On the robustness of topological quantum codes: Ising perturbation

0.52 MB·English

by  Mohammad Hossein Zarei

On the robustness of topological quantum codes:
Ising perturbation
MohammadHosseinZarei1
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PhysicsDepartment,CollegeofSciences,ShirazUniversity,Shiraz71454,Iran
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Abstract
p
- Westudythephasetransitionfromtwodifferenttopologicalphasestotheferromagneticphase
t
n byfocusingonpointsofthephasetransition.Tothisend,wepresentadetailedmappingfromsuch
a modelstotheIsingmodelinatransversefield. Suchamappingisderivedbyre-writingtheinitial
u Hamiltonianinanewbasissothatthefinalmodelinsuchabasishasawell-knownapproximated
q
phasetransitionpoint.Specifically,weconsiderthetoriccodesandthecolorcodesonsomevarious
[
latticeswithIsingperturbation. Ourresultsprovideausefultabletocomparetherobustnessofthe

1 topological codesandtoexplicitlyshow thattherobustnessofthetopological codes depends on
v triangulationoftheirunderlyinglattices.
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6 1 Introduction
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Oneoftheimportantproblemsinthecontrolofquantumsystemsisthedecoherencewhichemerges
.
1 fromundesirableinteractions[1]. Speciallyin quantumcomputationprocesses, itis veryimportant
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tofindaquantummemorywhichisrobustagainstemergenterrors.Inrecentyears,findingasolution
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for the decoherence problem has been one of the biggest challenges [2, 3]. One of the options to
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circumventthisproblemistopologicalquantumcomputationwheretopologicalnatureofaphysical
:
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systemisusedtoperformafaulttolerantquantumcomputation[4,5].
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X Thekeypropertyinthetopologicalquantumcomputationalmodelistopologicalorder. Topological
r orderisanewphaseofmatterwhichforthefirsttimewaspresentedbyWenin1980[6]. Itiscom-
a pletelydifferentfromordinaryordersuchasferromagneticordersothatthetopologicalorderscannot
be described by Landau paradigm and symmetry breaking theory [7]. Also there is no local order
parameterwhichrecognizestopologicalphase. Infacttopologicalorderisdescribedbyanon-local
order parameter and it is independent from local details of physical systems [8]. There are many
physicalsystemsincondensedmatterphysicswithtopologicalorder[9],forexampleinquantumHall
effects[10],high-temperaturesuperconductors[11,12,13]andfrustratedmagnetism[14,15,16].
Amongphysicalsystemswithtopologicalorder,latticemodelshavebeentakenintomoreconsider-
ation, becauseof theirsimplicity. In1997, Kitaev presenteda modelHamiltonianona toruswhich
couldencodethesubspaceoftwoqubitsinit’sgroundstates[4]. Thereafter,topologicalcolorcodes
wereintroducedbyBombinetal. [17,18,19,20]wheretheunitaryCliffordgatescouldbetopologi-
callyappliedwhilethiswasnotpossibleinthetoriccode.
1email:[email protected]
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Degeneracyof the groundstate in the topologicalcodesdependon topologicalnature of the model
andlocalsmallperturbationscannotliftsuchadegeneracy. Therobustnessofthedegeneracyofthe
groundstateisonlyguaranteedinsmallperturbationssothatwhenperturbationincreases,atacritical
value,aphasetransitionoccurs.Infact,thereisatopologicalphasetransition(TPT)wherethepointof
phasetransitionisameasureoftherobustnessofthecode.Therobustnessofthetopologicalcodesare
studiedagainstthermalperturbationsandlocalperturbations[21,22,23,24,25,26]. Asanexample,
robustnessofthetoriccodesagainstparallelandtransversemagneticfieldshavebeenstudiedwhere
magneticfieldcanbetreatedasanextrinsicfactorwhichperturbstopologicalorder[27,28,29,30].
If the perturbationis as Ising interaction, there is a phase transition between topologicalphase and
theferromagneticphase. SuchaproblemhasbeenstudiedforthetoriccodeinanIsingperturbation
(TCI) on a square lattice [31] and for the colorcode in Ising perturbation(CCI) on the honeycomb
latticein[32].
In this paper, we focus on the effect of the Ising perturbation on topological codes on various lat-
tices. WeshowthatsuchmodelsaremappedtoIsingmodelintransversefield(IT)byre-writingthe
Hamiltonian in a new basis. Since the approximatedphase transition points for the IT on different
latticesarewell-known[33,34,35],weprovideausefultabletocomparethephasetransitionpoints
oftopologicalcodesonvariouslattices.
Weconsiderdependenceofthephasetransitionpointandalsotherobustnessoftopologicalcodeson
triangulationof their underlyinglattices. In fact, for a pure topologicalmodel on a specific lattice,
therearesometopologicalpropertieswhichareindependentoftriangulationofthelattice[4]. When
theperturbationisadded,thesystemisnottopological. However,beforethephasetransitionoccurs,
topologicalpropertiesare still independentof triangulation. Since the pointof phase transition is a
propertyof the perturbedtopologicalcode, it is expectedthat the phase transition pointdependson
triangulation.Wederivepointsofphasetransitionforperturbedtopologicalcodesonvarioustriangu-
lationsandexplicitlyshowdependenceofphasetransitionpointsontriangulationofthelattice.
In the first step, we compare the robustness of CCI’s on two specific lattices. We show that they
aremappedtoIsingmodelintransversefieldontwodifferentlatticeswithdifferentphasetransition
points.ThesameresultisderivedfortheTCI’sonthreedifferentlattices. Weemphasizethatalthough
itiswellknownthatthephasetransitionpointinanordinaryquantumphasetransitiondependsonthe
underlyinglattice,suchaproblemhasnotbeenexplicitlystudiedforaTPT.
Inthesecondstep,wecomparetherobustnessoftheTCIandCCI.Suchaworkhasbeenperformed
forthetopologicalcodesinaparallelfield. In[24],theauthorshavecomparedtherobustnessofthe
colorcodeandthetoriccodeinparallelfield. Theyhaveconcludedthatthecolorcodeismorerobust
thanthetoriccodeagainstparallelmagneticfield. Weshowthatunderlyinglatticesofthesecodesare
importantinsuchaproblemsothattherobustnessofTCIonsquarelatticeislessthantherobustness
oftheCCIonhoneycomblatticewhiletheTCIonhoneycomblatticeisasrobustastheCCIonhon-
eycomblattice. Sotherobustnessoftopologicalcodesisverysensitivetotriangulationofthelattices
andwecannotcomparedifferenttopologicalcodeswithoutconsideringtheunderlyinglattices. The
structureofthispaperisasfollows:
InSection(2)wereviewtopologicalpropertiesoftwotopologicalcodes,thecolorcodesandthetoric
codes. InSection (3) we apply Isingperturbationto the topologicalcolorcodeson variouslattices.
Thenweapplythesameperturbationtothetoriccode. Inthissectionwepresentadetailedmapping
to re-write Hamiltonianof the models in a new basis so that the initial models are convertedto the
Isingmodelinthetransversefield. Finally,inSection(4),weprovideatableofthephasetransition
pointsandcomparetherobustnessofthetopologicalcodes.
2 Topological code states
Inthissection,wereviewsomeimportantpropertiesofthetopologicalquantumcodes. Specifically,
weconsidertwowell-knowntopologicalcodes,thecolorcodesandthetoriccodes.Topologicalorder
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ofthesemodelsisanon-localpropertywhichdoesnotdependontheunderlyinglattices. Forexample
thecolorcodescanbedefinedonthehoneycombandsquare-octagonallatticessothat,inbothcases,
topologicalorderisthesame.
2.1 Topologicalcolorcodes
Inthissubsection,wereviewthetopologicalcolorcodes(TCC).Ithasbeenshownthattheycanbe
used for topologicalimplementation of Clifford unitary gates and it is an advantage of these codes
comparedwiththetoriccodes[18].
The TCC models are defined on three-colorable lattices where one can color all faces (plaquettes)
of the lattice by threedifferentcolorsso thatany two neighborplaquettesare notin the same color
[19]. Here we consider the TCC modelon a honeycomblattice, but we emphasize thattopological
propertiesarethesameforallTCCmodels.Tothisend,considerahoneycomblatticeonatoruswhere
spinsliveonverticesofthelattice,seeFigure(1). Therearetwokindsofoperatorscorrespondingto
eachplaquetteofthelatticeasfollows:
Px = YXi , Pz = YZi, (1)
i∈P i∈P
whereX andZ denotethePaulioperatorsσ andσ ,respectively. Ifthenumberoftheplaquettesis
x z
N,thenumberoftheplaquetteoperatorswillbeequalto2N. Themostimportantproblemistofind
thestabilizedsubspaceofthecodewhichisdefinedasfollows:
L={|ψi | P |ψi=|ψi,P |ψi=|ψi}. (2)
x z
ThissubspacecanalsobeconsideredasthesubspaceofgroundstatesofaHamiltonianas:
H =−XPx−XPz. (3)
p p
BythefactthatP (1+P )=(1+P ),itissimpletocheckthatthefollowingstateisagroundstate
x x x
oftheHamiltonian(3):
|ψi=Y(1+Px)|00...0i, (4)
p
where|0iiseigenstateofthePaulioperatorZ.Alsotherearemanyconstraintsbetweentheplaquette
operators. Asithasbeenshowninfigure(1), thehoneycomblattice iscoloredbythreecolorssuch
as blue, green, red. According to the periodic boundry conditions on the torus, it is clear that the
followingconstraintswillbeholdontheplaquetteoperators:
Y Px = Y Px = Y Px , Y Pz = Y Pz = Y Pz, (5)
P∈R P∈G P∈B P∈R P∈G P∈B
whereP ∈ Rreferstotheplaqutteswhicharecoloredbyredandsoon. Hence,from2N plaquette
operators,only2N −4numberofthemareindependent. Becausethenumberofqubitsisalsoequal
to2N,degeneracyofthegroundstateissixteen-fold.
Inthefollowing,weshowthatallgroundstatesareconstructedbyapplyingsomenon-localoperators
tostate(4). Fordefiningthesenon-localoperators,weconsiderthatalledgesofthehoneycomblattice
canalsobecoloredbythreecolorssothateachedgewithaspecificcolorconnectstwoplaquetteswith
the same color, see figure(1). Then we define three kindsof color strings on the lattice where each
stringmovesontheedgesandplaquetteswiththesamecolorandtwoendpointsofthestringslivein
theplaquettes,seefigure(1). WedenotethesestringsbySR,SG,SB. Wedefinetwostringoperators
SC,SC correspondingtoeachstringintheformof:
z x
SzC = Y Zi , SxC = Y Xi, (6)
i∈SC i∈SC
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Figure 1: (Color online) In the color code on the honeycomb lattice, spins live on vertices of the
lattice. Thelattice is coloredby threedifferentcolors, red, blue andgreen. Each linkconnectstwo
plaquetteswith the same color and it is coloredby the same color, so all links are colored by three
colors,red,blueandgreen.Agreenstringmoveongreenlinksandithastwoendpointsintwogreen
plaquettes.
where the Pauli operators are applied on all qubits which live on the string SC and C can be each
one of three colors, red, green or blue. If two end pointsof a string are connectedto each other, it
generatesaclosedloopandwewillhaveacorrespondingloopoperator.Sinceeachplaquettesharesin
twospinswiththeloopoperator,itissimpletoshoweachloopoperatorcommuteswithallplaquette
operatorsintheHamiltonian(3). Byattentiontotopologyofthetorus,theloopoperatorsaredivided
to two generalclasses. One class containstrivialloops which are contractibleand anothercontains
non-trivialloopswhicharenon-contractible.
Importantpointisaboutnon-trivialoperators. Therearesixfundamentalnon-trivialloopswhichare
denotedby three colorsand two directionson the torus, see figure (2). Related to each kind of the
non-trivialloops,therearetwooperatorswhichareconstructedbyZorX operators.Wedenotethese
operatorsas:
LCx,σ = Y Xi , LCz,σ = Y Zi, (7)
i∈lC,σ i∈lC,σ
where l refers to a non-trivial loop and C denotes color of the loop and σ = {0,1} denotes
C,σ
directionofthelooparoundthetorus. Animportantpropertyofthenon-trivialloopoperatorsisthat
theycannotbewrittenasproductoftheplaquetteoperatorsintheHamiltonian(3). Butitissimpleto
showfromthreeloopoperatorsLR,σ,LG,σ andLB,σ,onlytwonumberofthemareindependentand
onecanbewrittenasproductoftwootheronesandsomeplaquetteoperators. Soweselectonlytwo
colorsoftheseloops,forexampleredandgreen.
Becauseofthetopologyofthetorus,twonon-trivialloopswhereoneisindirection0andanotheris
in direction1 crosseach other, see Figure (2), so thereis an anti-commutationrelationbetween the
non-trivialloopoperatorsasfollows:
{LR,0,LB,1}=0 , {LR,1,LB,0}=0 , {LB,0,LR,1}=0 , {LB,1,LR,0}=0. (8)
x z x z x z x z
Otherrelationsbetweentheseoperatorsarecommutation. Accordingtotheaboverelationsthenon-
trivialloopoperatorsgeneratea24dimensionalspacewhichcanencodethespaceoffourqubits.Also
wecanexplicitlyconstructallsixteenbasesofthisspacebythenon-localoperatorsinrelation(8). In
fact,itisclearthatthestate|ψias(4)isstabilizedbyLB,1,LB,0,LR,1,LR,0,becausetheycommute
z z z z
withoperatorsP thentheyare appliedtostate |00...0iwhichiseigenstateofthePaulioperatorZ.
x
Soallbasesofthedegeneratesubspaceareconstructedasfollows:
|ψ i=(LB,1)i.(LB,0)j.(LR,1)k.(LR,0)l|ψi, (9)
ijkl x x x x
wherei,j,kandlare0or1. Weshouldemphasizethatdegeneracyofthegroundstateinthismodel
isrelatedtothetopologyoftorusandthecolorablestructureofthelattice.
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l l l
R,0 B,0 G,0
l
R,1
l
B,1
l
G,1
Figure2: (Coloronline)Schematicallyweshowthreecoloredloopswhichwindaroundtorusintwo
differentdirections.
B
p
l0 l^0 Av
Figure3: (Coloronline)Inthetoriccodeonthehoneycomblattice,spinsliveonedgeofthelattice,
vertexoperatorsareappliedonthreespinswhichareneighborsofeachvertexandplaquetteoperators
areappliedonsixspinswhichareontheedgesofeachplaquette.Twonon-trivialloopsonlatticehas
beenshown,onemovesonedgesoflatticewhichhasbeendenotedbyredstringandanothermoves
onedgesofduallatticewhichhasbeendenotedbybluestring.
2.2 The toriccodes
Themostsimple ofthetopologicalcodesarethe toriccodeswhichcanbedefinedoneachoriented
graph[36]. HereweconsiderahoneycomblatticeonatoruswithN spinswhichliveontheedgesof
thelattice. Thetoriccodeonsuchalatticeisdefinedbythefollowingplaquetteandvertexoperators:
Bp =YZi , Av =YXi, (10)
i∈p i∈v
wherei ∈preferstospinswhichbelongtoplaquettepandi ∈vreferstospinswhichareneighbor
ofvertexv, see Figure(3). Thestabilizedsubspaceofthiscodecantreatedasthegroundstate ofa
Hamiltonianas:
HK =−XBp−XAv. (11)
p v
SinceA (1+A ) = 1+A ,itissimpletoshowthatthefollowingun-normalizedstateisaground
v v v
stateofthisHamiltonian:
|φi=Y(1+Av)|00...0i. (12)
v
Also there are two constraints between the plaquette and vertex operators in the Hamiltonian as
A = I and B = I. So the number of independentoperators is equal to N −2 and the
Qv v Qp p
groundstatehasfour-folddegeneracy.Tofindothergroundstatesweshouldconsidertwonon-trivial
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loopsl0,l1 aroundtwo directionsof the toruswhereloopscan be definedon the edgesof lattice as
wellastheedgesofduallattice wherethevertices,edgesandfacesofduallattice areinonetoone
correspondencewiththefaces,edgesandverticesofthelattice,respectively.Therearetwooperators
correspondingtoa non-triviallooplσ onthe latticeanda non-trivialloopˆlσ onthe duallattice, see
figure(3),intheformof:
Lσ,x = YXi , Lσ,z = YZi. (13)
i∈ˆlσ i∈lσ
Ononehandtheseoperatorscommutewiththe Hamiltonianso theyhavejointeigenstates, andfur-
thermorethereexistthefollowinganti-commutationrelationsbetweentheseoperators:
{L0,x,L1,z}=0 , {L0,z,L1,x}=0. (14)
Other relationsbetween these operatorsare commutation. Accordingto the aboverelationsall four
groundstatesofthemodelarederivedas:
(L0,x)i.(L1,x)j|φi, (15)
where i,j are 0 or 1, and |φi is a state as in Eq.(12). Thus in the toric codes, topologicalorder of
the groundstates is associated to theperiodicconditionsof the torusandit is independentfromthe
underlyinglattice.
2.3 The robustness ofthe topologicalcodes
The degenerategroundstate of the topologicalcodes can be used as a robust memory for quantum
computationgoals. Forexample,consideraperturbationasmagneticfieldonspinsinthetoriccode.
Suchaperturbationleadstosplittingbetweenthedegeneratestates. Theimportantpointinthiscaseis
thattwogroundstatesofthemodelareonlymappedtoeachotherbyanon-localoperatorasproduct
ofPaulioperatorsonanon-trivialloopwhichisrelatedtothetopologyofthemodel.Therefore,ifthe
perturbationissmallenough,suchasplittingcannothappen.
Itisclearthattherobustnessofthedegeneracyisliftedwhenperturbationincreases. Infactapertur-
bationeveninthefirstorderleadstoatransitiontermbetweenthegroundstateandtheexitedstates
intheperturbedenergylevels. Thisproblemleadstoaphasetransitionfromtopologicalordertoan
ordinaryorder. Finally,thetopologicalorderisliftedataphasetransitionpoint. Therefore,thepoint
ofphasetransitionisameasureoftherobustnessofthetopologicalcodes.
3 Ising perturbation on the topological code states
Inthissection,weconsiderthetopologicalcodesondifferenttwo-dimensionallatticeswithIsingper-
turbation. WeshowthatthesemodelsaremappedtotheIsingmodelintransversefieldbyre-writing
theinitialHamiltonianinanon-localbasis. Inageneralform,theHamiltonianofthetopologicalcode
withIsingperturbationisintheformof:
H =−Jh −kh (16)
Top I
whereh istheHamiltonianofthetopologicalcodeandh istheHamiltonianoftheIsingperturba-
Top I
tion. Atthefirst,letusconsidertwospecialregimesofthecouplings. IncasethatJ =0orK =∞,
wehavetheIsingmodelwhichhaveatwo-folddegenerategroundstatewhereZ componentsofall
spinsare upordown. Inthiscase system is in the ferromagneticphase anda localorderparameter
candescribephaseofthemodel[7]. NowweconsideracasethatJ = ∞orK = 0,sowehavethe
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topologicalmodelwhere the system is in a topologicalphase and any local orderparameter cannot
describesucha phase[6]. Since inthesetwo differentregimeswehavetwo differentphases, itisa
reasonableexpectationthatatacriticalratioofk/J aphasetransitionoccurs.
Were-writetheHamiltonian(16)inanewbasisandweshowthatinthisnewbasissuchaHamiltonian
ismappedtotheIsingmodelinthetransversefieldona2Dlattice.
3.1 The topologicalcolorcodeonthe honeycomb lattice
Our method in studying the topological codes in presence of the Ising perturbation is to re-write
Hamiltonianofthemodelinanewbasis. ItleadstoconverttheinitialmodeltoanIsingmodelinthe
transversefield. Similarmethodhasbeenalreadyusedinotherpapersaboutthetoriccodeonsquare
lattice[31]andthecolorcodeonthehoneycomblattice[32]. Inthissubsection,weexplainthemain
ideafortheTCCmodelonthehoneycomblatticethenweapplyittoothermodelsonvariouslattices.
Tothisend,weconsideraHamiltonianas:
H =−J(XPx+XPz)−KXZiZj, (17)
p p hi,ji
where hi,ji denotesneighborspins on the honeycomblattice. In this model, we have considereda
positivcouplingconstantJ fortheTCCHamiltonianandapositivcouplingconstantK fortheIsing
perturbation.
Atthefirst,bythefactthattheIsingperturbationcommuteswiththeoperatorsP intheTCCmodel,
z
we find thatgroundstate of the new modelis in a subspacewhich is stabilized byall operatorsP .
z
Were-writetheHamiltonian(17)insuchasubspacebychoosinganun-normalizedbasisasfollows:
|φr1,r2,r3,...,rNi=Y(1+(−1)rpPx)|00...0i. (18)
P
whereN isthenumberofplaquettesofthelatticeandr ’sarebinarynumberswhichareattributedto
p
eachplaquette. Infactwecanattributeavirtualspintoeachplaquettewherer denotesthevalueof
p
thevirtualspininaplaquettep. ThestatesinEq(18)arenon-localsothattheycorrespondtoanyonic
excitationsoftheTCCmodel.
The number of states as (18) is 2N and all of them are stabilized by the operators P . About the
z
above basis, we should emphasize that all these states are not independent, in fact there are con-
straints between the plaquette operators in different colors according to relation (5). For example
productofalloperatorsPRandPGisequaltotheIdentityandwefindthatPR.PG|φ i=
x x x x r1,r2,r3,...,rN
|φ i. Thisbasiswouldbecompleteifwe addsomenewstateswhichareconstructedby
r1,r2,r3,...,rN
applyingthenon-trivialloopoperatorsonthestate|φi.
Inthefollowingweshouldre-writetheHamiltonian(17)inthebasis(18). Asweexplained,allstates
(18)arestabilizedbyP ’s,soP |φ i = |φ iandsincethenumberofP ’sis
z z r1,r2,r3,...,rN r1,r2,r3,...,rN z
equaltoN,wefindthat P |φ i=N|φ i.
Pp z r1,r2,r3,...,rN r1,r2,r3,...,rN
ThenweapplytheoperatorPx onthestates(18). BythefactthatPx(1+(−1)rpPx)=(−1)rp(1+
(−1)rpPx),wefindthatPx|φr1,r2,r3,...,rNi = (−1)rp|φr1,r2,r3,...,rNi. Wecandescribethisfinalre-
sult in an interesting view so that effect of P in the basis (18) is equivalent to effect of the Pauli
x
operatorZpinabasisas|r1,r2,...,rNionthevirtualspins.
Interesting situation occurswhen we apply the Ising term on the basis (18). To this end, as we ex-
plainedintheprevioussection,wecancoloralllinksofthehoneycomblatticeinthreecolors. Thus
wedividetheIsinginteractionstothreepartscorrespondingtothecoloroftheirlinksas:
HI = X ZiZj + X ZiZj + X ZiZj, (19)
hi,ji∈B hi,ji∈G hi,ji∈R
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P P '
x x
Px
Figure4: (Coloronline)Left: AnIsingperturbationcorrespondingtoabluelinkdoesnotcommute
withtwoplaqutteoperatorsP andP′ intwoendpointsofthatlink. Right: Thevirtualspinswhich
x x
liveinblueplaquettesaredenotedbybluecircles.AnIsinginteractionZ Z betweentworealspinson
i j
bluelinksdoesnotcommutewithtwoplaquetteoperatorsP ontwoit’sendpointsanditisequivalent
x
toanIsinginteractionXpXp′ betweenvirtualspinswhichliveoncorrespondingplaquettes. Finally
wehaveanIsingmodelonatriangularlattice.
wherehi,ji∈ B denotessitesi,j whichareintwoendpointsofabluelinkandsoon. Forexample,
weapplytheIsinginteractiononthebluelinkstothestate|φ i. Eachlinkhi,jihasfour
r1,r2,r3,...,rN
neighborplaquettes and it is simple to show that correspondingoperator to hi,ji as Z Z does not
i j
commutewith two plaquetteoperatorsP correspondingto two plaquettesin two endpointsof that
x
link, see Figure(4, left). Therefore, we find that ZiZj(1+(−1)rpPx) = (1+(−1)rp+1Px)ZiZj,
whereP iscorrespondingoperatortooneoftheendpointsoftheedgehi,ji. Finally,weconclude
x
whentheIsinginteractioncorrespondingtoabluelinkofthelatticeisappliedtobasis|φ i,
r1,r2,r3,...,rN
itrisesupthevirtualspinsrp,rp′ correspondingtotwoblueplaquettesintwoendpointsofthatlink.
Therefore, we can describe the Ising interaction ZiZj as an Ising interaction XpXp′ on the basis
|r1,r2,...,rNi, wherep,p′ aretwo blueplaquetteswhichare connectedbya bluelink. Asyoucan
seeinFigure(4,right),ifwedenotethisIsinginteractionbetweentheneighborvirtualspinsbyalink,
wewillhaveatriangularlatticewhereallvirtualspinsinblueplaquettesliveontheverticesofthat
lattice.
TheaboveargumentcanalsoberepeatedfortheIsinginteractionsonthegreenandredlinkswhere,
correspondingtoeachcolor,wewillhaveanIsingmodelonatriangularlattice.
ByconsideringalltermsintheHamiltonian(17)inthenewbasis,wewillhaveanewHamiltonianin
theformof:
H =−JN− X (JZp+KXpXp′)− X (JZp+KXpXp′)− X (JZp+KXpXp′), (20)
{p}∈B {p}∈R {p}∈G
where {P} ∈ B refers to all virtual spins on the blue plaquettes and P,P′ refer to the neighbor
plaquettesinthesamecolorona triangularlattice. ForfindingthegroundstateoftheHamiltonian,
we use this fact that the final Hamiltonian in (20) on the virtual spins has been written as summa-
tionofthreeindependentHamiltonianonthreetriangularlatticeswithdifferentcolors. Therefore,it
is enoughto find the groundstate ofeach part, independently. Interestingresultis thateach partof
theHamiltonianisanIsingmodelinatransversefieldonatriangularlatticewhichaquantumphase
transitionhasbeenknownforitatJ/K ≈4.77[33].AtJ ≫K limitallvirtualspinsareinpositive
directionofZ andcorrespondinglytherealspinsareingroundstateofthecolorcode. AtK ≫ J
limit All virtualspins are in the groundstate of the Ising model on the triangularlattice and corre-
spondinglytherealspinsareinthegroundstateoftheIsingmodelonthehoneycomblattice.
8
Figure 5: (Color online) In the color code on Square-octagonallattice, spins live on vertices of the
lattice. Similar to thehoneycomblattice, we colorallplaquettesandlinksbythreedifferentcolors,
red,blueandgreen.
3.2 the colorcodeonthe square-octagonal lattice
The color codes can be defined on any three-colorablelattices. Definition of the stabilizers are the
sameforallofthemsothattherearetwokindsofstabilizerscorrespondingtoeachplaquetteP ,P
x z
whichareasproductofX orZ operatorsonthespinsofeachplaquette.
Inthissubsection,weconsiderthecolorcodeonasquare-octagonallatticeinaIsingperturbation,see
Figure(5). Inordertofindthephasetransitionpoint,similartotheprevioussection,were-writethe
Hamiltonian of the modelin the basis (18). All of argumentsare similar to the previoussection so
thatherewe dividetheHamiltonianto threepartscorrespondingto threecolorsoftheplaquettesof
thelattice. EachpartoftheHamiltonianinthebasis(18)isasanIsingmodelinthetransversefield
onthevirtualspinsinplaquetteswiththesamecolor. OnlydistinctioniskindoftheIsinginteraction
patternonthevirtualspinsineachcolor.
AsithasbeenshowninFigure(6),linksofthelatticewithgreencolorgenerateasquarelatticewhere
thevirtualspinsliveonit’svertices. Moreover,bluelinksgenerateasquarelatticewherethevirtual
spinsliveonit’svertices,seeFigure(7). Alsoweshouldbecarethattherearetwobluelinksbetween
eachtwoblueneighborplaquettes. ThisfinalpointleadstoanimportantdistinctionsothattwoIsing
interactionsinoriginalmodelontherealspinscorrespondtoanIsinginteractioninthesquarelattice
onthevirtualspins. Byconsideringasimilarargumentfortheredplaquettes,thefinalHamiltonian
onthevirtualspinswillbeasfollows:
H =−KN− X (JZp+KXpXp′)− X (JZp+2KXpXp′)− X (JZp+2KXpXp′), (21)
{p}∈G {p}∈R {p}∈B
where{P}∈GreferstoallvirtualspinsonthegreenplaquettesandP,P′areneighborplaquettesin
thesamecolorwhichgenerateasquarelattice. EachpartoftheaboveHamiltonianisanIsingmodel
inthetransversefieldonasquarelattice. Forthefirstpart,phasetransitionoccursatJ/K ≈ 3[34]
andfortwootherparts,phasetransitionoccursatJ/2K ≈3.WhentheIsingperturbationisincreased
byK, atK ≈ J/6forthe redandbluesublatticesa phasetransitionoccursfromtopologicalorder
to ferromagneticorderbutthe greensublattice staysintopologicalphase. AtK ≈ J/3topological
orderiscompletelyremovedandwehaveaferromagneticorder.Wecancomparethisresultwiththat
forthecolorcodeonthehoneycomblatticewherephasetransitionoccursatK ≈J/4.77.
3.3 The toriccode onthe honeycomb lattice
In this subsection, we study the toric code on a honeycomb lattice with the Ising perturbation. To
thisend,weconsiderthenearestneighborsofeachspinandweassumeanIsinginteractionbetween
them. AsithasbeenshowninFigure(8, left),ifweillustrateeachIsinginteractionbyalink,Ising
interactionpatternisdescribedbyatriangle-hexagonallatticewhichthespinsliveonit’svertices. In
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Figure6:(Coloronline)Thevirtualspinswhichliveingreenplaquettesaredenotedbygreencircles.
AnIsinginteractionZ Z betweentworealspinsongreenlinksdoesnotcommutewithtwoplaquette
i j
operatorsPx ontwoit’sendpointsanditisequivalenttoanIsinginteractionXpXp′ betweenvirtual
spins which live on corresponding plaquettes. Finally we have an Ising model on a green square
lattice.
Figure7: (Color online)Thevirtualspinswhich livein blueplaquettesare denotedbyblue circles.
Therearetwo IsinginteractionsZ Z betweenrealspinsontwobluelinkswhichconnecttwo blue
i j
plaquettes.BythefactthatthesetwoIsinginteractionsdoesnotcommutewithtwoplaquetteoperators
Px on two it’s end points, we find that they are equivalent to an Ising interaction XpXp′ between
virtual spins which live on corresponding plaquettes. Finally we have an Ising model on a square
lattice.
thisnewlatticethevertexoperatorsofthetoriccodecorrespondtothetriangularplaquettesandthe
plaquette operatorsof the toric code correspondto the hexagonalplaquettes. Finally the toric code
withtheIsingperturbationcanbedescribedbyaHamiltonianintheformof:
H =−J(XAv +XBp)−kXZiZj. (22)
v p hi,ji
BythefactthattheoperatorZ Z commutewiththeoperatorsB andthenon-trivialloopoperators
i j p
Lσ,Z ,wefindthatthegroundstateoftheaboveHamiltonianisinasubspacewhichisstabilizedby
B ’sandLσ,Z. Therefore,weselectthefollowingbasistore-writetheHamiltonian(22):
p
|φr1,r2,...,rNi=Y(1+(−1)rvAv)|00...0i. (23)
v
Then,wefindnewformoftheHamiltonian(22)inthisbasis.First,BythefactthatB ’scommutewith
p
alloperatorsA ,wefindthat B |φ i=N|φ i. Second,weapplytheoperators
v Pp p r1,r2,...,rN r1,r2,...,rN
Av,bythefactthatAv(1+(−1)rvAv)=(−1)rv(1+(−1)rvAv),wefindthatPvAv|φr1,r2,...,rNi=
Pv(−1)rv)|φr1,r2,...,rNi. Itisusefultodescribethefinalrelationinviewofvirtualspins.Tothisend,
if we insertvirtualspins on the verticesof the initial honeycomblattice, the state |φ i can
r1,r2,...,rN
treatedasproductstate|r1,r2,...,rNionthevirtualspins.Therefore,effectoftheoperatorAv onthe
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