyons,these operationsareshowntobedensein SU(N),with N asthenumberofanyonsinthesystem toprovideuniversalquantumcomputation.Thebraidgroupisgeneratedbygenerators Bn satisfyingtherelations

,(14) with φ = A2 A 2,whereunitarityisguaranteediftheprojections En areHermitian. A containsfoursolutions,allwith |

| = 1.Thefoursolutionsare A = e3πi/5 , e3πi/5 , e2πi/5 ,

and e2πi/5.Notethatthe R-matrixforFibonaccianyonsinEquation (2) contains e3πi/5 on someofthediagonals.Withthesolutionof A provided,onecanverifythat

ρA (Bn )ρA (B 1 n )= ρA (B 1 n )ρA (Bn )

ρA (Bn )ρA (Bm )ρA (Bn )= ρA (Bm )ρA (Bn )ρA (Bm ) if |n m| = 1 ρA (Bn )ρA (Bm )= ρA (Bm )ρA (Bn ) if |n m| > 1. (15)

Therefore,thequasicrystalprojectionoperatorscanbeusedtoconstructarepresentationofthebraidgroup.

Theusualstepfromquantumcomputationtotopologicalquantumcomputationcan nowbeperformedwithquasicrystalsbyfindinganembedding e ofan N-qubitspace (C2)⊗N intoasubspaceofthetilingspace.Theembeddingdoesnotneedtobeefficient, becauseitiswellknownthatthebraidgroupcanapproximateanyuniversalquantumgate toanydesiredprecision.Thecomputationalsubspaceofthetilingspacecanbegivenby fixingoneequivalenceclass (xi )n, n = 1, ,2N + 1 and i = 1, , d with d thenumberof sequenceswith (xi )2N+1 = 1.Werepresentthissubspaceusing TN,1 =(xi )n.Finally,to simulateaquantumcircuit,wecanhave

C2 ⊗N →e TN,1 U ↓↓ ρA (B)

C2 ⊗N →e TN,1 (16)

Explicitmatrixrepresentationsof ρA (B) canbeobtainedfromthealgebra AN (q) actingonthe N-qubitHilbertspace (C2)⊗N ,asubspaceofthetilingspace.Define E(q) actingon C2 ⊗ C2 as[55]

E(q)=[2] 1 q q 1e11 ⊗ e22 + qe22 ⊗ e11 + e12 ⊗ e21 + e21 ⊗ e12 (17) with eij thetwo-dimensionalmatrixunitsand Ei (q)= I ⊗ ⊗ I ⊗ E(q) ⊗ ⊗ I,where E(q) actsonthepositions i and i + 1ofthetensorproduct.

ForTQCwithaquantum–mechanicalquasicrystal,supposethatresearchersinthe futurecouldhavecompletecontrolofhowthequasicrystalisinflatedordeflated.The numberofpossibleinflation/deflationpathsinthetilingspace,whichgivestheHilbertspacedimension,istiedtothenumberofphysicaltiles,analogoustohowthenumber ofphysicalanyonsdefinethefusionHilbert-spacedimension.Thisallowsustoobtain adictionarybetweenconceptsrelatedtoFibonaccianyonsandTQCwithaquantummechanicalquasicrystal.Forconcretenessandsimplicity,considertheFibonaccichain, whichhastwoinflationrules

Toclarify,ourconventionsarethattheinflationrulesapplyaninflation.Itcanbe verifiedthatthesuccessiveapplicationofRuleAseededbySleadstothereverseofthe chainfoundbythesuccessiveapplicationofRuleB.If n arbitrarycombinationsofRuleA andRuleBareappliedfromtheseed,then 2n statescanbefound.However,theselead tovariousduplicatetilings,suchthat Fib(n + 2) uniquetilingsarefound.Forexample, withseed L,for n = 2 wehave{{L,SL,LSL},{L,SL,LLS},{L,LS,SLL},and{L,LS,LSL}} resultinginthreeuniquestates{LSL,LLS,SLL},or,intermsofthe (xi ), i = 1,2,3,describing theassociatedtilingspace,wehave{LSL,LLS,LLL}.TheassociatedBrattelidiagramis showninFigure 4,whichisequivalenttotheFibonaccianyondiagram[12]andthe AN (q) diagramfortheJones–Wenzlprojections[53].

Figure4. ABrattelidiagramfortheFibonaccichain(similarforthePenrosetilingwithfat(F)and thin(T)rhombus),whereeachpath, i,toanodegivesa xi,andthedifferentinflationlevels n are shown.Thenumberinparenthesesisthenumberofpathstothatnodeatlevel N, n = 1, , N, whichgivestheHilbert-spacedimensionfortheassociatedsubspacewithsequences (xi )N = L or S

Theanalogueofananyonicfusionprocessisgivenbytheoperationtoforgetthe Nthstepin (xi )n, n = 1, , N,leavingthesequences (xi )n with n = 1, , N 1.This sendsthesystemfromlevel N to N 1 ortheHilbertspaceofdimensionfrom F(n) to F(n 1) andisequivalenttoadeflationofthephysicalquasicrystal.SinceLisafixed length,thisoperationactingontheHilbertspaceassociatedwiththetwotilesLSwould leadtoLasadeflation,whichdecreasesthelengthofthechain.Whenperformingthe analogueofbraidinginthequasicrystal,onespecifiesabasisgivenbyinflation/deflation paths (xi )n anddecomposestheprojection En inadirectsumofprojectionsactingin lower-dimensionalsubspaces.FromEquation (14),thesubspaceactedinby En reachesa differentphase,whichrelatesto A andarescalingby φ.Inusualanyonicsystems,thebraid operationsinvolveabasistransformation.Thisselectstwoanyonstobefusedandapplies anoperationtothesetwoanyons,whichgivesaphase R andthenappliesaninversebasis transformation.Inquasicrystals,theprojection En directlyselectsthesubspacetobeacted onbyaphaseandrescaling.Table 1 summarizesadictionarythatcomparestheaspectsof Fibonaccianyonsandquantum–mechanicalFibonaccichainsforTQC. Table1. AdictionarycomparingconceptsrelatedtoFibonaccianyonsandTQCwithaquantum–mechanicalFibonaccichainisprovided.

Wehavealreadynotedthatcrystallographictheories,mainlyBloch’sandFloquet’s theories,donotextenddirectlytoquasicrystalsduetothelackoftranslationalsymmetry. WealsodiscussedanisomorphismbetweenanyonicandquasicrystallineHilbertspaces. Inthiscontext,itistemptingtoimportwell-developedtechniquesfromanyonicsystems forapplicationsinquasicrystalstoimplementTQC.Oneexampleistheso-calledgolden chain[56],whichmodelsFibonaccianyonsinonedimension.ThegoldenchainhasanaturalrealizationintermsoftheFibonacci-chainquasicrystal.ThelocalHamiltonian Hi acting onthe ithFibonaccianyononthechaindiscussedin[56]isimmediatelyidentifiedwith theprojections En,actingontheinflationlevel n, (x)n oftheFibonacci-chainquasicrystal, allowingaccesstothequantumquasicrystalgrowthandshrinkage.Adetailedanalysis ofthisHamiltonian(andotheranyonicHamiltonians)inthecontextofquasicrystalsand theirrelationshipwithquasicrystalHamiltonianscouldbediscussedinfuturework.

4.Implications

Conceptually,topologicalquantumcomputingisknowntohaveadvantagesover standardquantumcomputingforscalingduetohardware-levelerrorprotection.However, thephysicalimplementationoftopologicalphasesofmatterisabigchallenge.Onemain lineofresearchistoimplementlocalizedMajoranamodes,whichcanbehaveasabelian Isinganyons.Thislineofresearchhasseenamajorsetbackrecently,withamaingroupof researcherswithdrawingpapersthatclaimedexperimentalvalidationofabeliananyons, inparticulartheMajoranafermionexcitations[57,58].Additionally,non-abeliananyons needtobediscoveredtoimplementanyonicTQC.Thisopenstheopportunityfornew approachestotopologicalquantumcomputingthroughthediscoveryofnewhardware platformsthatcansupporttheanyonicquantuminformationprocessing.

Inthiswork,weinvestigatedlower-dimensionalquasicrystalsasaplatformforTQC. Insummary,weshowedthatquasicrystalsexhibitanyonicbehaviorandthatitstilingspaces canencodetopologicalquantuminformationprocessing.Considertwokeyingredients. First,notethatthefusionHilbert-spacerepresentationsofthe C∗-algebrasassociated withanyonicsystemspossessagrowingdimensionequaltothetilingHilbertspacesof quasicrystals,whichcanbedemonstratedthroughBrattelidiagramconstructions.Second, topologicalquantuminformationcanbeprocessedbyfindingasuitablecomputational subspaceofthetilingspaceswherethenecessaryoperationssuchasthebraidgroup transformationscanbeimplemented,forexample,usingtheexplicitrepresentationsofthe projection’sEquation(17).AdictionarycomparinginformationprocessingwithFibonacci anyonsandquantum-mechanicalFibonaccichainwasprovidedinTable 1.

Thenoveltyofourworkistheproposalofquasicrystalmaterialsasanaturalplatform fortopologicalquantumcomputing.Thesematerialsexhibitaperiodicandtopological order,andtheyarealreadyimplementedinlaboratoriesaroundtheworld.Moredifficult isthemanipulationofthetopologicalpropertiesoftilingspacesofquasicrystalsrequired forthetaskofquantuminformationprocessing,towhichourworkaddsfurthertheoretical understanding.Acompleteproposalforconcreteexperimentalimplementationremains anopenproblem.Oneideaistousegrapheneetchingwithaninnerquasicrystallayerto createthecircuitconnections,whereinflationcouldbeimplementedbydisconnectingalot ofconnectionsalongthechaininlinewithrecentadvancesinthefield[59–62].

AuthorContributions: Conceptualization,M.A.andK.I.;methodology,M.A.;software,M.A.,D.C. andF.F.;validation,D.C.andF.F.;formalanalysis,M.A.;investigation,M.A.,D.C.;writing—original draftpreparation,M.A.;writing—reviewandediting,M.A.andD.C.;visualization,D.C.;supervision, M.A.andK.I.;fundingacquisition,K.I.Allauthorshavereadandagreedtothepublishedversionof themanuscript.

Funding: Thisresearchreceivednoexternalfunding.

InstitutionalReviewBoardStatement: Notapplicable.

InformedConsentStatement: Notapplicable.

DataAvailabilityStatement: Notapplicable.

ConflictsofInterest: Theauthorsdeclarenoconflictofinterest.

References

1. Nielsen,M.A.;Chuang,I.L. QuantumComputationandQuantumInformation,10thAnniversaryed.;CambridgeUniversityPress: Cambridge,UK,2011.

2. BarbaraM.;Terhal,B.M.Quantumerrorcorrectionforquantummemories. Rev.Mod.Phys. 2015, 87,307.https://doi.org/ 10.1103/RevModPhys.87.307.

3. Kelly,J.;Barends,R.;Fowler,A.G.;Megrant,A.;Jeffrey,E.;White,T.C.;Sank,D.;Mutus,J.Y.;Campbell,B.;Chen,Y.; etal.Statepreservationbyrepetitiveerrordetectioninasuperconductingquantumcircuit. Nature 2015, 519,66–69. https://doi.org/10.1038/nature14270.

4. Djordjevic,I.B. QuantumInformationProcessing,QuantumComputing,andQuantumErrorCorrection:AnEngineeringApproach; AcademicPress:Cambridge,MA,USA,Elsevier:Amsterdam,TheNetherlands,2021.

5. Seedhouse,A.E.;Hansen,I.;Laucht,A.;Yang,C.H.;Dzurak,A.S.;Saraiva,A.Quantumcomputationprotocolfordressedspinsin aglobalfield. Phys.Rev.B 2021, 104,235411.https://doi.org/10.1103/PhysRevB.104.235411.

6. Breuckmann,N.P.;Eberhardt,J.N.QuantumLow-DensityParity-CheckCodes. PRXQuantum 2021, 2,040101.https://doi.org/ 10.1103/PRXQuantum.2.040101.

7. Wang,D.S.Acomparativestudyofuniversalquantumcomputingmodels:Towardaphysicalunification. QuantumEng. 2021, 3, e85.https://doi.org/10.1002/que2.85.

8. Pachos,J.K. IntroductiontoTopologicalQuantumComputation;CambridgeUniversityPress:Cambridge,UK,2012.

9. Wang,Z. TopologicalQuantumComputation;Number112;AmericanMathematicalSociety:Providence,RI,USA,2010.

10. Bartolomei,H.;Kumar,M.;Bisognin,R.;Marguerite,A.;Berroir,J.M.;Bocquillon,E.;Placais,B.;Cavanna,A.;Dong,Q.;Gennser, U.;etal.Fractionalstatisticsinanyoncollisions. Science 2020, 368,173–177.https://doi.org/10.1126/science.aaz5601.

11. Ding,L.;Wang,H.;Wang,Y.;Wang,S.BasedonQuantumTopologicalStabilizerColorCodeMorphismNeuralNetworkDecoder. QuantumEng. 2022, 2022,9638108.https://doi.org/10.1155/2022/9638108.

12. Marcolli,M.;Napp,J.QuantumComputationandRealMultiplication. Math.Comput.Sci. 2015, 9,63–84.https://doi.org/ 10.1007/s11786-014-0179-8.

13. Planat,M.;Aschheim,R.;Amaral,M.M.;Irwin,K.Universalquantumcomputingandthree-manifolds, Symmetry 2018, 10,773. https://doi.org/10.3390/sym10120773.

14. Planat,M.;Amaral,M.M.;Fang,F.;Chester,D.;Aschheim,R.;Irwin,K.Charactervarietiesandalgebraicsurfacesforthetopology ofquantumcomputing. Symmetry 2022, 14,915.https://doi.org/10.3390/sym14050915.

15. Baake,M.;Grimm,U. AperiodicOrder;CambridgeUniversityPress:Cambridge,UK,2013.

16. Bellissard,J.GaplabellingtheoremsforSchrödinger’soperators.In: FromNumberTheorytoPhysics;Luck,J.M.,Moussa,P.,Waldschmidt,M.,Eds.;LesHouchesMarch89;Springer:Berlin/Heidelberg,Germany,1992;pp.538–630.https://doi.org/10.1007/9783-662-02838-4_12

17. Kohmoto,M.;Sutherland,B.ElectronicStatesonaPenroseLattice. Phys.Rev.Lett. 1986, 56,2740.https://doi.org/10.1103/PhysRevLett.56.2740.

18. Sutherland,B.Self-similarground-statewavefunctionforelectronsonatwo-dimensionalPenroselattice. Phys.Rev.B 1986, 34, 3904.https://doi.org/10.1103/PhysRevB.34.3904.

19. Fujiwara,T.;Kohmoto,M.;Tokihiro,T.MultifractalwavefunctionsonaFibonaccilattice. Phys.Rev.B 1989, 40,7413(R). https://doi.org/10.1103/PhysRevB.40.7413.

20. Luck,J.M.Cantorspectraandscalingofgapwidthsindeterministicaperiodicsystems, Phys.Rev.B 1989, 39,5834. https://doi.org/10.1103/PhysRevB.39.5834.

21. Süt˝o,A.SingularcontinuousspectrumonacantorsetofzeroLebesguemeasurefortheFibonacciHamiltonian. J.Stat.Phys. 1989, 56,525–531https://doi.org/10.1007/BF01044450.

22. Benza,V.G.Bandspectrumoftheoctagonalquasicrystal:Finitemeasuregapsandchaos. Phys.Rev.BCondens.Matter. 1991, 44, 10343–10345.https://doi.org/10.1103/physrevb.44.10343.

23. Kaliteevski,M.A.;Br,S.;Abram,R.A.;Krauss,T.F.;Rue,R.D.;Millar,P.Two-dimensionalPenrose-tiledphotonicquasicrystals: fromdiffractionpatterntoband. Nanotechnology 2000, 11,274.https://doi.org/10.1088/0957-4484/11/4/316.

24. Florescu,M.;Torquato,S.;Steinhardt,P.J.Completebandgapsintwo-dimensionalphotonicquasicrystals. Phys.Rev.B 2009, 80, 155112.https://doi.org/10.1103/PhysRevB.80.155112.

25. Kalugin,P.;Katz,A.Electronsindeterministicquasicrystallinepotentialsandhiddenconservedquantities. J.Phys.AMath.Theor. 2014, 47,315206.https://doi.org/10.1088/1751-8113/47/31/315206.

26. Tanese,D.;Gurevich,E.;Baboux,F.;Jacqmin,T.;Lemaître,A.;Galopin,E.;Sagnes,I.;Amo,A.;Bloch,J.;Akkermans,E. FractalEnergySpectrumofaPolaritonGasinaFibonacciQuasiperiodicPotential. Phys.Rev.Lett. 2014, 112,146404. https://doi.org/10.1103/PhysRevLett.112.146404.

27. Gambaudo,J.M.;Vignolo,P.Brillouinzonelabellingforquasicrystals. NewJ.Phys. 2014, 16,043013.https://doi.org/10.1088/13672630/16/4/043013.

28. Macé,N.;Jagannathan,A.;Kalugin,P.;Mosseri,R.;Piéchon,F.;Criticaleigenstatesandtheirpropertiesinone-andtwodimensionalquasicrystals. Phys.Rev.B 2017, 96,045138.https://doi/10.1103/PhysRevB.96.045138.

29. Macé,N.;Laflorencie,N.;Alet,F.Many-bodylocalizationinaquasiperiodicFibonaccichain. SciPostPhys. 2019, 6,050. https://doi.org/10.21468/SciPostPhys.6.4.050.

30. Sen,A.;Perelman,C.C.AHamiltonianmodeloftheFibonacciquasicrystalusingnon-localinteractions:simulationsandspectral analysis. Eur.Phys.J.B 2020, 93,67.https://doi.org/10.1140/epjb/e2020-100544-y.

31. Baggioli,M.;Landry,M.EffectiveFieldTheoryforQuasicrystalsandPhasonsDynamics. SciPostPhys. 2020, 9,062. https://doi.org/10.21468/SciPostPhys.9.5.062.

32. Jagannathan,A.TheFibonacciquasicrystal:Casestudyofhiddendimensionsandmultifractality. Rev.Mod.Phys. 2021, 93, 045001.https://doi.org/10.1103/RevModPhys.93.045001.

33. Satija,I.I.;Naumis,G.G.ChernandMajoranamodesofquasiperiodicsystems. Phys.Rev.B 2013, 88,054204.https://doi.org/ 10.1103/PhysRevB.88.054204.

34. Ghadimi,R.;Sugimoto,T.;Tohyama,T.MajoranaZero-EnergyModeandFractalStructureinFibonacci-KitaevChain. Phys.Soc. Jpn. 2017, 86,114707.https://doi.org/10.7566/JPSJ.86.114707.

35. Varjas,D.;Lau,A.;Pöyhönen,K.;Akhmerov,A.R.;Pikulin,D.I.;Fulga,I.C.TopologicalPhaseswithoutCrystallineCounterparts. Phys.Rev.Lett. 2019, 123,196401.https://doi.org/10.1103/PhysRevLett.123.196401.

36. Cao,Y.;Zhang,Y.;Liu,Y.B.;Liu,C.C.;Chen,W.Q.;Yang,F.Kohn-LuttingerMechanismDrivenExoticTopologicalSuperconductivityonthePenroseLattice. Phys.Rev.Lett. 2020, 125,017002.https://doi.org/10.1103/PhysRevLett.125.017002.

37. Duncan,C.W.;Manna,S.;Nielsen,A.E.B.Topologicalmodelsinrotationallysymmetricquasicrystals. Phys.Rev.B 2020, 101, 115413.https://doi.org/10.1103/PhysRevB.101.115413.

38. Liu,T.;Cheng,S.;Guo,H.;Xianlong,G.FateofMajoranazeromodes,exactlocationofcriticalstates,andunconventional real-complextransitioninnon-Hermitianquasiperiodiclattices. Phys.Rev.B 2021, 103,104203.https://doi.org/10.1103/ PhysRevB.103.104203.

39. Hua,C.B.;Liu,Z.R.;Peng,T.;Chen,R.;Xu,D.H.;Zhou,B.Disorder-inducedchiralandhelicalMajoranaedgemodesina two-dimensionalAmmann-Beenkerquasicrystal. Phys.Rev.B 2021, 104,155304.https://doi.org/10.1103/PhysRevB.104.155304

40. Fraxanet,J.;Bhattacharya,U.;Grass,T.;Rakshit,D.;Lewenstein,M.;Dauphin,A.TopologicalpropertiesofthelongrangeKitaev chainwithAubry-Andre-Harpermodulation. Phys.Rev.Res. 2021, 3,013148.https://doi.org/10.1103/PhysRevResearch.3.013148.

41. Rosa,M.I.N.;Ruzzene,M.;Prodan,E.Topologicalgapsbytwisting. Commun.Phys. 2021, 4,130.https://doi.org/10.1038/s42005021-00630-3.

42. Sarangi,S.;Nielsen,A.E.B.Effectofcoordinationontopologicalphasesonself-similarstructures. Phys.Rev.B 2021, 104,045147. https://doi.org/10.1103/PhysRevB.104.045147.

43. Fan,J.;Huang,H.Topologicalstatesinquasicrystals. Front.Phys. 2022, 17,13203.https://doi.org/10.1007/s11467-021-1100-y.

44. Zhang,Y.;Liu,X.;Beli´c,M.R.;Zhong,W.;Zhang,Y.;Xiao,M.PropagationDynamicsofaLightBeaminaFractionalSchrödinger Equation. Phys.Rev.Lett. 2015, 115,180403.https://doi.org/10.1103/PhysRevLett.115.180403.

45. Elitzur,S.;Moore,G.W.;Schwimmer,A.;Seiberg,N.RemarksontheCanonicalQuantizationoftheChern–Simons-WittenTheory. Nucl.Phys.B 1989, 326,108–134.https://doi.org/10.1016/0550-3213(89)90436-7.

46. Trebst,S.;Troyer,M.;Wang,Z.;Ludwig,A.W.W.AShortIntroductiontoFibonacciAnyonModels. Prog.Theor.Phys.Suppl. 2008, 176,384–407https://doi.org/10.1143/PTPS.176.384.

47. Bratteli,O.Inductivelimitsoffinite-dimensional C∗-algebras. Trans.Am.Math.Soc. 1972, 171,195–234.https://doi.org/10.1090/ S0002-9947-1972-0312282-2.

48. Davidson,K.R. C∗-AlgebrasbyExample;FieldsInstituteMonographs;FieldsInstituteforResearchinMathematicalSciences: Toronto,ON,Canada,1996;ISSN1069-5273.

49. Hannaford,P.;Sacha,K.Condensedmatterphysicsinbigdiscretetimecrystals. AAPPSBull. 2022, 32,12.https://doi.org/ 10.1007/s43673-022-00041-8.

50. Connes,A. Non-CommutativeGeometry;AcademicPress:Boston,MA,USA,1994.

51. Sadun,L.Tilings,tilingspacesandtopology. Philos.Mag. 2006, 86,875–881.https://doi.org/10.1080/14786430500259742.

52. Tasnadi,T.PenroseTilings,ChaoticDynamicalSystemsandAlgebraicK-Theory. arXiv, 2002,arXiv:math-ph/0204022. https://doi.org/10.48550/arXiv.math-ph/0204022.

53. Jones,V.F.R.IndexforSubfactors. Invent.Math. 1983, 72,1–26.Availableonline: http://eudml.org/doc/143011 (accessedon1 January2022).

54. Kauffman,L.H.;Lomonaco,S.J.Braiding,Majoranafermions,Fibonacciparticlesandtopologicalquantumcomputing. Quantum Inf.Process. 2018, 17,201.https://doi.org/10.1007/s11128-018-1959-x.

55. Goodman,F.M.;Wenzl,H.TheTemperley-Liebalgebraatrootsofunity. Pac.J.Math. 1993, 161,307–334.https://doi.org/ 10.2140/pjm.1993.161.307.

56. Feiguin,A.;Trebst,S.;Ludwig,A.W.W.;Troyer,M.;Kitaev,A.;Wang,A.;Freedman,M.H.InteractingAnyonsinTopological QuantumLiquids:TheGoldenChain. Phys.Rev.Lett. 2007, 98,160409.https://doi.org/10.1103/PhysRevLett.98.160409.

57. Zhang,H.;Liu,C.X.;Gazibegovic,S.;Xu,D.;Logan,J.A.;Wang,G.;vanLoo,N.;Bommer,J.D.;deMoor,M.W.;Car,D.;etal. RetractionNote:QuantizedMajoranaconductance. Nature 2021, 591,E30.https://doi.org/10.1038/s41586-021-03373-x.

58. Gazibegovic,S.;Car,D.;Zhang,H.;Balk,S.C.;Logan,J.A.;DeMoor,M.W.;Cassidy,M.C.;Schmits,R.;Xu,D.; Wang,G.;etal.RETRACTEDARTICLE:Epitaxyofadvancednanowirequantumdevices. Nature 2017, 548,434–438 https://doi.org/10.1038/nature23468.

59. Zhang,Y.;Wu,Z.;Beli´c,M.R.;Zheng,H.;Wang,Z.;Xiao,M.;Zhang,Y.PhotonicFloquettopologicalinsulatorsinatomic ensembles. LaserPhotonicsRev. 2015, 9,331–338https://doi.org/10.1002/lpor.201400428.

60. Flouris,K.;Jimenez,M.M.;Debus,J.D.;Herrmann,H.J.ConfiningmasslessDiracparticlesintwo-dimensionalcurvedspace. Phys.Rev.B 2018, 98,155419.https://doi.org/10.1103/PhysRevB.98.155419.

61. Zhang,Z.;Wang,R.;Zhang,Y.;Kartashov,Y.V.;Li,F.;Zhong,H.;Guan,H.;Gao,K.;Li,F.;Zhang,Y.;etal.Observationofedge solitonsinphotonicgraphene. Nat.Commun. 2020, 11,1902.https://doi.org/10.1038/s41467-020-15635-9.

62. Saraswat,V.;Jacobberger,R.M.;Arnold,M.S.MaterialsScienceChallengestoGrapheneNanoribbonElectronics. ACSNano 2021, 15,3674–3708.https://doi.org/10.1021/acsnano.0c07835.