Self correction requires Energy Barrier for Abelian quantum doubles

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by  Anna Komar

Selfcorrection requires Energy Barrier forAbelianquantum doubles
Anna Ko´ma´r,1 Olivier Landon-Cardinal,1,2 and Kristan Temme1,3
1InstituteforQuantum InformationandMatterand WalterBurkeInstituteforTheoretical Physics,
California Institute of Technology, Pasadena, California 91125, USA
2Department of Physics, McGill University, Montreal, Canada H3A 2T8
3IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, USA
(Dated:January8,2016)
We rigorously establish an Arrhenius law for the mixing time of quantum doubles based on any Abelian
groupZd. Wehavemade theconcept of theenergy barrierthereinmathematically well-defined, itisrelated
tothe minimum energy cost the environment has to provide tothe system in order toproduce a generalized
Paulierror,maximizedforanygeneralizedPaulierrors,notonlylogicaloperators.Weevaluatethisgeneralized
6 energybarrierinAbelianquantumdoublemodelsandfindittobeaconstantindependentofsystemsize.Thus,
1 weruleoutthepossibilityofentropicprotectionforthisbroadgroupofmodels.
0
2
PACSnumbers:

n
a
J I. INTRODUCTION Thescalingenergybarrierofaquantummodelisintimately
related to the geometrical support of operators mapping a
6
Whetheritispossibletopreservearbitraryquantuminfor- ground state to a different orthogonal ground state, called

] mationoveralongperiodoftimeisaquestionofbothfunda- logical operators. For the 4D toric code, logical operators
h
mentalandpracticalinterest.Activequantumerrorcorrection are tensor productof single qubitoperatorsacting on a two-
p
- provides a way to protect quantum information but requires dimensionalsheet-likesubsetofqubits,similartothelogical
nt keeping track of and correcting the errors over a short time operatorofthe2DIsingmodelwhichflipsallspins.
a scale. Alternatively, quantum self-correcting systems would While the 4D Kitaev’s toric code is self-correcting, it re-
u passively preserve quantuminformationin the presence of a quires addressable long-range interactions if embedded in a
q thermal environmentwithout the need for external interven- lattice of lower dimensionality. Various attempts have been
[ tiononthesystem. Thedynamicsofthesequantum”memo- madetodecreasethedimensionalityofsuchaself-correcting
1 ries”wouldbesuchthattheprobabilityofanerroroccurring code,whileretainingalargeenergybarrierofthesystem[4–
v ontheencodedinformationisexponentiallysuppressedwith 6]. Atypicalshortcomingsofthesecodesincludessensitivity
4 systemsize,resultinginanexponentiallylongmemorytime. toperturbations[7]. Findingaself-correctingsysteminthree
2 Candidatesforself-correctionaretypicallysystemsgoverned dimensions(orlower)isstillanopenquestion.
3
byalocalHamiltonianwhosedegenerategroundspacestores Following the intuition based on the Arrhenius law, it is
1
quantuminformation. believed that quantum self-correction requires a scaling en-
0
. Assessingwhetherasystemisself-correctingrequiresesti- ergybarrier,i.e.,anenergybarrierthatisanincreasingfunc-
1
matingthescalingofitsmemorytimewithsystemsize. This tionofsystemsize. However,aformalrelationbetweenself-
0
difficultproblemisoftenreducedtoevaluatingtheenergybar- correction and a scaling energy barrier has not been estab-
6
1 rier, loosely defined as the maximal energy of intermediate lished and the Arrhenius law has only be proven for a few
: statesina sequenceoflocaltransformationstakingaground modelswhilethereareknowncounterexamples.Moreover,it
v
state to an orthogonalgroundstate, minimized over all such wasrecentlysuggestedthattheremightexistadifferentkind
i
X possible sequences. This sequence of excited states mimics of protection [8], one that does not require a scaling energy
r the evolution of the system under thermalization and decod- barrier,coinedentropyprotection. Theintuitionisthatwhile
a ing. Theintuition(andimplicitconjecture)isthatthesystem there exist paths in phase space mapping a ground state to
obeysthephenomenologicalArrheniuslawwhichrelatesthe anorthogonalgroundstatewhileonlyintroducingaconstant
memorytimet totheenergybarrier∆E∗ andtheinverse amountgoenergy,thesepathsmightnotbethetypical. Typ-
mem
temperatureβ 1/k T ical paths, however, might require the system to go through
B
≡
a scaling energybarrier. We could think of such a modelas
t eβ∆E∗ (1)
mem havinganeffectivefreeenergybarrier,i.e.,therearefreeen-
∝
The Arrheniuslaw is a usefulguidingprinciple. For clas- ergy valleys in the landscape between the two ground states
sicalmodels,onecanintuitivelyunderstandtheexponentially andinordertogetoutofsuchavalleythesystemwouldhave
long (classical) memory time of the ferromagnetic 2D Ising toovercomeaneffectivebarrier.
model by realizing that its energy barrier is proportional to In 2014, Brown et al. proposed a local 2D Hamiltonian
the linear system size. Indeed, to go fromthe all up state to whichseemedtorealizeentropyprotection[9]sinceitsmem-
the all down state, one needs to flip a macroscopic droplet ory time exhibits a super exponential scaling, albeit only in
ofspinswhoseenergyscale withitsperimeter. Forquantum a limited range of temperature. This model consists of a
models,themostwidelyknownexampleofaself-correcting toriccode-likestructure,whereinsteadofqubitsd-levelspins
quantummemory is the 4D Kitaev’s toric code [1–3] whose (qudits) are placed on the edges of a square lattice. This
energybarrierisalsoproportionaltothelinearsystemsize. model also corresponds to the quantum double of Z . Its
d
2
elementary excitations are d different electric and d differ- introducethe frameworkof ouranalysis: the constructionof
ent magnetic anyons. Specifically, in Ref. [9] d = 5, and Abelianquantumdoublesandthenoisemodelusedtosimu-
due to charge-flux duality, it is convenient to think only in latethethermalenvironment. InSec.IIIwepresentourmain
terms of e.g. electric charges. Then there are 5 different result: the upper bound on the mixing time and the formula
charges, grouped as: vacuum, light particle, heavy particle, forthegeneralizedenergybarrier,followedbyadiscussionon
heavyantiparticle,lightantiparticle.Particle-antiparticlepairs thephysicalinterpretationofthisresultinSec.IV.Wepresent
havethesamemass,furthermorethemassesaresetsuchthat thedetailsofthederivationoftheboundinSec.V.Finally,we
m > 2m to ensurethatthermalevolutionofthe sys- concludewithpossiblefuturedirectionsinSec.VI.
heavy light
tem favoursthe decay of a heavy particle into two lightpar-
ticles. The authors of Ref. [9] further introduce defect lines
to the system by modifying local terms of the Hamiltonian. II. FRAMEWORK
Whenalightparticlecrossessuchaline,itbecomesaheavy
one and vice versa. This construction results in fractal-like We now introduce the framework in which our result is
splitting of typical anyon-paths, resembling the fractal geo- valid. First, we introduce the systems of interests, i.e. the
metrical support of logical operators in Haah’s cubic code quantum double of Abelian groups. Second, we model the
[10,11].TheauthorsofRef.[9]numericallyobservedamem- thermalizationofsuchasystembytheDaviesmap.
orytimeforthisentropiccodesimilartothecubiccode,that
is, it growssuper-exponentiallywith the inversetemperature
(t exp(cβ2)). A striking difference between the cu- A. Abelianquantumdoubles
mem
∝
bic code and Brown’s entropic code is, however, that while
the former has an energy barrier that grows logarithmically Abelian quantum doubles are a special case of the quan-
with system size, the energy barrier of the entropic code is tumdoubleconstructionintroducedbyKitaev[1], wherethe
a constant, independent of system size. Thus, Brown’s en- quantumdoubleisbasedonthecyclicgroupZ . Thiswasthe
d
tropic code seems to have a better scaling of memory time modelinvestigatedin Ref. [9] with d = 5, and it is a gener-
thantheonepredictedbytheArrheniuslaw. However,itwas alized toric code construction(the toric code is the quantum
also remarked that the super-exponentialscaling did not re- doubleofZ )actingond-levelspinsorqudits.
2
main valid at arbitrarily low temperature, i.e. in the limit of
verylargeβ. Thus,Brown’sentropiccodearguesforthepos-
sibility of entropyprotectionbut failed to settle the question 1. GeneralizedPaulioperators
whetherentropycanprotectquantuminformationandleadto
a better scaling of than memory time than the one predicted We will choose a basis for the Hilbert space of a qudit to
byArrheniuslaw. be labelled by orthonormal states ℓ where ℓ Z =
{| i} ∈ d ∼
Here,wesettlethisquestioninthenegativebyprovingthat 0,...,d 1 .WeintroducethegeneralizedPaulioperators,
ascalingenergybarrierisnecessaryforself-correctionforany X{ k andZ−k,k} Z . Theyactonaquditaccordingto:
d
∈
quantumdoublemodelofanAbeliangroup,ageneralframe-
Xk ℓ = ℓ k , (2)
work which contains Brown’s entropic code. Thus, entropy | i | ⊕ i
cannotprotectquantuminformationintheabsenceofascal- Zk ℓ = ωkℓ ℓ , (3)
| i | i
ingenergybarrierforthosemodels.Technically,weestablish
where is the addition modulo d and ωℓ = exp(i2πℓ/d),
arigorousversionoftheArrheniuslawasanupperboundfor ⊕
ℓ Z are the dth roots of unity. The eigenvalues of the
the mixingtime of quantumdoublesof Abelian groups. We ∈ d
Z generalizedPaulioperatorbutalsotheX generalizedPauli
provethatthemixingtime–definedasthelongesttimeanini-
operatorarepreciselythedthrootsofunity.Inourconvention,
tial state takes to thermalizeto the Gibbsstate– and thusthe
theidentityisageneralizedPaulioperatorswithk = 0. One
memorytimeareupperboundedbypoly(N)exp(2βǫ)where
canstraightforwardlyderivethefollowingusefulidentities
N isthesizeofthesystemandǫisthegeneralizedenergybar-
rier. Werigorouslydefineǫbyanaturalquantityarisingfrom X† =Xd−1 Z† =Zd−1 Zk′Xk =ωkk′XkZk′. (4)
ouranalysiswhichstraightforwardlyextendstheintuitiveno-
tion of energy barrier. Finally, we evaluate the generalized
energybarrierandshowthatitisindependentofsystemsize 2. Hamiltonian
ortemperaturefortwo-dimensionalAbelianquantumdouble
models. Asourboundholdsforanytemperature,thismeans
We now define the Hamiltonianof the quantumdoubleof
thatAbelianquantumdoublesdon’tallowforentropyprotec-
Z on 2N d-level spins or qudits located on the edge of a
d
tion, i.e., their memorytime can at most scale exponentially
two dimensional square lattice with N vertices. We define
withinversetemperature.Ourresultsarebasedonthemethod
a (generalized) Pauli operator to be a 2N-tensor product of
presented in Ref. [12] and are a generalizationof the results single-qudit(generalized)PaulioperatorXk orZk, k Z .
d
therein,wheretheauthorhasderivedasimilarArrheniuslaw ∈
For convenience, we will henceforth omit the (generalized)
boundandenergybarrierforanycommutingPaulistabilizer
modifier. We note theset ofPaulioperatorsactingnon-
M
codesinanydimensions. P
triviallyonatmostM 2N qudits. Thequditsonwhicha
≤
Thepaperisorganizedasfollows. InSec.IIAandIIBwe Paulioperatoractsnon-triviallyareits(geometrical)support.
3
Z
-k Xk Xk Xk Xk
Z Z
Z Z Xk Xk X-k Xk X-k
Z Z Z k k -k X-k X-k
Z Z
X Z Z
X X Z2 Z Z2 Z2
X FIG.2: (a)Anyonpaircreatedfromvacuum,(b)oneoftheanyons
movedand(c)thepairfusedbacktovacuum,allthewhileapplying
localoperators.
FIG.1: (a)StaroperatorA(v)inblueandplaquetteoperatorB(p)
inred. (b)ExamplesofanyonpathsforanAbelianquantumdouble
(d>2). asquarelatticewithperiodicboundarycondition,i.e.,atorus,
thegroundspaceisd2-degenerateandcanbeusedtoencode
quantuminformation. ThepositivenumbersJa(v) andJb(p)
v p
ThelocalinteractionsoftheHamiltonianwillbePauliop- fornon-zeroaandbcanphysicallybeinterpretedasmassesof
eratorssupportedonfourquditsneighbouringeitheravertex thedifferentexcitationsofthemodel,whichwenowdiscuss.
vofthelatticeforstaroperatorsA(v)oraplaquettepforpla-
quetteoperatorsB(p), seeFig.1(a). Astar(andaplaquette)
is the union of four edgesor, equivalently,quditslocated on
3. Excitationsandsyndromes
thoseedges. Itisconvenienttolabelthequditsaroundastar
+orplaquette(cid:3)usingthecardinalpoints: East,South,West
EveryspectralprojectorPa(v)andQb(p)arepairwisecom-
andNorth.ThestaroperatorA(v)forvertexvis v p
muting. Moreover, they commute with the Hamiltonian.
A(vi)=XE⊗XS⊗XW† ⊗XN† (E,S,W,N)=+v (5) Tthheuqsu,iatnitsucmonnvuemnibeenrtstaol=abealaneannedrgby=eigebnvedcetofirn|eψdibuysing
v p
{ } { }
andtheplaquetteoperatorB(p)forplaquettepis
a = ψ A(v) ψ (11)
v
h | | i
B(p)=Z Z† Z† Z (E,S,W,N)=(cid:3) . (6) b = ψ B(p) ψ . (12)
E ⊗ S ⊗ W ⊗ N p p h | | i
The eigenvalues of star and plaquette operators are the dth Usingtheterminologyofquantumerrorcorrection,wedefine
rootsofunity,inheritedfromthesingle-quditPaulioperators. thesyndromeof ψ by
The projector unto the eigenvalue ωa of the star operator at | i
vertexvis e( ψ )=(a,b) ZN+N. (13)
| i ∈ d
1 d−1 Hence,theHamiltoniancanbediagonalizedusingthediffer-
Pa(v) = (ωaA(v))k. (7)
v d entsyndromevalues,i.e.,
k=0
X
Similarly,theprojectoruntotheeigenvalueωaoftheplaquette H = ǫ(a,b)Π(a,b), (14)
operatoratplaquettepis (Xa,b)
d−1 wheretheexplicitformulafortheenergiesǫ(a,b)andprojec-
1
Qb(p) = (ωbB(p))k. (8) tors Π(a,b) can be found in Sec. VA. Let us try to draw a
p d
k=0 physicalpicturewhichwillhelpintuition.
X
Thesyndromeofanenergyeigenvectorisabookkeepingof
Notethatthoseprojectorscommutesinceeverystaroperator
thedifferentexcitationsateveryvertexandplaquette.The+1
commutewitheveryplaquetteoperator.
eigenvectorsofPa(v)fora(v)=0haveapoint-likeexcitation
TheHamiltonianoftheZd quantumdoubleis[1,9] located on the vevrtex v which6 we call an electric charge (or
d−1 d−1 chargeon)of type a. Similarly, the +1 eigenvectorsof Qpb(p)
H = Ja(v)Pa(v)+ Jb(p)Qb(p), (9) for b(p) = 0 have a point-like excitationlocated on the pla-
v v p p 6
v a=0 p b=0 quettep whichwe calla magneticflux(orfluxon)of typeb.
XX XX
ThegroundstatesoftheHamiltonianhavesyndrome(0,0).
where Ja(v) and Jb(p) are non-negative numbers. We set Physically,the point-likeexcitationscan(i) becreatedout
v p
v,pJ0 = J0 = 0 such that a ground state Ω is a com- ofthevacuumbyapplyingalocaloperatoronagroundstate,
∀ v p | i
mon+1eigenvectorofallP0andQ0 (ii)propagateonthelatticeand(iii)annihilatebacktothevac-
v p
uumbyapplyinga localoperator. Thiscanbe understoodat
v,p P0 Ω =Q0 Ω =+ Ω . (10) thelevelofthesyndrome.Consideronagroundstateandthen
∀ v| i p| i | i applyageneralizedPaulioperatorXkonaquditlocatedona
ThegroundspaceisdegeneratewheneverthisHamiltonianis horizontaledge(seeFig.2). Thiswillmodifytheeigenvalues
definedonamanifoldwithnon-zerogenus. Forinstance,on oftheplaquetteoperatorsNorthandSouthofthathorizontal
4
edge,denotedB(p )andB(p ). Indeed,theresultingstate whereH isthe(Lamb-shifted)systemHamiltonianH =
N S eff eff
will be a +1 eigenvector of the spectral projectors Q−k and H + S†(ω)S (ω)andtheLiouvillianis
pN system α,ω α α
Qk . Physically,Xk createdamagneticfluxoftypek (resp.
pS P
k) on the South (resp. North) plaquettes. In other words, 1
−Xkcreatedapairofconjugatemagneticfluxesoutofthevac- L(ρ)= γα(ω) Sα(ω)ρSα†(ω)− 2{Sα†(ω)Sα(ω),ρ}+ .
uum. Similarly,Zk wouldcreate apair ofconjugateelectric Xα,ω (cid:18) (cid:19)
(21)
chargesoutofthevacuum.WeassigntoanygeneralizedPauli
Theoperatorsgoverningtheevolutionofthesysteminenergy
operatorσ thesyndromee(η)ofthestateσ Ω
η η| i space are the spectraljump operators,Sα(ω). Theytake the
system from energy eigenstate ǫ′ to another eigenstate with
e(η)=e(ση|Ωi). (15) energyǫ=ǫ′+ω,andhavetheform
Inourexamples,
S (ω)= Π(a,b)S Π(a′,b′). (22)
α α
e(Xk) = (a=0,b=[0,...,0,k, k,0,...,0]) (16) ǫ(a,b)−Xǫ(a′,b′)=ω
−
e(Zk) = (a=[0,...,0,k, k,0,...,0],b=0).(17) TheyaretheFouriertransformsofS (t)(thetime-dependent
− α
operatoractingonthesystemduetoitscontactwiththether-
Givenanyenergyeigenvector ψ andanygeneralizedPauli malbath):
| i
operatorη,thesyndromeofthestateσ ψ isobtainedby
η
| i
S (t)= eiǫ(a,b)tΠ(a,b)S Π(a′,b′)e−iǫ(a′,b′)t.
α α
e(σ ψ )=e(η) e( ψ ). (18)
η| i ⊕ | i (a,bX),(a′,b′)
(23)
ThisverysimpleadditionrulestemsfortheAbelianstructure Theratewith whicha state ofthe systemis takentoanother
ofthegroupZdandisrelatedtothefusionrulesoftheexcita- state ω far in energy, by applying the jump operator Sα(ω)
tionsofthisAbeliantopologicalmodel. due to its coupling to the thermal bath is the transition rate
Fluxonsandchargeonsturnouttobe(Abelian)anyons,i.e., γ (ω). Thesetransitionratesobeydetailedbalance
α
quasi-particleswhicharenotbosonicnorfermionic.Yettheir
anyonicnaturewillnotbeessentialinourwork.However,we γ (ω)=eβωγ ( ω). (24)
α α
−
will from now on use the term anyon to designate a generic
point-like excitation (either a fluxon or a chargeon). More- ThisLiouvilliandrivesanystatetowardstheGibbsstate
over, chargeons and fluxons are related by an exact duality
whichmapsthe lattice to the duallattice. Thus, itwill often ρ e−βHsystem (25)
G
∝
beconvenienttofocusonasingleanyontype,e.g.,chargeons
in order to simplify our discussion and notations. Also, we whichisitsuniquefixedpoint: (ρG)=0.
L
would like to introduce a single index s which labels either ApplyingtheDaviesmaptoaZd quantumdoubleweneed
theverticesortheplaquettes,i.e.,s = v/p. Thus,anyanyon to choose an operator basis for the jump operators S . For
α
(chargeonorfluxon)islocatedonasite(vertexorplaquette). d = 2, apossible choiceis thePauligroup,whileford > 2
itisthegeneralizedPauligroup. Weshouldbecareful,since
although the elements of the Pauli group are Hermitian, the
elementsofthegeneralizedgrouparenot: X† = Xd−1. We
B. Thermalnoisemodel
cancircumventthisproblembyeitherwritingtheinteraction
terms in the full Hamiltonian as σ B† + σ† B
Themodelusedinourworktosimulatethethermalization j,α′ ⊗ α j,α′ ⊗ α
withσ = ZlXm,thusS = ZlXm asintheZ
processofthequantumdoubleistheDaviesmap[13,14],the j,α′=(l,m) j j j,α′ j j 2
case, or by constructing Hermitian jump operators: S =
goldstandardforsimulatingthethermalizationofmanybody j,α′
systems [3, 9, 11, 15]. The system is coupled to a bosonic 1/√2(σj,α′ +σj†,α′). Independentofwhichchoicewemake,
ourresultsinthefollowingsectionsarethesame.
bathandtheHamiltonianof system+bath reads
{ }
H =H +χ S B +H , (19)
full system α α bath
⊗ III. GENERALIZEDENERGYBARRIER
α
X
whereB istheoperatoractingonthebathandS Sj is Weestablishaformerlyill-definedlinkbetweentheenergy
an operaαtoracting on spin j of the system. We coαn≡siderα′the barrierof a system andits mixingtime forAbelian quantum
weakcouplinglimit,withχ 1. doubles. WeprovearigorousArrheniuslawupperboundfor
The density operator of th≪e system, noted ρ, evolves ac- themixingtime(Sec.IIIB,detailsoftheproofinsectionV),
cordingtothemasterequation andgiveaproperdefinitionfortheenergybarrierappearingin
thatbound(Sec.IIIA).InsectionIIICweevaluatethisenergy
dρ barrier for Abelian quantum doubles in two dimensions and
= i[Heff,ρ]+ (ρ), (20) finditisaconstantindependentofsystemsizeortemperature.
dt − L
5
A. Definitionofthegeneralizedenergybarrier ση is
ǫ(η)= min maxǫ(ηt η) (30)
Recallfromtheintroductionthattheenergybarrierisintu- {σηt}→ση t |
itively related to the decompositionof operators acting non-
triviallywithinthegroundspace(logicaloperators)intoase- ThegeneralizedenergybarrieroftheHamiltonianHis
quence of local operators. Surprisingly, the generalized en-
ergybarrierarisingfromouranalysisisrelatedtotheenergy ǫ(H)=maxǫ(η). (31)
η
costofbuildinganarbitraryPaulioperator. Thisseemstogo
againstintuitionsince an arbitraryPaulioperatorση 2N Wenowintroducethemixingtime,anupperboundonthe
∈ P
can create an extensive amount of energy. However, excita- quantummemorytime, and thenintroduceour boundwhich
tions which appear in the final error configuratione(η) cre- relatesitto the generalizedenergybarrierthrougha formula
atedbythePaulioperatorwillnotcontributetowardsthegen- similartotheArrheniuslawgiveninEq.(1).
eralizedenergybarrier: onlyintermediateexcitationscreated
inthesequentialconstructionofthisfinalerrorconfiguration
do.Notethatifηisalogicaloperator,thegeneralizedenergy B. Arrheniusupperboundonthemixingtime
barriercoincideswiththeintuitiveenergybarrier.
The idea is thus to consider sequences of Pauli operators
We define the mixing time as the time scale after which
σ which sequentiallybuild the operatorσ by applying
ηt η the evolution of any initial state of the system becomes
Pauli operators acting on a single qudit. We call such a se-
ε = e−1/2-indistinguishablefromtheGibbsstate definedby
(cid:8) (cid:9)
quencealocalerrorspath.Indeed,wethinkofηastheindex Eq. (25). Theε = e−1/2 value is chosenso the relationship
of the final error which we sequentiallybuild throughsingle
between the mixing time and the gap of the Liouvillian will
quditerrorssuchthattheerroratsteptisindexedbyηt.
haveaconvenientform,andtheexactvaluewon’tmodifyei-
therthequalitativeaspectofourcalculationsorthescalingof
Definition1(Localerrorspath) A local errors path
theboundobtainedonthemixingtime.
σ isasequenceofPaulioperatorssuchthat
ηt t≥0
{ }
Definition4((ε)-mixingtime) The (ε)-mixing time of a Li-
σ =I (26)
ηt=0 ouvillian(whosefixedpointistheGibbsstateρ )is
G
locality t P σ =P σ (27)
∀ ∃ ∈P1 ηt+1 · ηt t (ε)=min t t′ >t eLt′ρ ρ <ε ρ . (32)
convergence ση,T t>T σηt =ση (28) mix { | ⇒|| 0− G||1 ∀ 0}
∃ ⇒
Atanyintermediatestept≤T,thePaulioperatorσηt will whereweusedthetracenorm,||A||1 = Tr √A†A ,tomea-
createasyndromee(ηt)correspondingtoapatternofanyons. surethe(in)distinguishabilityoftwoquantuhmstatesi.
Ateverysite,onlytheenergyofananyonwhosechargeisdif-
ferentfromtheoneinthesyndromee(η)contributetowards Definition5(Mixingtime) The mixingtime ofa Liouvillian
theenergybarrier. Formally,wedefinetheadditionalenergy isitsε=e−1/2(ε)-mixingtime.
oftheerrorindexedbyηtwithrespecttotheerrorindexedby
Looselydefiningthequantummemorytimeasthemaximal
ηas
timeafterwhichonecanrecoverinformationabouttheinitial
Definition2(Additionalenergy) Letηt andη beindicesof groundstate, we immediatelysee it isupperboundedby the
two Pauli operators. The additionalenergy of the error σ mixingtime. Indeed,theGibbsstatetreatsallgroundstateon
ηt
thesamefootingandthusinformationabouttheinitialground
withrespecttothereferenceoperatorσ is
η
state hasdisappeared. We donotprovidea formaldefinition
ofthequantummemorytimeinthiswork.
ǫ(ηt|η)= Jses(ηt) 1−δes(ηt),0 1−δes(ηt),es(η) miOxiunrgmtiaminertehsruolutgrhelaatreeslathtieognesnimeriallairzetodtehneerAgryrhbeanrriiuesrltaowt.he
s
X (cid:0) (cid:1)(cid:0) ((cid:1)29)
Theorem6(Arrheniusboundonmixingtime) For any
Note that in Eq. (29), summands do not contribute if AbeliangroupZ , foranyinversetemperatureβ, themixing
d
e (ηt) = 0,i.e., iftheintermediateerrordoesnotcreateex- time oftheDaviesmapLiouvillianofthequantumdoubleof
s
citations on site s but also if es(ηt) = es(η), i.e., if the in- Zd isupperboundedby
termediate error creates the same excitation on site s as the
referenceerrorσ .
η
Wearenowinpositiontodefinethegeneralizedenergybar- tmix βNµ(N)eβ(2ǫ¯+∆) , (33)
≤O
rierofanerrorση andthenoftheHamiltonian. (cid:16) (cid:17)
where2N isthenumberofquditsinthesystem,∆isthegap
Definition3(Generalizedenergybarrier) Let σ of the system Hamiltonian, ǫ¯is the generalized energy bar-
η 2N
∈ P
be a Pauli operatorand σ denotean arbitrary localer- rierandµ(N)definedbyEq.(36)isthelengthofthelongest
ηt
{ }
rorspathconvergingtoσ . Thegeneralizedenergybarrierof optimallocalerrorspath.
η
6
ThederivationofthisresultcanbefoundinsectionV.We ateerror,thereisauniqueelementaryerrorundercon-
willnowshowthatforAbelianquantumdouble,ǫ¯isbounded struction. The other elementary errors are either not
byaconstantindependentofsystemsizeinSec.IIICandthat constructedyet, oralreadyconstructed. Ineithercase,
µ(N)isboundedby8N(d 1)inSec.IIID.Therighthand theydonotcontributetowardstheenergybarrier.
−
sideofEq.(33)hasadependenceonalowpowerofN,that
does not qualitatively modify the scaling of the mixing time Anyelementaryerrorcanbeinterpretedasanyonsde-
•
northebehaviourofthesystemwhenconsideredasa candi- caying and fusing together, thus forming a fully con-
date for a quantummemory. The importantphysicalquality nected, directed graph with weighted edges. Due to
ofthisboundistheArrheniuslawscaling. Thisscalingisset chargeconservation,thisgraphisaflow.
bythegapofthesystemHamiltonianbut,moreinterestingly, Theerrorisatensorproductofsingle-quditPaulioper-
bythegeneralizedenergybarrier,whichwenowevaluate. ators which create pair of conjugateanyonsout of the
vacuum, fuse and move anyons. Thus, to every ele-
mentaryerror,wecanassociateagraphwithweighted
C. ThegeneralizedenergybarrierisaconstantforAbelian edges, often referred to as string-nets in the literature.
quantumdoubles
SuchagraphisdepictedonFig.3.
A terminal vertex, i.e., a vertex of valency 1, is an
Wewillnowevaluatethegeneralizedenergybarrierofany
anyon.Itisconvenienttolabelterminalverticesbytheir
2D quantum double of an Abelian group and show that it is
anyoniccharge,i.e.,thevalueofthesyndromeoftheel-
aconstant,independentofsystemsize,moreprecisely2J .
max ementary error on that site. Other vertices correspond
WhilethiswasknownfortheZ case[12],weextendittoany
2 toworldlinesofanyons.Verticesarelinkedbyanedge
Z quantumdouble.Fromnowon,weconsideraZ quantum
d d of weight k if the errors between them correspond to
double, with arbitrary d. Furthermore, we henceforth omit
moving an anyon of charge k along the orientation of
the ’generalized’modifierin (generalized)energybarrierfor
theedge. Anedgeofweightk connectingsiteitosite
simplicity.
j isequivalenttoanedgeofweightd kconnectingj
ToevaluatetheenergybarrieroftheHamiltonian,givenby −
toi.
Eq. (31), we want to bound the barrier of an arbitrary Pauli
operator, given by Eq. (30). Thus, we aim to exhibita local At this point, we have built a directed graph satisfy-
errors path where the additional energy of any intermediate ingweightconservationateveryvertex.Indeed,weight
errorisaconstant. Todoso,wewillusethefollowingstrat- conservationinthegraphisequivalenttochargeconser-
egy.Wewillfirstturnthefinalerrorsyndromeintoaweighted vationoftheanyonsinthisAbeliantopologicalmodel.
directed graph intuitively correspondingto the worldlinesof Suchagraphiscalledagraphflow.
anyons. Then,wewilldecomposethatgraphintocyclesand
trees. Cyclescorrespondtopairofconjugateanyonsappear-
ingoutofthevacuum,propagatingandthenfusingbacktothe
vacuum. Treesrepresentpropagationofanyons,whoseposi- 2. Decomposingthegraphintocyclesandtrees
tion (resp. word lines) correspond to terminal vertex (resp.
edges)ofthetree. Finally,usingdifferenttechniquesforcy- Wenowuseawell-knownresultfromflowtheory:anyflow
cles and trees, we show how to build the error of each type can be partitioned into three sets: a rotational and an irro-
bymovingatmostoneanyonatatime ina way thatthead- tational flow and a harmonic component (Helmholtz-Hodge
ditionalenergyofanyintermediateerrorinvolveatmosttwo decomposition)[16].
localtermsoftheHamiltonian,resultinginanenergybarrier On this discrete geometry of a graph, the rotational flow
ofatmost2Jmax. consist of loops (a.k.a cycles), the irrotational flow consists
oftreeswhichcanbethoughtasunionofstringsandthehar-
monicpartconsistofirrotationalflowsonthenon-contractible
1. Graphcorrespondingtoanerrorconfiguration cycles.
Thisdecompositioncanbephysicalinterpretedintermsof
Any error is the product of elementary errors whose anyonswhichwenowdoinordertoevaluatetheenergybar-
•
supports are disjoint. The energy barrier of the error rier. Remembertherulesoftheadditionalenergy,definedin
isthelargestenergybarrierofitselementaryerrors. Definition. 2 : an intermediate errorhas additionalenergyif
an anyon at a given site is not the anyon created by the ref-
Let’s consider an arbitrary error, i.e., a Pauli operator.
erenceerror. The goalis nowto buildthe referenceerrorby
Its geometrical support, i.e., qudits on which it acts
introducingaslittleadditionalenergyaspossible.
non-trivially,splitsintoconnectedcomponents.Wecan
decompose the global operator into a product of ele-
mentary operators, each of which is supported on one
connected component. No terms of the Hamiltonian 3. Evaluatingtheenergybarrier
hassupportwhichintersecttwoconnectedcomponents.
Thus, we can choose the local errors path so that ele- Loops correspond to a particle-antiparticle pair appearing
mentaryerrorsarebuiltsequentially. Inanyintermedi- out of the vacuum, then propagating and eventually fusing
7
Z 1 Z 1
Z Z
Z Z
Z 4 Z
Z Z
Z2 4 Z2
Z3 Z2 Z3 Z2
Z3 Z Z3 Z3 3 Z3 Z Z3 Z3 2
2 2
3 3 3 3
1 sink 1 source (cid:1) (cid:3)
(cid:2) (cid:4)
3 3
2 2 2 2
4 source 4 =-1 sink
2
3
Z2
3 2 Z3 Z3 Z2
Z3 Z3 Z3 Z3
FIG. 3: Illustration of the steps towards constructing the optimal
canonical pathforquantumdouble Z5: (a)thesupport oftheerror
canbepartitionedintothreeconnectedcomponents,colouredinred,
blueandgreen; (b) eachelementaryerror canbeinterpretedasfu-
sions,decaysandmovingofanyons;(c)anelementaryerrorcanbe FIG.4: Decompositionofatreeintostringsforthequantumdouble
mappedontoafullyconnecteddirectedgraphwithweightededges of Z5. The weight of each edge of the graph is represented by a
(reversing theorientationof an edge changes theweight fromk to colourcoding.
d−k);(d)thiscanbeinterpretedasaflowofcharges;(e)theflow
canbepartitionedintoarotational(blue)andirrotational(red)part;
and(f)thesedifferentpartitioningsoftheflowareallequivalent to siteoftheroot. Wesequentiallyconnecteachleaftotheroot.
applyingthesamecombinationofoperatorsineitherloopsorstrings. Duringanystep,therewillbeatmosttwoviolations,onefor
thesiteoftheconjugateanyonbeingmovedtothesiteofthe
rootand one for the anyonat the rootwhich mightnothave
back to the vacuum. Such a configurationcan be createdby the anyonic charge it should have in the error configuration.
movingtwoanyons. Thus, loopshaveanenergybarriercor- At the end of the procedure, the charge of the anyon at the
respondingtotheenergyoftwoanyons. rootwillbetheoneitshouldhaveinthereferenceerrorsince
the totalanyonicchargeofthe tree is zero. Thus, treeshave
Wenowexplainhowtoconstructanerrorwhosesupportis
anenergybarriercorrespondingtotheenergyoftwoanyons,
atree.
similartotheenergybarrierofloops.
We can consider the tree to be a superposition of strings,
Wehavethusproventhefollowingresult:
eachstringcorrespondingtoapairofconjugateanyonswhich
hasbeencreatedoutofthevacuumandthenpropagated.The Theorem7(EnergybarrierofAbelianQuantumDoubles)
terminalvertexofthetreecorrespondtoanyons,conveniently For any d, the generalized energy barrier ǫ of the quantum
labelledbytheiranyoniccharge.Forconvenience,chooseone doubleofZ isatmosttheenergyoftwoanyons,i.e.,
d
oftheseterminalverticestobetherootofthetree. Otherter-
minal vertices will now be called ”leaves”. The root is con- ǫ(HZd)≤2Jmax (34)
nected to each leaf by a path whose weight is the anyonic
charge of the leaf. Each such path is a string operator con-
necting an anyon (at the leaf) to its conjugate anyon (at the D. Lengthofthelocalerrorspath
root).SeeFig.4foragraphicalexample.
We construct the error corresponding to the tree by itera- Wehaveestablishedintheprevioussubsectionsthattheop-
tivelychoosingarandomleafandthenapplyingthesequence timallocalerrorspathconsistsofpartitioningtheerrorintoa
ofgeneralizedPaulioperatorswhichcreatethecorrectanyon productoferrors,eachofwhichissupportedeitheronaloop
atthesiteoftheleafandthenmoveitsconjugateanyontothe oraunionofstrings. Thequestionremainswhatisthelength
8
µ(N), i.e., the number of steps before the local errors path (cid:1)
σ convergesto the referenceerrorσ . Formally,define
ηt η (cid:2)
σ to be the number of operators needed to converge
(cid:8) ηt(cid:9)
to a reference error ση. We only consider optimal local er- (cid:3)
(cid:12)(cid:8) (cid:9)(cid:12)
r(cid:12)orspat(cid:12)h,i.e.,thosewhichrealizeσηwiththeminimalenergy
(cid:4)
barrier. Wethendefinetheoptimallocallengthofanerrorto
be
µ(η)= min σ (35)
{σηt}→ση ηt FIG.5: Mergingofloopsataquditinitiallyaffectedbythreeloops
maxtǫ(ηt|η)=ǫ(η)(cid:12)(cid:12)(cid:8) (cid:9)(cid:12)(cid:12) fisorretphreeqseunatnetdumbydaoucbolleouorfcZo5d.inTgh.eweightofeachedgeofthegraph
and the quantity which enters in the bound of mixing time,
Eq.(33)isthemaximaloptimallocallengthoferrors
2. Strings
µ(N)=maxµ(η) (36)
ση
Aunionofstrings–afterremovingtheloopsfromthestruc-
Notethatthisisconstrainedoptimization:wechoosetofirst
ture–formatreewithseveral”leaves”,theleavescorrespond-
minimizetheenergybarrierandthenlookatthelengthofthe
ingtotheendpositionofanyons.
localerrorspathrealizingthatminimum. Thischoiceisdic-
Itwillbenecessarytointroduceaprocedureto”prunethe
tatedbythefactthattheenergybarrierenterstheexponential
tree”, i.e., decompose a tree into a superposition of subtrees
inEq.(33)whereasthemaximaloptimallocallengthoferrors
withoutintroducingnewanyons.Thispruningprocedurewas
µ is only a multiplicativeconstant. Nonetheless, µ is an ex-
notnecessarytoprovethatthegeneralizedenergybarrierisat
tensivequantitysinceforanyerror,µ(η)islower-boundedby
most 2J , but will proveuseful to boundthe length of the
twicethesizeofthesupportoftheerror(thefactortwocomes max
canonicalpath.
fromapplyingthe X andZ partof theerrorindependently).
The pruning procedure identifies subtrees that can be re-
Thus,4N µ.
≤ moved from the original tree. Those subtrees should have
However, in orderto minimize the energybarrier, a given
leaves whose anyonic charge sum to zero modulo d so that
qudit could be affected multiple times by single-qudit oper-
theycanberemovedwithoutaffectingtheroot. Beforeiden-
ators applied between two intermediate errors. In the lan-
tifyingthosesubtrees,itisconvenienttofirst”fattenthetree”
guage of graph, a given edge of the graph could belong to
by connecting every leaf of weight k to the root through a
a large number of loops and trees. Indeed, one has to be
string of weight k. The pruningprocedurethen proceedsby
carefultoavoidsuchaphenomenon. Herewewillshowthat
visiting every vertex of the tree (for instance using a post-
the loop part of the error can be constructed with a path of
order depth first search [28]). At every vertex of the tree, it
lengthatmost4(d 1)N whilethestringpartwithapathof
− checkswhetherthereexistsasubsetofedgeswithzero-sum.
lengthatmost4(d 1)N,thusthemaximaloptimallengthis
− If so, the pruning procedure removes the subtree generated
µ(N) 8(d 1)N.
≤ − by the correspondingleaves. After visiting everyvertex, the
pruningstops. Thetreeisnowdecomposedintosimpletrees.
Simple trees have at most d 1 leaves since any set of d
1. Loops
−
anyons contains a subset whose sum is zero modulo d (see
Thm.10). Theirdepthisthusboundedbyd 1Also, every
Givenaqubit,wewanttoboundthenumberofloopswhich −
vertexofasimpletreebelongstoatmostd 1strings.
actnon-triviallyonthatqudit. A priori,the numberofloops −
Thus,afterpruning,everyquditbelongstoatmost(d 1)
couldbeverylarge. However,wecanuseasimpleprocedure −
loopsoftypeX and(d 1)loopsoftypeZ. Sincethereare
to reduce it. The idea is to look at the weight of all edges −
2N qudits,
overlappingthatquditandtoidentifysubsetsofthoseweights
whichsumto0 modulod. Inthatcase, we canfusethecor-
µ 4(d 1)N. (38)
responding anyons to the vacuum and get new loops which strings ≤ −
do not affectthe qudit. We call this proceduremerging. An
exampleofmergingispresentedonFig.5.
E. Theeffectofdefectlines
This procedure can be repeated on every qudit indepen-
dently. Thequestionisthentoboundthenumberofloopsat
ForAbelianquantumdouble,itispossibletolocallymod-
theendofmerging. InAppendixA,weinvestigatethisques-
ifytheHamiltonianinordertointroducedefectlines,suchas
tion using multiset theoryand find that the maximalnumber
in the work of Brown et al. [9]. Defect lines are character-
ofloopsthatcanremainaftermergingisd 1(seeThm.10).
− izedbyaninvertibleelementM Z andanorientation. An
Thus,aftermerging,anyquditbelongstoatmost(d-1)loops d
∈
anyonoftypek Z crossingadefectlineoftypeM along
oftypeXand(d-1)loopsoftypeZ.Sincethereare2N qudits, d
∈
the orientation (resp. against the orientation) will be trans-
µ 4(d 1)N. (37) formedintoananyonoftypeM k(resp.M−1 k). Whatdo
loops
≤ − · ·
we meanby ”transformed”? Considertwo vertices(v ,v )
− +
9
onthelattice,oneoneachsideofthedefectlinesuchthatthe 2. Globallyconsistentlabellingofanyontypesinthepresenceof
orientation points from v to v . There exists a local Pauli defectlines
− +
operatorwhichmapsa+1eigenstateofPk toa+1eigenstate
vi
ofPM·k. Inotherwords,anexcitationwilllocallyatenergy Theonlyremainingquestionis:knowingthatthedefinition
vi
J becomeanexcitationcarryingenergyJ . of the energy barrier is the same with defect lines, does the
k M·k
In [9], Brown et al. proposed a local 2D Hamiltonian evaluationof the energybarrier detailed in the previoussec-
which seemed to realize entropy protection [9]. This model tionsgothroughthesameway?Themainissueishowtolabel
is the quantumdoubleof Z ; and dueto charge-fluxduality, theexcitations.Indeed,duetothepresenceofdefectlines,the
5
we’reallowedtothinkonlyin termsofe.g. electriccharges. locallabellingoftheanyontypeisnotconsistentglobally.
Thenthereare5differentcharges,groupedas: vacuum,light Hereweexplainhowtorecoveragloballabellingofanyon
particle, heavy particle, heavy antiparticle, light antiparticle. types, under one technical condition we call consistency of
Particle-antiparticle pairs have the same mass, furthermore defectlines. We definethe consistencyofthedefectlinesof
m > 2m to ensure that duringthe thermalevolution a model by requiring that when we create a pair of anyons
heavy light
of the system it is favorable for the heavy particles to decay from vacuum, then take one of them around any loop any-
into two light particles. In order to favor the occurrence of whereonthelattice,theyfusebacktovacuumwitheachother.
heavyparticles,theauthorsofRef.[9]introduceddefectlines Should that transparencycondition be violated, the intersec-
of type M = 2 to the system. When a light particle crosses tionofdefectlineswouldbecomeasinkandasourceforsin-
such a line, it becomes a heavy one and vice versa. Thus, gle anyons, which we forbid. Furthermore, we do not know
theexcitationsinthemodelaretypicallylightparticleswhich howtheHamiltonianofsuchapathologicalmodelwouldbe
propagatefreelyuntiltheyeventuallycrossa defectline, ac- writtendowninaformsimilartoEq.(9).
quire mass, and then decay into two light particles. It was Thus, we consider consistent defect lines. Our goal is to
observed numerically in [9] that the memory time seems to take the globallyinconsistent, localanyonsyndromeswhich
behaveliket exp(cβ2)oversomerangeofparameters isarecordoftheeigenvaluesoftheA(v)/B(p)star/plaquette
mem
butseemstofailf∝orlargeβ. Canourboundshednewlighton operators at each site, and translate them to a consistent,
this model? To thatend, we now analyzethe effectof those globallabelingofanyons.Thistranslationisobtainedthrough
defectlinesonourbounds. aglobaldictionaryT2 Zd usingtheformula
∈
eglobal(ξ)=(T ) elocal(ξ), (41)
2
·
where ismultiplicationditwiseandmodulod. Todefinethe
1. SyndromesfortheHamiltonianwithdefectlines ·
globaldictionary,theideaistolabeleachregionenclosedby
defect lines. The anyon types will be defined in one (arbi-
Onecouldwonderwhetherthedefinitionoftheenergybar- trary)referenceregionandallotherregionswillcarryalabel
riergivenbyEq.(31)shouldbechangedduetotheintroduc- to translate the local anyon type within its region to what it
tion of defect lines. It does not. However, the Hamiltonian wouldbeinthereferenceregion(globalsyndrome).
changed and thus the syndromes of Pauli errors will change For instance, for Z , with M = 2 an anyon type a in the
5
too. GivenaPaulierrorξ ,itssyndromewithrespect reference region might become: a or 2a or 4a or 8a(= 3a
2N
to the new Hamiltonian en∈ewP(ξ) is related to the syndrome mod 5), depending in which region we observe it. We can
enodefectlines(ξ)ithadintheabsenceofdefectlinesbysimply nametheseregions,e.g. L=1,2,4and3intheaboveexam-
multiplyingthesyndromebythedefectlinestringT Z2N, ple.Wheneverweobserveananyonwhoselocaltypeisbina
1 ∈ d
regionwithalabelL,weknowthatanyonwouldhavealocal
typeb′ = L−1binthereferenceregion(oranytrivialL = 1
region).Thus,theT dictionaryisdefinedforeverysiteby
enew(ξ)=(T ) enodefectlines(ξ) (39) 2
1
·
(T ) =L−1 fors regionwithlabelL. (42)
2 s
∈
wheremultiplicationisunderstoodditwiseandmodulod.The
defectlinestringT Z2N isdefinedforeverysitesby
1 ∈ d
3. Evaluationofthegeneralizedenergybarrierandmaximum
lengthoftheoptimallocalerrorspath
M ifsnear(andonthe” ”sideof)adefectline
(T ) = −
1 s (1 otherwise Finally,introducingdefectlinesdoesn’tchangetheallowed
(40) anyon fusion/decay processes either, since the fusion rules
Therefore,thereisaconsistentwaytogetthesyndromesof are the same as before in every region. Whenever a parti-
thequantumdoubleZ withdefectlines,andwecanusethis cle crosses a defectline it is essentially just renamed, i.e., it
d
new set of syndromesto work throughthe same steps in the doesn’t leave behind a charge at the defect line. Using the
derivationas we did forthe quantumdoubleswithoutdefect factthatthesyndromescanbemadeconsistentwiththepro-
lines. These two derivations will essentially be identical – cedure of tracing all anyonsback to the L = 1 regions, any
exceptforthedifferentdefinitionsofthesyndromes–andwe errorcanstillbemappedontoagraphflowofanyons,andthe
willarrivetothesameformulafortheenergybarrier. planforconstructinganygeneralizedPaulierrordescribedin
10
Sec. IIIC still works. Thus, the value of the energy barrier
This work.
andthemaximumlengthoftheoptimalcanonicalpathisun- Brown et al.
changedaswell: ǫ¯=2Jmaxandµ(N)≤4N2+4(d−1)N. Eoefnx ehtrnaatnprcooeplamicti eonnt
Therefore, as neither the definition of the energy barrier, Expected
scaling in the
northestructureoferrors,northeoptimalcanonicalpathfor absence of
enttropic
a certain error, nor the length of this path is changed by de- enhancement
fectlines,theArrheniuslawbounditselfisunchangedbythe Numerical
exploration of
defectlines. Brown et al.
FIG.6:Conceptualscalingofthelogarithmofthequantummemory
timetmemasafunctionoftheinversetemperatureβ.
IV. DISCUSSION
A. Possibleimprovements however,theprobabilityofsuchaworldlineareentropically
suppressed.
Applyingthe resultofthe presentpaperto thismodel, we
We briefly review some possible improvements on our
canseethatitsmemorytimeisupperboundedbyastrictAr-
bounds, indicate possible avenues to achieve those improve-
rhenius law, with an energy barrier that has no dependence
mentsandconjecturewhattheoptimalboundswouldbe.
ontemperatureorsystemsize,evenwhenincludingtheeffect
The polynomial dependence of the Arrhenius bound on
of permuting type defect lines. Since our bound is valid for
mixing time can probably be improved. Indeed, we expect
any value of the inverse temperature β, we can see that the
that better techniques would allow to get rid of the N pref-
exp(cβ2) scaling observed in Brown’s entropic code needs
actor in Eq. (33). However, the polynomial dependence of
tobreakdownforsufficientlylowtemperatures,asthemem-
the length of the longestoptimallocal errorspath µ N is
∼ ory time can’t exceed our bound. This breakdown at low
tight since one can find errors whose length are of the order
temperature was forecasted in [9]. Indeed, for low temper-
ofthenumberofqudits. Thus,weexpectthemixingtimeto
aturethe thermalprocessresultingin fractal-likeworldlines
scale with system size. The extensivenessof mixing time is
ofanyonsisnottypicalanymore,sincetheenvironmentcan’t
coherentwiththeintuitionthatsomesystemrelaxlocally.
provide the energy required for a light particle to become
However,thequantummemorytimemightbemuchshorter
heavy. Rather, a lightparticleneara defectlinewon’tcross,
than the mixing time. A dramatic example is the three-
but linger there until it meets with another particle and fuse
dimensionaltoriccodewhosequantummemorytime iscon-
with it either to vacuumor to a heavy particle. If fused to a
stant whereasits mixing time is exponentiallylong. Indeed,
heavy particle, that heavy particle can then cross the defect
oneofthelogicaloperatorisstring-likewhereastheotherlog-
lineandlowertheenergybybecomingalightparticle.
icaloperatorissupportedona2Dsheetofqudits.Theexpec-
The low temperature behaviour of Brown’s entropic code
tation value of the sheet-like logical operator thermalizes in
agreeswiththefactthatourbounddoesn’tallowittohavea
exponentialtimewhereastheexpectationvalueofthestring-
better than exponentialmemory time. The scaling observed
like logical operator is short-lived. We expect the quantum
inRef.[9]ismostlikelylimitedtotheregiondiscussedthere,
memorytimeof2DAbelianquantumdoubletobeaconstant,
andneedstobreakdownfortemperaturesoutofthatregion.
independentofsystemsize.
One question that remains open is whether the super-
exponentialbehaviourtheyobserveisanartefactoftheircon-
struction, e.g., of the decoder or a physical property of the
B. Implicationforentropyprotection model. Indeed, one could imagine that the introduction of
defectlines doeschangethe thermalbehaviourof the model
In[9],authorsinvestigatethequantummemorytimeofan oversometemperatureregion. Thesuper-exponentialscaling
Abelian quantum double with d = 5 with defect lines. By couldthenbeunderstoodasanentropicenhancement. While
tuning the masses of anyons, they obtain a thermaldynamic thisenhancementdoesnottranslateintoaqualitativelydiffer-
inwhichthetypicalworldlinesofanyonshaveafractalstruc- entscalingatlowtemperature,itcouldintroduceamultiplica-
ture.Indeed,heavyparticles,ratherthanpropagate,will(with tivegaininsidetheexponentialscaling.
highprobability)decayintotwolightparticlespropagatingin- Ourbound,however,ismoregeneralthantoonlyexclude
dependently;while lightparticleswill eventuallycross a de- thepossibilityofentropicprotectionforthespecificconstruc-
fectline, becomea heavyparticle(atan energycost), which tion of Brown et al. The result presented in the present
then decays into two light particles. Brown et al. numeri- paper means that entropic protection doesn’t exist for any
callyobserveasuper-exponentialscalingofthememorytime Abelianquantumdoubles(withorwithoutpermutingtypede-
which they explain to be the result of this fractal structure fectlines);inordertohaveaself-correctingmemorybasedon
of the world lines of excitations. It is called ”entropic pro- suchmodels,oneneedsascalingenergybarrier.However,we
tection” as the world lines only have a fractal structure and should remark that a scaling energy barrier does not always
thusthere’sascalingenergybarrierforatypicalworldineof ensure self-correction, as seen in the example of the welded
anyons. There are, in fact, world lines taking the system to code[17]whichisexpectedtohaveamemorytimewhichis
an orthogonalgroundstate with onlya constantenergycost, independentofsystemsize[15].