Can long-range interactions stabilize quantum memory at nonzero temperature?

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by  Olivier Landon-Cardinal

Can long-range interactions stabilize quantum memory at nonzero temperature?
Olivier Landon-Cardinal, Beni Yoshida, and John Preskill
Institute for Quantum Information and Matter and Walter Burke Institute for Theoretical Physics,
California Institute of Technology, Pasadena, California 91125, USA
David Poulin
D´epartement de Physique, Universit´e de Sherbrooke, Qu´ebec, Canada
(Dated: March 11, 2015)
A two-dimensional topologically ordered quantum memory is well protected against error if the
energy gap is large compared to the temperature, but this protection does not improve as the
systemsizeincreases. Wereviewandcritiquesomerecentproposalsforimprovingthememorytime
by introducing long-range interactions among anyons, noting that instability with respect to small
5 localperturbationsoftheHamiltonianisagenericproblemforsuchproposals. Wealsodiscusssome
1 broader issues regarding the prospects for scalable quantum memory in two-dimensional systems.
0
2
PACSnumbers: 03.67.Pp,03.67.Lx

r
a
M I. INTRODUCTION knownmodelsofself-correctingquantummemoryrequire
at least four spatial dimensions [1, 25]. (A recently pro-

0 posedmodelbasedonalow-distortionthree-dimensional
Protecting quantum information from damage will be
1
embedding of a four-dimensional code [26] has not been
essential for future quantum technologies. Thus, devis-

shown to be perturbatively stable.) For a recent review
] ing a self-correcting quantum memory, capable of stor-
h of quantum memories at nonzero temperature, see [27].
ing qubits for a very long time without active error cor-
p rection [1–4], would be a scientific milestone with far- Tosuppresserrorsarisingfromthepropagationofther-
-
t reachingimplicationsforthescalabilityofquantumcom- mally produced anyons, Hamma et al. proposed cou-
n
puting and the security of quantum communication pro- pling the toric code to massless bosonic fields, inducing
a
u tocols. long-rangeattractiveinteractionsamongtheanyons[28];
q We define a self-correcting quantum memory as a however, their scheme requires the coupling strength to
[ quantum many-body system with a local Hamiltonian diverge with the system size. Pedrocchi et al. pro-

such that (i) the ground state is degenerate in the limit posed an alternative scheme, in which coupling anyons
2
v ofalargesystem,(ii)atasufficientlysmallnonzerotem- tothree-dimensionalmasslessbosonsinducesaninfrared
2 perature, quantum information stored in the degenerate divergent renormalization of the anyon chemical poten-
1 ground state is protected from error for a time which tial, forcing the rate for thermal anyon production to
1 grows without bound as the system size increases, (iii) approach zero as the system size grows [29–31]. Another
4 both of these features are stable with respect to generic suggestionisthatdisorderinatwo-dimensionaltopolog-
0 small local static perturbations of the Hamiltonian. ical medium could localize the anyons and hence extend
.
1 Foraquantummemorytobeuseful,onewillalsoneed the memory time [32, 33].
0 apracticalmethodforstoringaquantumstateinmemory Inthispaper,weemphasizeageneralissueconfronting
5
and reading it out, but we do not impose that criterion schemesforstabilizingatwo-dimensionalquantummem-
1
inourdefinition. Rather,weregard(ii)assatisfiedifitis ory by coupling to massless bosons — the criterion (iii)
:
v possible in principle to recover the stored quantum state typically fails because generic small local perturbations
Xi fromathermallyfluctuatingstate. Topologicallyordered introduce a mass gap for the bosons and hence compro-
systemshaveaperturbativelystableground-statedegen- mise the scheme. We consider the model of Pedrocchi
r
a eracy[5,6],andwouldmeetallthreeofourcriteriaifthe et al. [31] as an instructive example, but our arguments
topologicalorderpersistsatnonzerotemperature[1,4,7– applymoregenerally,andwealsoassesssomealternative
9]. proposals for enhancing memory times. We offer no rig-
Knownmathematicalmodelsoftwo-dimensionaltopo- orousargumentsexcludingself-correctingmemoryintwo
logically ordered media (such as the toric code [10]) typ- or three dimensions, but we hope our discussion helps to
ically do not satisfy (ii) [11–14]. The rate per unit area clarify the question. Nor do we mean to deny the possi-
forthermalproductionofanyonsissuppressedbytheen- bility that clever engineering of interactions or disorder
ergy gap, but does not depend on the system size; once inanyonicsystemscouldsubstantiallyenhancethemem-
produced,thermalanyonscanpropagatewithoutfurther ories times of anyonic systems; our focus is on whether
energypenalty,quicklydestroyingthequantuminforma- arbitrarilylongmemorytimesareachievableasamatter
tion stored in the ground state [3, 4, 15–19]. Three- of principle.
dimensional models have been proposed in which the This paper is organized as follows. In Section II, we
quantum memory time is enhanced because the prop- discusshowHamiltonianperturbationsandthermalfluc-
agation of excitations is impeded [20–24], but so far the tuationscancorrupttheinformationstoredinaquantum
2
memory, and in Section III we recall some proposals for affected by thermal fluctuations, and computing the fi-
enhancing the storage time. In Section IV, we discuss delityofasuitablychosendecodingmap. Sincethispro-
the idea of stabilizing a two-dimensional topologically gram is quite difficult to carry out, and in any case the
ordered system at nonzero temperature by introducing results depend on the details of the environment and de-
long-rangeinteractionsbetweenanyons,usingthe“toric- codingprocedure,weusuallyassess(ii)usingsimplercri-
boson model” introduced in [31] as an explicit example, teria.
and in particular we explain how to recover the conclu-
sions of [31] using an effective field theory description of
the system. In Section V and VI, we argue that the in- 1. Energy and free energy barriers
finite energy barrier of the toric-boson model, and hence
itsthermalstability, areunstablewithrespecttogeneric
One way to simplify the story is to imagine that the
perturbations of the system’s microscopic Hamiltonian.
environment applies a sequence of local quantum opera-
Section VII contains some concluding remarks and open
tions to the quantum memory. After many such steps,
questions regarding self-correcting quantum memory.
we envision cooling the memory, so that it relaxes back
totheground-statespace,andthenwecomparethisfinal
statetotheinitialstatedepositedinthememory. Wesay
II. LOGICAL ERRORS IN A QUANTUM
theerrorsequenceis“harmful”ifthefinalstatedeviates
MEMORY
significantly from the initial state.
Each error sequence has an energy height, the max-
Toclarifytheconceptofself-correctingquantummem-
imal value of the expectation value of the Hamiltonian
ory, we recall in this Section some ways for a quantum
attained during the sequence, and we say that the mem-
memory to fail, either due to imperfect control of the
ory’s energy barrier is the lowest energy height for any
Hamiltonian or due to thermal fluctuations.
harmful error sequence. Naively, a high energy barrier
means a long memory time, because only rare thermal
fluctuations will be able to surmount the barrier and
A. Spectral instability
damage the stored quantum state.
An Ising ferromagnetic in two or more spatial dimen-
Desideratum (iii) of our definition of a self-correcting sions is a thermally stable classical memory storing one
memory requires the degenerate ground space to be bitofinformationinthesignofitsmagnetization;forthe
stable with respect to small local perturbations of the magnetizationtoflip,thermalfluctuationsmustproduce
Hamiltonian. If adding a small local perturbation can a large droplet of flipped spins with size comparable to
liftthedegeneracy,thentheenvironmentmightcoupleto the system size, surmounting an energy barrier scaling
thislocaloperator,producingfluctuationsintherelative like the droplet’s surface area. The divergent memory
phases of energy eigenstates, hence driving decoherence time, though, is not perturbatively stable. A small mag-
in the energy-eigenstate basis. This failure mechanism netic field favors one sign of the magnetization; hence
relies only on the properties of the nearly-degenerate bubbles of flipped spins, arising as thermal fluctuations
ground space, not the properties of the excitations sup- in the disfavored phase at a constant rate per unit time
ported by the system. and volume, destroy the stored classical information in a
Alternatively,thememorycouldfailbecauseaweaklo- constant time. The three-dimensional toric code, on the
cal perturbation destroys the energy gap separating the other hand, is an example of a self-correcting classical
groundspacefromtherestoftheenergy-eigenstatespec- memory, as the one bit detected by its membrane-like
trum. When the energy gap collapses, the structure of logical operator is robust against small local perturba-
the ground space may undergo a qualitative change, al- tions of the Hamiltonian.
lowing the information stored in the ground state to be As in the classical case, to ensure a long storage time
corrupted. for a quantum memory we would like the energy barrier
Weusetheterm“spectralinstability”torefertoeither todivergeasthesystemsizeincreases. However,adiver-
of these failure modes. Sufficient conditions for spectral gent energy barrier is not by itself sufficient for thermal
stability,basedontopologicalorder,wereidentifiedin[6] stability. Haah’s three-dimensional cubic code [20] has a
and then generalized in [34]. logarithmically divergent barrier and Michnicki’s welded
code [22, 23] has a polynomially divergent barrier, yet
in both cases the memory time is bounded above by a
B. Thermal relaxation constant independent of the system size at any nonzero
temperature [21, 27].
Desideratum (ii) of our definition requires the stored These examples caution us that even if harmful error
quantuminformationtobewellprotectedfromerrorata sequences are strongly Boltzmann suppressed, the sup-
sufficientlysmallnonzerotemperature. Ideally,wewould pression can be overwhelmed if the harmful sequences
assess whether (ii) is satisfied by modeling the thermal have high entropy. We should consider not just the en-
environment, simulating how the stored information is ergy cost, but rather the free energy cost of encoded er-
3
rors. In the case of the cubic code, for example, the R requires energy scaling like logR; thus thermal par-
Boltzmann suppression dominates when the system size ticle propagation is suppressed and the memory has a
is sufficiently small; hence the logarithmically growing logarithmically growing energy barrier. Nevertheless, as
energy barrier implies that the memory time τ increases noted in Sec. IIB1, the free energy cost of a logical er-
withsystemsizeLasτ ∼exp(cβlogL),wherecisacon- ror becomes independent of system size for a sufficiently
stant and β is the inverse temperature. However once L large system at a fixed temperature.
islargerthanacriticalsystemsizeL∗ ∼ec(cid:48)β,entropybe- In the three-dimensional welded code [22, 23], v > 2
comes dominant, and the memory time actually declines three-dimensionaltoriccodesubblocksaresewntogether
with increasing system size. The memory time is opti- at each weld. Anyons propagate freely within each sub-
mized for L∼L∗, yielding τ <ec˜β2 for some constant c˜; block, but for an anyon to pass through a weld from
criterion (ii) for self correction is not satisfied. onesubblocktoanother, atleastv−2additionalanyons
Conceivably, entropic effects might enhance the mem- must be created. Therefore the energy barrier is high
orytimeundersomecircumstances,perhapsallowingthe if the code has many such subblocks. However, logical
memorytobeselfcorrectingeventhoughtheenergybar- errors can arise at nonzero temperature from a process
rier is finite, as suggested in [18]. A model in which en- in which v thermal anyons propagating in distinct sub-
tropy suppresses harmful error sequences was proposed blocks meet at a weld and fuse to the vacuum [27]. This
in [35] (see also Sec. IIIB). process is entropically enhanced because the fusion can
occur at any point along the weld; the resulting logical
errorrateisindependentofsystemsizeandthereforethe
2. Anyon propagation code is not self correcting.
Typical topologically ordered systems in two dimen-
sions support anyons — pointlike quasiparticle excita- III. IMPROVING THE STORAGE TIME
tions obeying exotic statistics. If the system resides on a
torus, a logical error occurs when a pair of anyons is cre-
Having discussed how a quantum memory can fail, we
ated,andthenonememberofthepairpropagatesaround
now briefly consider some possible ways to improve the
a homologically nontrivial cycle of the torus before rean-
stability of a quantum memory and hence enhance its
nihilating with its partner. Thus the energy barrier is
storage time.
the (constant) energy cost of creating and separating a
pair of anyons.
Atnonzerotemperature,thermalfluctuationsproduce
A. Anyon localization
a nonvanishing density of anyons per unit area. We can
imagine decoding the state by bringing pairs of anyons
together to annihilate, returning the system to a ground Evenforatopologicallyorderedsystemwithaspectral
state. Starting with an initial noisy encoded state, we gap, in which small local perturbations split the ground
may allow the system to evolve for a while in contact state degeneracy by an amount which is exponentially
with a heat bath, before applying the decoding map to suppressed for a large system size, a quantum memory
the final configuration. By combining the anyon world mayfailduetosmalllocalperturbationsoftheHamilto-
lines arising from the decoding of the initial state, the nian. Theproblemisthat,althoughinformationencoded
decoding of the final state, and the thermally induced in the ground space may be well protected in principle,
anyon propagation, creation, and annihilation between iftheperturbationisunknownitmaybedifficulttopre-
the initial and final time, we obtain a cycle, a set of pare precisely the ground state of the perturbed system.
closed loops. If this cycle is homologically nontrivial, a For example, if the ideal system has the toric code
logical (encoded) error occurs. Hamiltonian, we might write to the memory by prepar-
When the anyons propagate a distance comparable to ingoneofthetoriccodegroundstates. ButiftheHamil-
the typical (constant) distance between particles in the tonian is slightly deformed by a weak perturbation, the
thermalanyongas,alogicalerrorwilloccurwithnonneg- ground state of the ideal Hamiltonian is a coherent su-
ligibleprobability. Therefore, wedonotnormallyexpect perposition of excited states of the perturbed Hamilto-
a two-dimensional quantum memory to be self correct- nian; this superposition dephases quickly, destroying the
ing, unless supplemented by an additional mechanism to encoded information in a constant time independent of
impede the propagation or creation of anyons. systemsize[36]. Iftheperturbationwereexactlyknown,
Thethree-dimensionalversionofthetoriccodealsohas theninprinciplewecoulddecodethestatebyrunningthe
pointlike “anyon” excitations, and a logical error occurs Hamiltonian evolution backward in time. But if (more
when an anyon propagates across the system. Therefore realistically) the perturbation is unknown, then an un-
this code, too, has a constant energy barrier and is not correctable logical error is likely to occur. This error
self correcting. The cubic code [20] has pointlike exci- mechanism operates even at zero temperature; the er-
tations, but reaching a configuration in which one par- ror arises not from thermal fluctuations but rather from
ticle is separated from all others by a distance at least imperfect control of the system’s Hamiltonian.
4
Onewaytoevadethisproblemistorespondtothede- code,subjecttoamasterequationwhosejumpoperators
formationoftheHamiltonianbyadjustingtheprocedure arePaulioperators;inthatcasethejumpstransformen-
forencodinganddecodingthestate. Underagenericper- ergy eigenstates to energy eigenstates, so that the con-
turbation, the static pointlike anyons of the exact toric vergence to a fixed point can be analyzed using classical
code become propagating dressed anyons with nonzero methods [14].
size. Using a physical device that detects these dressed
anyons, we can prepare a ground state of the perturbed
system,whichwillstorequantuminformationreliablyat B. Entropically suppressed propagation
zero temperature. In principle we could read out the en-
codedstateatalatertimeusingdressedstringoperators. Brownetal. describedananyonmodelinwhichanyon
Alternatively, these zero-temperature errors can be propagation is impeded by entropic considerations [35].
suppressed by introducing disorder in the system Hamil- In their model, a two-dimensional topologically ordered
tonian, ensuring that in low-lying excited states the mediumisdividedintodomainsseparatedbydefectlines.
anyons are spatially localized rather than freely prop- Each domain supports both heavy and light anyons,
agating [33]. In fact, localization may suffice to make where the decay of a heavy anyon into two light anyons
the memory time exponentially large in the system size, is allowed by both kinematics and the fusion rules. Fur-
whentheerrorsareduetounknownHamiltonianpertur- thermore, upon crossing a defect line, a heavy anyons is
bations [32, 37]. However, at any nonzero temperature, transformed into a light one, and a light one is trans-
single-particle localization does not prevent anyons from formed into a heavy one.
diffusing at a nonzero rate due to thermally activated Propagation of light anyons across defect lines is sup-
hoppingthroughthedisorderedsystem. Thusatnonzero pressed at low temperature, because an energy boost is
temperature,thoughlocalizationmayenhancethemem- needed to cross to the other side. Furthermore, it is en-
ory time by a constant factor, we still expect the logical tropically favored for heavy anyons to decay to pairs of
errors to occur after a constant time independent of sys- light ones. Therefore, anyons cannot easily travel across
tem size — the memory is not self correcting. the system and the memory time is correspondingly en-
hanced. Numerical evidence suggests that if the system
We note that a disordered topologically ordered sys-
sizeisproperlyoptimized,thememorytimeτ scaleswith
temmayexhibitmany-body localization,inwhichanyons
are spatially localized even in highly excited states of the inverse temperature β as τ ∼ ecβ2, just as in the
the disordered Hamiltonian [38, 39]. However, many- three-dimensional cubic code, even though the energy
body localization is a property of a perfectly isolated barrier is constant rather than logarithmically growing.
Hamiltonian system. Even a many-body-localized two- This is a notable improvement compared to the scaling
dimensional topological phase would fail to be self cor- τ ∼ecβ inthetoriccode. Thoughnotselfcorrecting,this
recting, because anyons would still be able to diffuse system illustrates how in some cases favorable entropic
when the system is in contact with a thermal bath. effects can significantly enhance the memory time.
Forthecaseofatwo-dimensionalcommutingHamilto-
nianlikethedisorderedtoriccodemodel,thefindingthat
C. Long-range anyon interactions
topologicalorderisdestroyedatnonzerotemperaturecan
be formulated rigorously. Hastings [7] showed that the
Gibbs state of a (not necessarily translation-invariant) Since logical errors in a two-dimensional topologically
2DcommutinglocalHamiltoniancanbe(approximately) orderedquantummemoryarisefromanyonpropagation,
transformedtoaproductstatebyaconstant-depthlocal any mechanism that impedes anyon propagation can in
quantum circuit. To analyze the memory time, we can principleenhancethememorytime. Asidefromlocaliza-
draw guidance from studies of the disordered 1D Ising tion due to disorder as discussed above, there is another
model, whose stability properties are similar to those of mechanism that could limit how far anyons can travel
the 2D toric code. For a one-dimensional classical spin — attractive interactions between anyons. The interac-
system with Hamiltonian tions may impose a hefty energy penalty on harmful er-
ror sequences, encouraging confinement of the anyons to
(cid:88) small dilute clusters which do not produce logical errors.
H =− J Z Z (1)
Ising i i i+1
For the system to be truly self correcting, though, these
i
interactions must have infinite range. Otherwise only a
where the nearest-neighbor couplings {J } are positive constantamountofenergywouldbeneededforananyon
i
but otherwise arbitrary, the memory time τ is indepen- tobreakawayfromacluster,leavingitfreetopropagate
dent of system size, scaling like τ ∼ exp(2βJ¯) where unmolested by attractive forces applied by other anyons,
J¯≡ (J +J )/2 is the average of the minimal and thusallowinglogicalerrorstooccurataconstantrateat
min max
maximal values of the coupling [40]. Thus the mem- any nonzero temperature.
ory time is not significantly enhanced in the regime For a system with a local Hamiltonian, long-range
J (cid:29) J where defects are strongly localized. The forces arise from the exchange of gapless excitations.
max min
conclusionτ =O(1)alsoappliestothe(disordered)toric Hamma et al. [28] proposed a model in which the anyon
5
density provides a source for a massless scalar field. The energy cost of creating an anyon pair is 4J, and
Scalar exchange produces long-range attractive interac- theanyonsarenoninteracting. Alogicalerroroccursifa
tionsbetweenanyons,buteachanyonalsohasalogarith- pair is created and one anyon propagates around a non-
micallyinfrareddivergentnegativeselfenergy, rendering trivialcycleofthetorusbeforetheanyonsreannhihilate.
the system unstable. The divergent self-energy can be Therefore the (constant) energy barrier is 4J. The typ-
cancelledbyintroducingotherfieldswithcarefullytuned ical separation between particles in the thermal anyon
couplings,butthentheHamiltoniancontainstermswith gas, and hence also the memory time, scale like the in-
infrared divergent coefficients. verse Boltzmann factor e2βJ.
Improving on [28], Pedrocchi et al. proposed an al-
ternative model with a local Hamiltonian and finite cou-
plings,inwhichtheanyonicchemicalpotentialisinfrared
divergent [31]. Thus the energy barrier, the cost of
creating a distantly separated anyon pair, diverges with B. Coupling the toric code to massless bosons
system size, as does the memory time. We review this
scheme in Sec. IV. It works, if the Hamiltonian is care-
Now, following Pedrocchi et al. [31], we consider cou-
fully tuned. But as we explain in Sec. V it does not
pling the two-dimensional toric code to an auxiliary
meet our criteria for self correction, because the diver-
three-dimensional bosonic system which induces long-
gent energy barrier is unstable with respect to generic
range attractive forces among the plaquette variables.
small perturbations of the local Hamiltonian.
The Hamiltonian proposed by [31] is
We also note that schemes for “active” rather than
“passive” error correction have been proposed, inspired (cid:88)
H =−A W (a +a†)+H ,
by the idea that long-range attractive interactions might r r r boson
r
impede the propagation of diffusing anyons [41]. In a
(cid:88)
scheme recently discussed in [42], the anyon motion is Hboson = (cid:15)qa†qaq, (3)
guidedbyaclassicalfield,whichisregularlyupdatedus- q
ing a local evolution rule as the anyons propagate. Such 1 (cid:88)
a = √ a eiqs.
active error correction schemes, though effective and in- q N s
s
teresting, do not meet our criteria for self correction for
several reasons. For one thing, to achieve an arbitrar- Here the index r labels the sites of the two-dimensional
ily low logical error rate, the scheme requires a “speed topologically ordered medium and s labels the N sites
of light” (rate of information propagation in the classi- of a three-dimensional lattice which contains the two-
cal field) which diverges relative to the “speed of sound” dimensional lattice; a and a† are the annihilation and
r r
(rate of anyon motion), a requirement not satisfied by a creationoperatorsforbosonicmodesresidingattheloca-
systemwithalocalHamiltonianandboundedcouplings. tion of plaquette operator W , and q labels the discrete
r
Furthermore, the dissipative dynamics of the field which Fourier modes for a bosonic field defined on the three-
controls the anyons may not be realizable by any Hamil- dimensionallattice. Weassume(cid:15) >0,becauseabosonic
q
tonian system in contact with a thermal bath. mode with negative (cid:15) would be unstable. The precise
q
behavior of (cid:15) for large q does not matter for analyzing
q
the long-distance physics of the model, but the behav-
IV. TORIC CODE COUPLED TO A SCALAR ior of (cid:15)q for small q does matter. We assume a gapless
FIELD spectrum for the Fourier modes with (cid:15)q ∼q2 for small q.
We will call the Hamiltonian in Eq. (3) the “toric-boson
model.”
A. Toric code
Couplinganyonstothethree-dimensionalmasslessbo-
To prepare for our discussion of models with long- son field produces an attractive force between anyons,
range anyon interactions, let’s first recall Wen’s formula- with potential energy inversely proportional to separa-
tion [43] of Kitaev’s toric code model [10]. Consider an tion. After a polaron transformation (i.e., completing
L×L square lattice with one qubit per site, where L is the square) the Hamiltonian may be rewritten as [31]
an even integer. The Hamiltonian is
(cid:88) (cid:88)
H =− J W W + (cid:15) a˜†a˜ , (4)
r,r(cid:48) r r(cid:48) q q q
(cid:88)
H =−J W , W =X Z X Z , (2) r,r(cid:48) q
Toric r r r,1 r,2 r,3 r,4
r
where
where r labels lattice plaquettes and W acts on the
four qubits contained in plaquette r (X rand Z denote J = A2 (cid:88) 1eiq(r−r(cid:48)) ∼ A2 ; (5)
Pauli matrices). On the torus (i.e. a square with peri- r,r(cid:48) N (cid:15)q |r−r(cid:48)|
q
odicboundaryconditions),themodelhasfourdegenerate
ground states which can encode two qubits. here N is the number of lattice sites in the three-
6
dimensional bosonic system, means that the spontaneous breakdown of the Z global
2
symmetry, and hence the self-correcting behavior, hold
a˜ =a − √A (cid:88)W eiqr, (6) at arbitrarily high temperature.
q q r
(cid:15) N
q r
andtheconclusionJ ∼A2/|r−r(cid:48)|appliesforasymp-
r,r,(cid:48)
toticallylargeseparation. Notethata˜ anda˜† bothobey C. Derivation using effective field theory
q q
ordinarybosoniccommutationrelationsbecausetheW ’s
r
residing at distinct plaquettes are commuting operators. Because the conclusion µ = O(L) follows from only
The long-range behavior of the potential energy, scaling the long-distance properties of the Hamiltonian Eq. (4),
like∼1/RwhereRistheseparation,arisesfromthepole itcanberecoveredusinganeffectivefieldtheorydescrip-
in1/(cid:15) ,whichdominatestheFourierintegralforlargeR. tion which captures the infrared behavior of the model.
q
The effective Hamiltonian becomes
Because the Hamiltonian in Eq. (4) is the sum of an
anyonic part H and bosonic part H , which are com- (cid:90) (cid:90)
pletely decoupleAd, the energy eigenstaBtes can be written HEFT = 12 d3x (∇φ)2−A d2yw(y)φ(y)+··· , (8)
as |ψ(cid:105) = |ψ (cid:105)⊗|φ (cid:105), where |ψ (cid:105) is an eigen-
qubit boson qubit
state of HA and |φboson(cid:105) is an eigenstate of HB. Since where φ(x) is a suitably normalized scalar field. The
Jr,r(cid:48) > 0, the interactions between plaquette variables 1(∇φ)2 term in the Hamiltonian density arises from the
in Eq. (4) are “ferromagnetic,” favoring alignment. HA d2ispersionrelation(cid:15) ∼q2forsmallq;higher-orderterms
hasaZ globalsymmetry,underwhichW changessign, q
2 r inq correspondtotermswithmorederivativesinH ,
EFT
which is spontaneously broken at zero temperature —
which are not shown because they produce only small
there is one ground state sector with W = 1 for all r
r correctionsforlong-wavelengthmodesofthescalarfield.
and another sector with W = −1 for all r, where both
r Thesourcew(y)describestheanyonconfiguration. Tobe
sectors have a topological degeneracy as well. The spon-
compatiblewiththeconventionsofSec.IVB,weassume
taneous breakdown of the global Z symmetry persists
2 w(y) takes the positive value w >0 in the ground state
0
at sufficiently small nonzero temperature, and we may
of the two-dimensional topological medium, and has the
assume without loss of generality that the thermal ex-
valuew(y)=w −2n(y)whenanyonsarepresent,where
0
pectation value (cid:104)W (cid:105) is positive .
r n(y) is a suitably smoothed and normalized anyon den-
As emphasized in [31], the crucial property of Eq. (4) sity. We assume w(y) = 0 outside the two-dimensional
isthattheenergycostofcreatingasingleisolatedanyon medium.
is infrared divergent, i.e., increases with the system size. The energy is minimized by a solution to the static
To compute this energy cost, we follow [31] and suppose classical field equation, which is Poisson’s equation
that the bosonic bath is at any moment in a Gibbs state
withrespecttotheeffectiveHamiltonian(cid:104)α|H|α(cid:105),where ∇2φ(x)=−Aw(x). (9)
|α(cid:105) is the instantaneous anyonic state, i.e., an eigenstate
of all operators W . In that case, this energy cost of cre- The solution is
r
atingasingleisolatedanyononlydependsontheanyonic (cid:90) (cid:16) (cid:17)
part H of the Hamiltonian in Eq. (4) (see [31] for the φ(x)=A d2y 1 w(y)+φ (x) (10)
A −∇2 0
detailedcalculation). Thedivergencearisesbecauseeach x,y
plaquette interacts with many other distantly separated
where φ (x) is a solution to the homogeneous field equa-
0
plaquettes on the two-dimensional lattice; flipping the tion, and the Green function 1 is determined by im-
sign of one W relative to all the others increases the −∇2
r posing appropriate boundary conditions. We assume
potential energy by an amount
that the two-dimensional anyonic source for the three-
(cid:88) dimensionalscalarfieldφhasboundedsupport,andthat
µ∼ J ∼O(L). (7)
r,r(cid:48) φ vanishes at spatial infinity.
r(cid:48) The boundary condition justifies an integration by
parts with no surface term, and using the field equation
This effective chemical potential µ is O(L) because we
we may obtain an alternative expression for the minimal
sumthe1/RinteractionenergyovertheL×Llattice. In
energy of the field configuration in the presence of an
contrast to the standard toric code, where the energy of
anyon source:
anisolatedanyonis2J andthethermaldensityofanyons
at inverse temperature β scales like e2βJ, in the model (cid:90) (cid:90)
Eq. (4) the thermal density scales like ecβL (where c is a E[w] = − d3x1φ∇2φ−A d2yw(y)φ(y)
2
constant), and correspondingly the memory time grows
(cid:90) (cid:90)
exponentially with L at any nonzero temperature. The = A d3xw(x)φ(x)−A d2yw(y)φ(y)
infrared divergent chemical potential and the resulting 2
self-correcting behavior of Eq. (4) were the main con- (cid:90)
= −A d2xw(x)φ(x) (11)
clusions of [31]. In fact, the infinite chemical potential 2
7
with the integral now supported only on the two- The bosonic ground state |φ (cid:105) of Eq. (4) satisfies
boson
dimensional surface where the source field w is nonzero.
Inserting Eq. (10) into this expression, one finds a˜q|φboson(cid:105)=0 (19)
(cid:90) (cid:16) (cid:17) for all q, which implies
E[w]=−A2 d2xd2yw(x) 1 w(y). (12)
2 −∇2
x,y
(cid:104)a (cid:105) ∼ A (cid:88) 1 (cid:88)eiq(r(cid:48)−r) =O(L) (20)
The boundary condition determines the Green function r 0 N (cid:15)
q
q r(cid:48)
to be
where (cid:104)a (cid:105) denotes the ground-state expectation value,
(cid:16) (cid:17) 1 r 0
1 = , (13) and we have substituted W = 1. The bosonic mode
−∇2 x,y 4π|x−y| r(cid:48)
a has an unbounded occupation number in the anyon
r
groundstate,whichiswhythecostofcreatinganisolated
and so we obtain
anyon at site r is divergent.
(cid:90)
E[w]=−1 d2xd2yw(x)J(x,y)w(y), (14) Additional terms in the bosonic Hamiltonian, which
2
aregenericallypresent,willpreventtheoccupationnum-
where ber from diverging. Suppose, for example, the Hamilto-
nian contains a term
A2
J(x,y)= , (15) (cid:88) (cid:88)
4π|x−y| V =(cid:15) nr =(cid:15) a†qaq, (21)
r q
as in Eq. (5).
wheren =a†a isthenumberoperatorforthemodelo-
To compute the effective anyon chemical potential, we r r r
calizedatsiter. Wemayassume(cid:15)>0becauseotherwise
write w(y) = w −2n(y), finding that E[w] contains a
0
thebosonicfieldwouldbeunstable,anindicationthatwe
term linear in the anyon density
are expanding around the wrong bosonic vacuum. Now
(cid:90) the perturbation V, which disfavors a large occupation
d2xµ(x)n(x) (16)
number,competeswiththeHamiltonianEq.(4),andthe
ground-stateexpectationvalueofa takesthefinitevalue
r
where
(cid:90) (cid:104)a (cid:105) ∼ A (cid:88) 1 (cid:88)eiq(r(cid:48)−r) =O((cid:15)−1/2), (22)
µ(x)=2w0 d2yJ(x,y). (17) r 0 N q (cid:15)+(cid:15)q r(cid:48)
because the perturbation removes the pole at q =0.
Assumingforsimplicitythatxisthecenterofadiskwith
radius L/2, we obtain For this particular perturbation the bosonic Hamilto-
nian is quadratic and therefore exactly solvable, but the
(cid:90) 2π (cid:90) L/2
µ = 2w0A2 dθ rdr4π1r = w02A2L, (18) cfroanrceldusdioivnertgheantceahgoelndesrimcopreertguernbeartailolyn. rIenmtohveesefftheectiivne-
0 0
fieldtheorylanguage,thesolutiontothestaticfieldequa-
recovering the linear divergence found in Eq. (7). tion Eq.(10) becomes
(cid:16) (cid:17)
φ(x)=A(cid:82) d2y 1 w(y)
−∇2
x,y
=Aw (cid:82) d2y 1 =O(L) (23)
V. PERTURBATIVE INSTABILITY OF 0 4π|x−y|
MASSLESS BOSONS
in the anyonic vacuum such that w(y) = w in the two-
0
dimensionaltopologicalmediumandw(y)=0elsewhere.
Having seen that the model in Eq. (3) has an infrared
The infrared divergent behavior of φ(x) arises because
divergent energy barrier, and hence an exponentially
the scalar field is exactly massless in Eq. (8). Gener-
increasing quantum storage time at sufficiently small
ically a nonzero mass term will be present, perturbing
nonzero temperature, we next consider the stability of
the Hamiltonian according to
the energy barrier with respect to small perturbations of
the system’s local Hamiltonian. For assessing perturba- (cid:90)
H →H + d3x 1m2φ2. (24)
tive stability, it is important to note that the divergent EFT EFT 2
anyon chemical potential arises because the response of
thescalarfieldtotheanyonvacuumisinfrareddivergent. We may assume m2 > 0, because otherwise the scalar
A small perturbation of the Hamiltonian can tame this fieldwouldbeunstable,indicatingthatweareexpanding
divergence, rendering the energy barrier finite. around the wrong bosonic vacuum. The static vacuum
8
solution becomes ceptions to this rule, and we should consider whether
theseexceptionscanbe exploitedbyquantumengineers.
(cid:90) (cid:16) (cid:17)
φ(x)=Aw d2y 1 =O(m−1); (25) In particular, the masslessness of Goldstone bosons and
0 −∇2+m2
x,y gauge bosons follows from general symmetry principles.
now the y integral converges because the Green function
decays like e−m|x−y| for large separation.
A. Goldstone bosons
The effective field theory approach is useful for ana-
lyzing the energy barrier of a quantum memory because
whether the energy barrier diverges or not hinges on the Aspontaneouslybrokenexactcontinuoussymmetryis
behavior of the model in the far infrared, where the field alwaysassociatedwithanexactlymasslessGoldstonebo-
theory approximation should be reliable. The degrees son,whosemasslessnessispreservedbyanyperturbation
of freedom described by the field theory may be emer- thatrespectsthesymmetry(andalsomaintainsitsspon-
gent, not corresponding in any very direct with the mi- taneous breakdown). If φ(x) is a Goldstone field, then
croscopic degrees of freedom of the underlying Hamil- the symmetry acts according to
tonian model. But just as the divergent energy barrier
φ(x)(cid:55)→φ(x)+constant, (28)
in the model Eq. (8) is realized only when the free pa-
rameter m2 of the effective field theory is tuned to zero,
disallowing a mass term 1m2φ2 in the effective Hamilto-
we may anticipate that a divergent energy barrier arises 2
nian.
fromtheexchangeofmasslessbosonsonlyon(atbest)a
The symmetry not only forbids a mass term for the
codimension-one surface in the space of all Hamiltonian
Goldstone boson, but also constrains its coupling to
models. A generic perturbation moves the Hamiltonian
other fields. For example, a term −Aφ(x)w(x) describ-
offthissurface,destroyingtheinfiniteenergybarrier. To
ing the coupling of the Goldstone field to the anyonic
evade this conclusion, we need models in which bosons
density w(x) is not allowed, but a derivative coupling
with the appropriate couplings remain gapless when the
−A∇φ(x)w(x)iscompatiblewiththesymmetry,andcan
microscopic Hamiltonian is deformed.
bereexpressedasAφ(x)∇w(x)afterintegratingbyparts.
Not just the infinite energy barrier but also the topo-
(This term violates rotational invariance, but could be
logical order of the two-dimensional medium might be
allowed in an anisotropic system.) Then, arguing as in
threatened by small perturbations of the Hamiltonian.
Sec. IVC, we find in place of Eq. (12) that the energy
In addition to coupling to plaquette variables, the scalar
of the bosonic ground state in the presence of a source
field could couple to individual qubits; for example the
becomes (in D spatial dimensions)
perturbed Hamiltonian might contain terms such as
(cid:16) (cid:17)
V =(cid:15)(cid:48)(cid:88)(ar+a†r)(Xr,1+Xr,2+Xr,3+Xr,4), (26) E[w]=−A22 (cid:82) dDxdDy∇w(x) −∇12 x,y∇w(y)
r =−A2 (cid:82) dDxw(x)2, (29)
2
If in the ground state the expectation value of the field
scales like (cid:104)a (cid:105) ∼ m−1 ∼ (cid:15)−1/2 as in Eq. (22) and (25), the infrared-finite integral of a local source-dependent
r 0
then an effective magnetic field energy density. Because they are derivatively coupled,
the exchange of Goldstone bosons generates short-range
(cid:15)(cid:48) contact interactions between anyons rather than a long-
B ∼ √ (27)
(cid:15) range forces.
Perturbations that explicitly break the continuous
isappliedtothequbits. Althoughenhancedbythelarge symmetry would gap out the Goldstone bosons in any
strength of the field, this magnetic field is still pe√rtur- case. But even if we are willing to impose a symmetry
batively small compared to the energy cost ∼ 1/ (cid:15) of thatprotectsthemasslessGoldstonebosons,therewould
creating an isolated anyon, so it is plausible that the be no resulting divergent energy barrier to suppress log-
topological order would survive. In any case, we expect ical errors.
the topological order to persist within a region of finite
volume in the space of Hamiltonians.
B. Gauge bosons
VI. STABILIZING MASSLESS BOSONS In some cases, exactly massless deconfined U(1) gauge
bosons can emerge from an underlying spin model with
The scheme of Pedrocchi et al. [31] for stabilizing a no fundamental gauge symmetry. From an effective field
quantum memory requires massless bosonic fields, and theory viewpoint, since the gauge symmetry is not an
we have argued that under generic conditions bosonic exactfeatureoftheunderlyingmicroscopicHamiltonian,
fields acquire mass unless the Hamiltonian is tuned to nothing seems to forbid adding a gauge non-invariant
a codimension-one surface. But there are important ex- term to the Hamiltonian of the long-distance effective
9
field theory, which would gap out the gauge bosons. But stability of the long-range interactions, we may consider
if the microscopic local Hamiltonian energetically favors the consequences of long-range forces between “anyons”
a sector of the theory which satisfies gauge constraints, in D dimensions described by Eq. (4), where
these constraints could be robust against generic small
1
perturbations of the microscopic Hamiltonian [44]. A
J ∼ . (30)
modelwithsuchfeaturescansupportastable“Coulomb r,r(cid:48) |r−r(cid:48)|α
phase,”wherea“photon”remainsexactlymasslesswith-
For 0 < α < D, the cost of creating an isolated anyonic
out any need to impose symmetries or carefully tune the
excitationdivergeswiththesystemsize,sothemodelhas
Hamiltonian.
a quantum memory time which increases rapidly with
Emergent photons can mediate long-range forces be-
system size, even at arbitrarily high temperature. The
tween charged particles, which are attractive for parti-
precisescalingofthelifetimeofthememorywithsystem
cles of opposite charge, and repulsive for particles of like
size is discussed in [45]. For α>D, energy cost remains
charge. Chesi et al. suggested that long-range repulsive
finite, and the memory time is bounded above by a con-
interactionsamonganyonsinatwo-dimensionalmedium
stant at any nonzero temperature. The case α = D is
couldproduceadivergentfreeenergybarrier,becausefor
marginal, with a logarithmically divergent energy bar-
a nonzero anyon density the cost of introducing an addi-
rier.
tionalisolatedanyonwoulddivergewithsystemsize[29];
By similar reasoning, the one-dimensional Ising model
thus theanyondensity shouldapproach zero inthe ther-
with 0<α<1 is a good classical memory at arbitrarily
modynamic limit.
hightemperature. Itisalsoknownthatthissystemhasa
However, anyon models in which gauge-mediated in-
finite-temperature phase transition for 1<α<2, and is
teractions are repulsive for each anyon pair do not seem
magnetically disordered at any nonzero temperature for
to be self consistent. Because the photon couples to a
α>2 [46, 47].
locally conserved charge, anyon pairs which can be lo-
cally created from the vacuum must have opposite “elec-
tric charges.” At nonzero temperature, anyons coupled
to emergent photons would form a plasma with charge VII. DISCUSSION
screening and no divergent energy barrier
It may be possible in principle for anyons to carry The main thrust of this paper is contained in Sec. IV-
dipole moments which couple to emergent photons. In VI, where we have described schemes for stabilizing a
that case, if the gauge field is three-dimensional there two-dimensional topologically ordered quantum memory
would be a long-range interaction with potential energy at nonzero temperature using long-range interactions,
fallingofwithdistanceRlike1/R3. Evenifinasuitably andhavearguedthatsuchschemeshaveasignificantand
engineeredHamiltonianthe3Dgaugefieldcouplestothe generic drawback — the Hamiltonian needs to be pre-
plaquette variables of the 2D toric code, these interac- cisely tuned for the memory time to scale favorably with
tions decay too rapidly to produce an infrared divergent thesystemsize. Ourargumentsareincomplete,andleave
energy barrier. many open questions, including the important and en-
For the 3D toric code, a logarithmically divergent ticingquestionwhetherself-correctingquantummemory
chemical potential µ ∼ logL due to dipolar interactions is achievable in a suitably engineered three-dimensional
isconceivable. It’snotclear,however,howtoarrangethe system. Thisconcludingsectioncontainsafewquestions
neededcouplingbetweenthetopologicalmediumandthe and remarks which may help to guide future research on
gauge theory; see Sec. VIIE for further discussion. self-correcting quantum memory.
C. Other long-range forces A. Quasi-topological phases
Massless scalars can also be stabilized by imposing In mathematical models of topologically ordered sys-
both supersymmetry and chiral symmetry – the chiral tems, we usually assume the system has an energy
symmetryenforcesthemasslessnessofafermionspecies, gapseparatingthetopologicallydegenerategroundstate
and the supersymmetry requires the scalar and fermion fromtherestoftheenergyspectrum. Thenwecanimag-
to be degenerate. Massless spin-two bosons (gravitons) ine integrating out all quasiparticle excitations to obtain
can be stabilized by general covariance. If we could find an effective theory with no propagating excitations, a
spin models in which supersymmetry and/or general co- topological quantum field theory (TQFT) which accu-
varianceareemergentandrobustwithrespecttogeneric rately describes the physics of the system up to correc-
local perturbations of the microscopic Hamiltonian, that tions which are exponentially small in the system size.
would be very interesting from the perspective of funda- Butstrictlyspeakingthispicturedoesnotnecessarilyap-
mental physics, apart from the potential applications to ply to the topologically ordered systems observed in the
self-correcting quantum memory! laboratory,whichmaysupportgaplessphotonorphonon
Putting aside the microscopic origin and perturbative excitations. Furthermore, disordered systems may have
10
many low-lying localized bulk quasiparticle excitations size larger than L (T), where [11, 49]
0
with energy well below the topological gap.
logL (T)≈∆/T. (31)
Bonderson and Nayak proposed the term quasi- 0
topological phase for a system that has some of the fea-
For L > L (T) the system is likely to contain propa-
tures of a topological phase, but also has gapless excita- 0
gating thermally excited anyons, causing rapid decay of
tions and therefore cannot be precisely described at low
stored quantum information. But just as the quantized
energy by a TQFT [48]. For example, Laughlin states
Hall conductance is a robust physical phenomenon, even
are quasi-topological; their fractionally charged quasi-
though it disappears in the thermodynamic limit at any
particles couple to the massless photon, and correspond-
nonzero temperature, so a topological quantum memory
ingly the topological degeneracy on the torus is split by
with storage time enhanced by the inverse Boltzmann
an amount that scales polynomially rather than expo-
factor e∆/T may be a valuable resource, even though it
nentially with the system size. Furthermore, the anyon
fails to meet our criteria for self correction.
braiding statistics is modified by nonuniversal dynami-
cal phases which are not described by the TQFT. Bon-
derson and Nayak also proposed the term strong quasi-
topological phase for a system in which some sector of C. Self correction and fast propagation
the theory decouples from the gapless excitations and
can be accurately described by a TQFT. For example, The Lieb-Robinson bound, which applies to any local
the Moore-Read state has a strongly quasi-topological Hamiltonian whichis a sumof finite-normterms, asserts
sector. Although the Ising anyons carry electric charges that information propagates at finite speed. This bound
which produce non-topological exchange phases, the dis- is violated by the toric-boson model Eq. (3); therefore
tinguishable fusion channels for a system of many Ising thismodelcannotarisefromanunderlyingmodelwhose
anyons span a topologically protected Hilbert space well Hamiltonian is a sum of terms, each with bounded norm
described by the Ising TQFT. and bounded range.
The toric-boson model, the toric code coupled to an When the toric-boson model on a L×L lattice is per-
exactly massless scalar field as in Eq. (4), is a strong turbedbyasuitablelocalHamiltonian,anyonscanprop-
quasi-topological phase. One might have thought that, agate at speed O(L). Consider dividing the lattice into
just as distantly separated anyons have a potential en- two domains separated by a horizontal domain wall, as
ergy scaling algebraically with their separation, so large in Fig. 1. The lower half A is in the phase where plaque-
punctures on the lattice, which we may think of as fat- ttes take the value W =+1, and the upper half B is in
r
tened anyons, should also have an algebraically decaying the phase with W = −1. At time t = 0, two anyonic
r
interaction, and corresponding the splitting of the de- excitations(eachwithW =+1)arecreateddeepwithin
r
generacy on the torus should be algebraically decaying the B phase, and at the same time magnetic fields are
as for Laughlin states. But in the exactly solvable toric- appliedtospinsonaverticallinewhichextendsfromthe
boson model, the topological degeneracy on the torus is lower anyon to the domain wall, and beyond into the A
exact. Deforming the model with a small local pertur- phase. This anyon is subjected to a long-range force at-
bation might conceivably destroy the topological order tracting it toward the A phase. Were no magnetic field
as discussed in Sec. V, but in the “weak magnetic field” applied, the configuration with an anyon pair would be
regime of Eq. (26) and (27) the perturbation may pro- an exact energy eigenstate and would not evolve. But
duce only an exponentially small splitting of the ground the applied magnetic field gives the lower anyon a finite
state degeneracy, just as for the toric code without long- effective mass, so it can respond to the attractive force
range interactions. by falling vertically. By the time it reaches the domain
wall, this falling anyon has accelerated to speed O(L).
Thus anyons can travel at unbounded speed in the limit
of infinite system size, showing that the Lieb-Robinson
B. Topological order at low temperature in a finite
bound does not apply. When the anyon reaches the do-
system
main wall, it plunges through, becoming an anyon with
W = −1 in the A phase. Forces attracting it back to-
r
Our focus throughout this paper has been on the scal- ward the B phase slow the anyon down until it comes to
ingofthememorytimewithsystemsize,andinparticu- restdeepwiththeAphase,andthenproceedstooscillate
laronwhetherarbitrarilylongmemorytimesareachiev- between positions in the A and B phases.
able as a matter of principle. But we should emphasize Since the divergent anyon chemical potential of the
that a two-dimensional topologically ordered system of toric-boson model is directly related to the long-range
fixed size could be a useful quantum memory if the tem- forces which can accelerate anyons to unbounded speed,
perature T is far below the topological gap ∆. one wonders whether this connection holds more gener-
The signatures of topological order, such as quantized ally. Generalized Lieb-Robinson bounds have been de-
Hall conductance and nontrivial topological entangle- rivedforsystemswhichhavelong-rangeinteractionsscal-
ment entropy, disappear on an L×L lattice for system ing with separation r as r−α, showing that in D dimen-