Table Of ContentSurface Code Threshold in the Presence of Correlated Errors
E. Novais1 and Eduardo R. Mucciolo2
1Centro de Ciˆencias Naturais e Humanas, Universidade Federal do ABC, Santo Andr´e, SP, Brazil
2Department of Physics, University of Central Florida, Orlando, Florida 32816, USA
(Dated: September12, 2012)
We study the fidelity of the surface code in the presence of correlated errors induced by the
couplingofphysicalqubitstoabosonicenvironment. Bymappingthetimeevolutionofthesystem
afteronequantumerrorcorrectioncycleontoastatisticalspinmodel,weshowthattheexistenceof
anerrorthresholdisrelatedtotheappearanceofanorder-disorderphasetransitioninthestatistical
modelin thethermodynamic limit. This allows ustorelate theerror threshold to bathparameters
and to thespatial range of thecorrelated errors.
2
1
0
The surface code is considered one of the best quan- is not essential to our results, but makes the calculation
2
tumerrorcorrectioncodes(QEC)toimplementonphys- more concise. In addition, we assume that the bath is
p
ical devices [1–3]. This steams from two major points: initially at zero temperature and that it is reset to this
e
S first,allsyndromesandoperationscanbeperformedwith temperature at the end of the QEC cycle.
spatially local operators; second, all threshold estimates
0 Our results show a sharp transition between two dis-
showthat,forsufficiently largelattices,the errorthresh-
1 tinct noise regimes. Onone hand, below a fictitious crit-
old is the highest known for two-dimensional architec- ical “temperature” (which is related to microscopic pa-
] tures with only nearest-neighbor interactions [4, 5].
h rameters of the bath), the dissipation due to the bath
p The error threshold is usually defined for stochastic cannot be suppressed by the encoding. On the other
- error models. By assuming that errors are independent hand, above this critical “temperature”, we show that
t
n events and assigning a probability p to each of these if the thermodynamic limit is taken, the effects of the
a events, it has been shown numerically that the quantum bosonic environment become irrelevant and the logical
u
q informationencodedin the surface canbe faithfully pro- qubit is fully protected.
[ tected when p is below a critical value. Although this Eventhoughthisinvestigationfocusesonquantumin-
resultis firmly establishednumerically,there aretwobig formation protection, the physical problem we consider
1
v open questions that still need to be addressed. First, has a much broader appeal. In essence, it amounts to
7 stochasticerrormodelsareapproximationstorealitythat a lattice gauge system interacting with a scalar bosonic
5 sometimes cannot be justified. In fact, most studies so fieldintwodimensions[13]. Inthis language,the nonex-
1 farlackedamicroscopicdescriptionoftheinteractionbe- istence of a quantum error threshold can be understood
2
tween physical qubits and the environment. Second, the as the lifting of the ground state topological degener-
.
9 same locality of operations and syndromes that makes acy. Our discussion of the error threshold can therefore
0
the surface code powerful also makes it more susceptible be recast as a quantum phase transition. This fits into
2
1 to correlated errors. Thus, a discussion of the tradeoff our earlier discussionof the thresholdtheorem as resem-
: between locality of operations and correlated errors is bling a quantum phase transition [8]. This analogy is
v
long overdue. In this Letter, we address both issues by nontrivialsince the errorthreshold is in essence a driven
i
X employing a more realistic error model. We consider a dynamical problem, far from the equilibrium conditions
r Caldeira-Leggetttype of environmentwhere freely prop- required to describe phase transitions in statistical me-
a
agating bosonic modes couple linearly and locally to the chanics. Thisworkturnswhatwasananalogyintoawell
physicalqubits. Sucha modelhasa verystrongphysical defined map. Therefore, we believe that the results pre-
motivation since in most experimental implementations sented here transcend our original motivation and com-
photons and phonons couple to the two-level systems plement the recent discussion in Ref. 12.
making up the physical qubits. This model also plays
The model – We consider physical qubits ~σ
a fundamental role in our understanding of decoherence { i}i=1,...,N
(i.e., spin 1/2 systems) located on the links of a square
[6] and its interplay with QEC [7–12].
lattice. The QEC code is defined by an encoding pre-
Consider a logical qubit in a quantum memory. In a scription and a set of stabilizer operators [14]. The sta-
QECcycle,thelogicalqubitispreparedandlefttofreely bilizer operators of the surface code are easily labeled
evolve during a certain time interval. Then, syndromes when we define stars and plaquette operators (see Fig.
are extracted and, if necessary, suitable error correction 1). Star operators are the product of the four σx op-
operations are implemented to bring the logical qubit erators of qubits adjacent to a vertex of the square lat-
backtoitsoriginalstate. Inouranalysis,weevaluatethe tice, A♦ = i σix. Similarly, plaquette operators are
fidelity of a logical qubit after such a QEC cycle under the product of∈f♦our σz of quits located at the edges of
the assumptionofnonerrorsyndromes. This assumption a tile of theQsquare lattice, B = σz. Each pla-
(cid:3) i (cid:3) i
∈
Q
2
(a) (b) Thus, we can introduce a fictitious inverse tempera-
ture β = 1 λω0 2 and rewrite the intermediate factor
2 v
that contains the two-spin interaction in Eq. (4) as
A e−βPr6=sJrsσ(cid:0)rxσsx,(cid:1)where Jrs represents an effective anti-
ferromagnetic coupling.
To simplify the notation, we assume that the system
ispreparedinitiallyinthe logicalstate ¯ andthe boson
B X field initial state is the vacuum, |↑i
Z
ψ =(GF ) 0 . (6)
0 z
| i | i ⊗| i
FIG. 1. (a) Star (A) and plaquette (B) operators. (b) Ex- We then let the system evolve under the Hamiltonian
amplesoflogical X¯ andZ¯ operators. Thecrossesandshades H until a time ∆, when an error correction protocol is
indicate thequbitsinvolved in therespective operator.
performed flawlessly.
The assumptions – Since we are only allowing for bit-
quette and star is a stabilizer that has to be measured flip errors,the syndrome outcome for the star stabilizers
at the end of a QEC period. The logical operators are is trivial. For the plaquettes, in principle, all possible
defined as X¯ = σx, where Γ is a path along the syndromes should be considered. However, it is useful
i Γ i
center of the plaquet∈tes crossing horizontally the lattice, to look at the most benign evolution and assume that
and Z¯ = σzQ, where Γ¯ is a path along the edge of all plaquette syndromes return a nonerror. This non-
i Γ¯ i
the plaquette∈s crossingthe lattice vertically. Finally, the errorsyndromeprovidesanupperboundtotheavailable
codewordsQof the surface code are also easily written, computational time and also substantially simplify the
calculationsinceit removesfromconsiderationwhichre-
¯ =GFz , ¯ =X¯ ¯ , (1) coveryoperationshouldbeperformedtosteerthesystem
|↑i | i |↓i |↑i
where G = √21N(cid:8)A ♦(1+A♦), NA is t(cid:9)he total number baIcnk tQoEtChetchoemorpyu,taittioisnasltabnadsaisr.d to focus only on the
ofpossiblestars,and F istheferromagneticstatealong
Q | zi qubit system evolution and disregard any change to the
the positive z direction, namely, F = N .
| zi i=1|↑ii,z environment’s state, even though the latter is a also a
The Hamiltonian we consider is written as
Q quantumsystemcapableofsustainingcorrelations. Ifno
H =H +V, (2) extrastepistakentodissipatethosecorrelations,theen-
0
vironment will keep a memory of events that happened
where H is a free bosonic Hamiltonian, H =
0 0 between the QEC periods. Keeping track of such bath-
k=0 ωka†kak, and V = λ2 rf(r)σrx, where r denotes induced, long-time correlations between QEC cycles in
the6spatial location of a qubit and f is a local bosonic
a fidelity calculation is a difficult task even for simple,
P P
operator,
nontopological logical qubit systems [9, 11]. Thus, to
f(r)= LωD0 |k|s eik·ra†k+e−ik·rak . (3) ptorotcheeeQdEwCithprtohteoccoall.cuInlaatdiodni,tiwonetcoonpsriodjeerctainngetxhteraqustaenp-
kX=6 0 (cid:16) (cid:17) tum computer wave function back to the logical Hilbert
Here, D is the bathspatialdimension, ωk =v k, andω0 space, we assume that at the end of the QEC step the
| |
isamicroscopiccouplingthatmakesf dimensionless(we environment is reset to its ground state. Hence, from
adoptunitssuchthat~=1). Thiserrormodelleadstoa this point on, we are excluding from the calculation any
remarkably simple evolution operator in the interaction spatialcorrelationbetweenQECperiods,aswellasmem-
picture: for a time interval ∆, we have [11] ory and spatial correlations between the time evolution
Uˆ(∆)=e−λ22NGrr(∆)e−λ22 Pr6=sΦrs(∆)σrxσsx oinfgbarsassumanpdtiokne,tsw.eWcanhecnonwveenaiednotplyt rtehwisrietexttrhaesniomnpelrirfoyr-
:e−iλPrFr(∆)σrx :, (4) syndrome projector as
×
w0hFerre(∆N)Fsi(s∆t)h0e ,toFtar(l∆n)um=ber∆odftqfu(rb,itts),, ΦGrrss((∆∆)) == P′ =|ψ0ihψ0|+X¯|ψ0ihψ0|X¯. (7)
h | | i 0
Grs(∆)+ 0∆dt1 0t1dt2[f(r,t1),Rf(s,t2)],and::standsfor Using Eq. (7), it becomes now straightforward to write
normal ordering. For ohmic baths, the correlation func- an expression for the logical qubit fidelity just after the
tion for rR=s taRkes the simple form [15] syndrome extraction,
6
Φrs(∆)= ω0 2 arcsh |rv−∆s| + i2π, 0<|r−s|<v∆, F = |2A+| 2, (8)
v iarcsi(cid:16)n v∆(cid:17) , 0<v∆< r s. |A| |B|
(cid:16) (cid:17) r s | − |
(cid:16)| − |(cid:17) (5) where = ψ0 Uˆ(∆)ψ0pand = ψ0 X¯Uˆ(∆)ψ0 .
A h | | i B h | | i
3
Fidelity calculation – Thus,ourtasknowisreducedto (See Fig. 2.) Here, B is a product of plaque-
j (cid:3)j
evaluate and . Using Eq.(4),it is straightforwardto ttes that do not touch a logical error Z¯ and F is
A B Q γ | xi
show that the ferromagnetic state in the x basis. Splitting the
terms of the sums in Eqs. (13) and (14) between these
and A=χ(cid:10)Fz(cid:12)(cid:12)e−βHG2(cid:12)(cid:12)Fz(cid:11), (9) Btw=o g2χrNo(uTp+s−ofTs−ta)t,ews,hwereerewrite A = 2χN(T+ +T−) and
=χ F X¯e β G2 F , (10) = S e−βH S . (16)
B z − H z T± XS± (cid:10) ±(cid:12) (cid:12) ±(cid:11)
where (cid:10) (cid:12) (cid:12) (cid:11) (cid:12) (cid:12)
(cid:12) (cid:12) Thus, we have to evaluate the “free energies” of the two
= Jrsσrxσsx, (11) groups of states (see Fig. 2).
H The double sum in [see Eq. (11)] runs over lat-
Xr6=s tice points r and s insHide and outside the path of the
and χ = e−λ22NGrr(∆). Notice that although we chose liongtoicatlwoopseertast,onraZm¯γe.ly,Thrus,=lettus breauk thaensde psoint=s
to start with a microscopic model of the environment to v w , where t {an}d v{bγe}lo⊕ng{toγ}Z¯ wh{ile}u
makeaconnectiontophysicalimplementations,wecould a{ndγ}w⊕{doγn}ot. Furtheγrmore, lγet us factor thγe sum oveγr
as well have started by imposing an effective two-body γ
all S into a sum over paths γ and a sum over con-
interaction between qubits such as that defined by . +
H figurations Sγ compatible with a logical operator along
Clearly, when β 0, we have 0 and 1. A +
→ B → F → this path, namely, = . After some sim-
perturbative expansion for small β is the standard route S± γ S±γ
to discuss the error threshold. This “high-temperature” ple algebra, we obtPain T± =P Pγe−βǫγ S±γ z±γ, where
expansion will be discussed elsewhere. Here we follow a ǫγ = Sγ tv Jtvσtxσvx Sγ aPnd P
different route. To understand the opposite limit, β ±| | ±
∞, we need to rewrite |Fzi in the x basis, → zγ(cid:10)= SP+γ e−βPu=6 wJuwσux(cid:11)σwx∓βPwhγwσwx S+γ , (17)
±
|Fzi= N |↑ii,x√+2|↓ii,x . (12) lwoigtihcahlγwoDp=erat(cid:12)(cid:12)(cid:12)ortJZ¯tγw.is tWoeinctarondsueceetahbatoutnhdea(cid:12)(cid:12)(cid:12)eryffetEcetrmofrtehpe-
iY=1(cid:18) (cid:19) resented by tPhe effective local magnetic field hγw.
This dual representation corresponds to a “low- We are now in the position to state our definition of
temperature” expansion and it is suitable for describing the quantum error threshold. We define the critical pa-
the regime where errorcorrelationsare strong. Inserting rameter βc as the value of β that separates the regime
Eq. (12) into (9) and (10), we obtain where = 1 from the regime where < 1 in the ther-
F F
modynamic limit (N ).
χ
A= 2N e−βEshS|G2|Si, (13) Phasetransition–T→he∞evaluationofzγ inEq.(17)isa
S formidabletaskandageneralanswerma±yonlybeachiev-
X
χ
= e−βEs S X¯G2 S , (14) able through numerical simulations. Hence, we now re-
B 2N h | | i strict our considerations to more manageable effective
S
X
qubit interactions. Let us first consider the case of
respectively, where S is an element of the x basis and
| i
Es = S S is its “energy”. Notice that Es may have J, r,s nearest neighbors,
an imahgi|nHa|ryipart. Jrs = (18)
(0 otherwise,
If we had unrestricted sums in Eqs. (13) and (14), we
would be essentially discussing a two-dimensional Ising where J is real. Such case is of physical relevance. Any
model. However G2 projects S onto the subspace of measurement or operation on the plaquettes and stars
| i
positive stars, and among other things remove time re- canintroduceshort-rangecorrelatederrors,regardlessof
versalstateof Sγ fromthesum. Thisrestrictionmakes the presence of an environmental bath. For an ohmic
the computation±of and nonstandard. Nevertheless, bath, it corresponds to set v∆ to the order of the lattice
the action of X(cid:12)(cid:12)¯ is(cid:11)tAo introBduce a sign between two dis- spacing.
tinctclassesofstates. Tobetterunderstandthis,weneed Toshowthatanorder-to-disordertransitionindeedex-
a more convenient way to write the states S in the re- ists, let us consider Eq. (17) with (18) in the absence of
| i
strictedsubspaceofpositivestars. Itisnothardtoprove the field hγ. Using Eq. (15), we obtain
w
thatthesestatesfallintotwogroups, S and S ,
+
{| i} {| −i}
where
S = B F and S =Z¯ S . (15) zS±γ =*Fx(cid:12)(cid:12) j B(cid:3)jeβPu6=wJuwσuxσwx j B(cid:3)j(cid:12)(cid:12)Fx+,
| +i (cid:3)j| xi | −i γ| +i (cid:12)(cid:12)Y Y (cid:12)(cid:12) (19)
Yj (cid:12) (cid:12)
(cid:12) (cid:12)
4
(a) (b) develops,leadingtotheinsensibilitytotheboundaryfield
and, therefore, =1.
F
There are some roadblocks to study the general case
represented by the correlator in Eq. (5). The first one
is its imaginary part. It is expected that any imaginary
partwouldhaveanegativeeffectonthefidelity(aconclu-
sionthatcanbereachedbyapplyingSchwarzinequality).
Even if we disregard this imaginary part, the magnetic
model givenby the realpart ofEq. (5)may leadto frus-
tration. However, since the real part of the correlator
FIG. 2. Example of states that contribute to the S+ (a) and is a slow growing function, it is likely that the model is
S− (b) sums [see Eq. (15)]. The light (dark) circles indicate controlledbyboundaryeffects(asmanyotherfeaturesof
qubits in + (−) σx eigenstates. The dark line indicates the topologicalsystems are). Only throughfuture numerical
pathofthelogicalerrorZ¯γ,whichisactivein(b)andinactive work it will be possible do address these issues.
in (a). In order to gain some insight into the effects of long-
range correlations,we arbitrate a stripped versionof the
ohmic model. Consider the case where qubits on one of
where we have multiplied all spins in one of the sublat-
the sublattices interacts only with qubits on the other
ticesby 1tomakethe modelferromagnetic. Theprod-
− sublattice thought the realpart of Eq. (5). This interac-
ucts over plaquettes introduce a sum over loops where
tion preserves the bipartite nature of the lattice and we
the qubits at the these loops have negative eigenvalues
can again map it onto a ferromagnetic model. We can
(they are located at the edges of the shaded regions in
repeat all step of the calculation done in the case with
Fig. 2). Let us label these qubits by a and b and use c
nearest neighbor interactions. Then, the “energy” cost
and d to label qubits with positive eigenvalues. Thus
of a loop with length ℓ is equal to (n 2)Jℓ, where n is
−
the number of sites connectedto anylattice site through
zS±γ =eβPa6=bJabeβPc6=dJcde−βPa6=cJac. (20) the interaction (i.e., n is related to the range of the in-
teraction). The transition happens at
At “low temperatures”, the first two factors in Eq. (20)
dominate. The third factor leads to excitations above lnµ
β . (23)
the ferromagnetic ground state whose “energy” is EF. c ≈ 2(n 2)J
Therefore, we can go back Eq. (17) and write (for hγ = −
w
Notethatβ explicitly depends onthe interactionrange.
0) c
In a strict sense, if all qubits in the lattice are in the
“causality cone” (and therefore participate in ) there
H
zγ =eβEF 1+ e−2βPa∈loop,c∈/loopJac . (21) is never a threshold. Although this could seem a dismal
result, it also shows us that a a finite QEC period intro-
XS±γ lXoops duces an infrared cutoff that creates a finite transition
“temperature”. The smallest of such transition “tem-
Then,the“energy”costofaloopwithlengthℓisequal
peratures” corresponds to the case of nearest neighbors,
to 4Jℓ. The scaling of the number of loops with ℓ is well
known: N µℓ, where µ is the lattice connectivity con- Eq. (18). Finally, we note that, under the same assump-
ℓ
∼ tions that we used for the ohmic bath, superohmic envi-
stant (e.g., µ 2.638 for a square lattice) [16]. Hence,
the contributio≈n of loops with length ℓ to zγ goes ronmentsarelocalandthereforewillalwaysyieldafinite
Sγ
± transition “temperature”. Conversely, subohmic baths
as µℓe 8βJℓ. Clearly, a tradeoff between “energy” mini-
− P will require an infrared cutoff to yield a finite threshold.
mization and “entropy” maximization takes place and a
In conclusion, in this Letter we studied the fidelity of
phase transition is expected at β =β , where
c a logical qubit encoded with the surface code after one
complete QEC period. We derived a nontrivial mapping
lnµ
βc . (22) toanstatisticalmechanicalproblemandprovidedanan-
≈ 8J
alyticalexpressionfortheerrorthresholdforsomecorre-
For β > β the system is in its ordered phase. Thus, we lated models. This mapping provides a promising route
c
expect the system to be sensible to the boundary mag- forexploringfaulttoleranceoftopologicalquantumerror
netic field direction, leading to < 1 after the QEC correctioncodesinthepresenceofrealisticenvironments.
F
period. In particular, there is an obvious result to be WethankM.CorreaandR.Paszkohelpfuldiscussions
stated. At the zero “temperature” limit, β , only and C. Chamon and R. Raussendorf for insightful com-
→ ∞
the “groundstate”contributesto thesums (i.e. onlythe ments. E.N. was partially supported by INCT-IQ and
sum survive). Hence, = and = 1. Con- CNPq (Brazil). E.R.M. was supported in part by the
T− |A| |B| F 2
versely,forβ <β (thereforeλ<λ ),adisorderedphase ONR and NSF (USA).
c c
5
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