Table Of ContentCodeword Stabilized Quantum Codes for
Asymmetric Channels
Tyler Jackson∗†, Markus Grassl‡§, and Bei Zeng∗†¶
∗Department of Mathematics & Statistics, University of Guelph, Guelph, Ontario, N1G 2W1, Canada
†Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
‡Institut fu¨r Optik, Information und Photonik, Universita¨t Erlangen-Nu¨rnberg,91058 Erlangen, Germany
6 §Max-Planck-Institutfu¨r die Physik des Lichts, Leuchs Division, 91058 Erlangen, Germany
1 ¶ Canadian Institute for Advanced Research, Toronto, Ontario, M5G 1Z8, Canada
0
2
n
Abstract—We discuss a method to adapt the codeword stabi- where
a
J lized(CWS)quantumcodeframework totheproblemoffinding
1 aPsayumlimeertrroicr qmuoadnetlusmforcodaems.plWituedefodcuasmpoinngthneoicseorraensdponpdhiansge A0 =(cid:18)10 √10 γ(cid:19), A1 =(cid:18)00 √0γ(cid:19), (4)
2 −
damping noise. In particular, we look at codes for Pauli error
models that correct one or two amplitude damping errors. for some damping parameter γ.
]
h Applying local Clifford operations on graph states, we are able IthasbeendemonstratedthatdesigningQECCadaptivelyto
p toexhaustivelysearch forall possiblecodesuptolength9. With specific error modelscan result in better codes [8], [11]–[13],
- a similar method, we also look at codes for the Pauli error [23],[24],[30],[31]andfault-tolerantprotocols[1].Although
t model that detect a single amplitude error and detect multiple
n mostofthesecodesare indeedCWScodes,therehasbeenno
phase damping errors. Many new codes with good parameters
a are found, including nonadditive codes and degenerate codes. systematic constructionapplying the CWS framework.In this
u
IndexTerms—codeword stabilizedquantumcode, nonadditive work,wefillthisgapbydevelopingamethodforfindingCWS
q
[ code,asymmetriccode,amplitudedampingchannel,phasedamp- codesforasymmetricchannels.Ourmethodleadstomanynew
ing channel codes with good parameters, including nonadditivecodes and
1
degenerate codes. These results demonstrate the power of the
v I. INTRODUCTION
CWS framework for constructing good QECC.
3
Codeword stabilized (CWS) quantum codes constitute the
6
by far most general systematic framework for constructing
7 II. ERROR MODELS
5 quantum error-correcting codes (QECC) [6], [7], [9]. It en-
Dependingonthenoisemodelindifferentphysicalsystems,
0 compassesstabilizercodes[4],[5],[14],[32],aswellasmany
. nonadditivecodeswithgoodparameters[23],[28],[35].Over we obtain different asymmetric quantum channels. We start
1
withtheamplitudedampingchannel withKrausoperators
0 thepastyears,ithasbeenexploredinvarioussettingsandhas EAD
giveninEq.(4),whichmodelsrealphysicalprocessessuchas
6 been applied in many different cases, leading to promising
1 results [2], [15]–[17], [20], [25], [26], [33]. spontaneous emission. If the system is at finite temperature,
: then the noise modelwill notonly containthe Krausoperator
v Most of the QECC constructedso far are for the depolariz-
i ing channel A1 corresponding to emission, but also A†1 corresponding to
X
absorption [27]. Notice that
p
r DP(ρ)=(1 p)ρ+ (XρX+YρY +ZρZ), (1)
a E − 3 √γ √γ
where the Pauli X,Y,Z errorshappen equally likely. (Here ρ A1 = 2 (X +iY), A†1 = 2 (X −iY). (5)
denotes the density matrix representing the state of the quan-
tum system.) The most general quantum channels allowed by Hence, the linear span of the operators A1 and A†1 equals the
quantum mechanics are completely positive, trace-preserving linear span of X and Y. We can then equivalently formulate
the error modelby using the Pauli operatorsX and Y, which
linearmapsthatcanberepresentedintheKrausdecomposition
(ρ)= E ρE† with E†E =I [27]. happenwith equalprobability.That is, if a code is capable of
EtheOanseymePxmakmetrpiklcePgaekunleircahliaznPinngeklthwkehidckehposelanrdizsinρgtochannel EDP is tcoArr†1e-cetrirnogrst. X- and t Y-errors, it can also correct t A1- and
Furthermore, notice that
(1 p p p )ρ+p XρX+p YρY +p ZρZ, (2)
− x− y− z x y z γ 2
where the Pauli X,Y,Z errors happen with probabilities A0 =I − 4(I −Z)+O(γ ), (6)
p ,p ,p , respectively [21]. Other asymmetric channelsstud-
x y z while A1 depends linearly on √γ. This then results in an
iedintheliteratureincludetheamplitudedampingchannel[8]
asymmetry between the probabilities p =p and p that the
x y z
AD(ρ)=A0ρA†0+A1ρA†1, (3) Pauli X,Y errors or the Pauli Z error, respectively, happen.
E
Apart from amplitude damping, another common noise in Z-errors up to weight r will allow to correct r/2 Z-
⌊ ⌋
physical systems is dephasing, with Kraus operatorsgiven by errors. The error set is
√1 pI and √p(I Z)/2,or equivalently,in termsof I and
Z w−ith pz >0 and±px =pz =0 [27]. In general, the system E{3} ={I}∪{Xi,Yi: i=1,...,n}∪Zr, (10)
undergoes both amplitude damping and dephasing, resulting
where isthesetofallPauliZ operatorsuptoweight
in a wide range for the parameters p =p and p . r
x y z Z
r. A code that detects this error set in fact detects both
Therefore,inthisworkweconsiderthefollowingasymmet-
an arbitrary error and r phase errors.
ric Pauli channel
(ρ)=(1 (2p +p ))ρ III. ALGORITHMTO SEARCH FORCWS CODES
AS xy z
E −
+pxy(XρX+YρY)+pzZρZ, (7) A QECC Q is a subspace of the space of n qubits (C2)⊗n
(here we focus on quantum systems of dimension q = 2,
whereX andY happenwithequalprobabilityp =p =p .
x y xy but the approach can be generalized to qudits of dimension
In terms of Eq. (7), the asymmetric Pauli error model
q > 2). For a K-dimensional code space spanned by the
corresponding to amplitude damping is given by p γ
and p γ2. This is different from the amplitude dxaym∝ping orthonormal basis {|ψii: i = 1,...,K} and an error set E,
z ∝ thereisaphysicaloperationdetectingalltheelementsEµ
error model in, e.g., [11], [18], [23], [24], [31], where the ∈E
(as well as their linear combinations) if the error detection
Kraus operators A0 and A1 are used. The main reason that condition [3], [22]
we use the Pauli Kraus operators as our error sets is that this
enables us to use the CWS framework to construct codes. ψ E ψ =c δ , c C, (11)
i µ j µ ij µ
WithintheCWSframework,inordertotransformthequantum h | | i ∈
error detection condition into a classical condition, it is more is satisfied. The notation ((n,K)) is used to denote a QECC
convenient to use Pauli errors, as we will discuss in Sec. III. with length n and dimension K.
In other words, since A0 and A1 are not Pauli operators, the Ourgoalistofindgoodcodesdetectingtheerrorsets {j},
CWS framework does not directly apply. Furthermore,due to for each of the three cases. For each code length n, weEseek
Eq. (5) and Eq. (6), our error modeldoes notonly correctthe thelargestdimensionK ofCWScodesforeacherrorset {j},
errors A0 and A1, but the resulting codes will be stronger in j =1,2,3.ThisisdonethroughamaximumcliquesearchE[7],
the sense that A†1 can be corrected as well. by using the algorithms and programs developed in [34].
Inthisworkweconsiderthreespecificcasesforasymmetric
codes, as listed below. We use Xi,Yi,Zi to denote the Pauli A. The CWS framework
X,Y,Z operators on the ith qubit. Notice that our method
An ((n,K)) CWS code Q is described by two objects: 1)
for generating the error sets is very general and can be
A stabilizerS thatis anabeliansubgroupof then-qubitPauli
straightforwardly generalized to deal with different relations
group, has order 2n, and does not contain I; the group S
between p , p , and p .
x y z −
is called the word stabilizer. 2) A set of K n-qubit Pauli
1. Codes correcting a single amplitude damping error: to
operators W = w : ℓ = 1,...,K , which are called the
ℓ
improve the fidelity of the transmitted state from 1 γ { }
to1−γ2,oneonlyneedstocorrectasingleA1 error−and wbyorSd,opi.eer.,atsorSs.T=hereSisafournaiqlluesquanSt.umThsetacteod|SeiQstaibsiltihzeedn
detectasingleA0 error[18].IntermsofPaulioperators, spanned by th|eibasis|viectors given∈by w =w S .
ℓ ℓ
the corresponding error set is given by | i | i
According to Eq. (11), the code Q detects the error set
E
{1} = I Xi,Yi,Zi, XiXj,YiYj,XiYj,YiYj , if and only if hwi|E|wji = cEδij for all E ∈ E. When E
E { }∪{ } consists of Pauli matrices, this error-detecting condition can
(8)
be written in terms of S and w as below [9]:
where i,j =1,...,n. A code that detects this error set i
in fact also corrects a single A† error. For all E ,
1 ∈E
2. Codes correcting two amplitude damping errors: based
i=j: w†Ew / S (12)
on the analysis on the single error case above, the error ∀ 6 i j ∈±
set is given by
and
E{2} ={EµEν: Eµ,Eν ∈E{1}}. (9) (cid:16)∀i: wi†Ewi ∈/ ±S(cid:17) or (13)
A code that detects this error set in fact also corrects i: w†Ew S or (14)
two A†1 errors. (cid:16)∀ i i ∈ (cid:17)
3. Codes detecting both a single amplitude damping error i: w†Ew S . (15)
and multiple dephasing errors: detecting X ,Y ,Z (cid:16)∀ i i ∈− (cid:17)
i i i
{ }
suffices to detect an arbitrary single qubit error (includ- If condition (13) holds for all E different from identity,
∈ E
ing a single amplitudedampingerror),and detectingall then the code Q is nondegenerate, otherwise it is degenerate.
B. The CWS standard form Recall that the single-qubit Clifford group is generated by
the Hadamard operator H and the phase operator P as given
Every ((n,K)) CWS code can be transformed, by local
below [4], [14]
Clifford operations, into a standard form [9], where the word
operators take the form wℓ = Zcℓ and the word stabilizer 1 1 1 1 0
has generators of the form Si = XiZri, for some choices of H = √2(cid:18)1 1(cid:19), P =(cid:18)0 i(cid:19). (19)
classicaln-bitstringscℓandri.HereZcℓ =Zcℓ,1 ... Zcℓ,n. −
⊗ ⊗ Since overall phase factors can be ignored, we only need
In the standard form, any n-qubit Pauli error, which can
be written in the form E = ZvXu for some classical n-bit to consider the action of the Clifford group on the Pauli
strings v and u, can be tran±slated to classical errors via the matrices X,Y,Z modulo phase factors. The Clifford group
map acts as the permutation group S3 on three letters (we use
n 1,2,3 to denote X,Y,Z, respectively). The group S3 has
Cl (E = ZvXu)=v (u) r . (16) order six, with the elements given by (in cycle notation)
S i i
± ⊕
Mi=1 id,(123),(132),(12),(13),(23) ,whereiddenotestheiden-
{ }
Nowforthewordoperators Zcℓ: cℓ ,theerrordetec- tity permutation. All error sets E{j}, j =1,2,3, are invariant
{ ∈C} withrespecttointerchangingX andY.Henceitissufficientto
tionconditionrequiresthattheclassical binarycode detects
C consideronerepresentativefromeachofthethreerightcosets
all errors from Cl ( ), and that for each E
S
E ∈E of(12),givenby id,(12) , (13),(132) ,and (23),(123) .
{ } { } { }
Cl (E)=0 (17) Soeffectively,weonlyneedtotest,e.g.,thethreepermutations
S
6
c c id,(13),(23) .
or ℓ: Z ℓE =EZ ℓ. (18)
{ }
∀ Therefore, for each of the error sets {j}, we have three
E
If Eq. (17) holds for all E , the CWS code is nondegen- cases for each qubit i: no permutation, permute Y and Z,
∈ E
erate, otherwise it is degenerate. or permute X and Z. To search for a length n code, this
will reduce the total number of error sets from 6n to 3n
C. Local Clifford operations
for each graph state to be tested. Compared to codes for the
To get to the standard form, one needs to apply local depolarizingchannel, the search space is enlarged by a factor
Clifford (LC) operations of the form L = L , where L of3n,duetotheasymmetrybetweenp andp .Nevertheless,
i i i xy z
are single-qubit Clifford operations [9]. ThNis transforms the wecanstillhandlethesearchforsmalln,inparticularforthe
stabilizer S and word operators w to the standard form, error sets {j}, j =1,2,3, up to length n=9.
ℓ
{ } E
but at the same time also changes the error model.
ForthedepolarizingchannelgiveninEq.(2),theerrorsetis IV. RESULTS
invariant under LC operations, since in this model essentially As described above, in our search algorithm, we start from
all single-qubit errors happen equally likely. Therefore, in or- theCWSstandardformandtransformtheerrorset {j} byLC
E
dertosearchforaCWScode,onecansimplyusethestandard operations.Thishasnoeffectonthecodeparameters((n,K))
form by starting from a stabilizer of the form Si = XiZri, found. However, to present the CWS codes found, we fix the
which correspondsto a graphstate [19]. For a fixed length n, error set {j} and equivalently transform the CWS standard
E
itissufficienttoconsiderallgraphstatesuptoLCequivalence form into a general CWS code.
as classified in [10]. This results in an exhaustive search for
A. Codes correcting a single amplitude damping error
all possible CWS codes of length n.
Being able to restrict the search to graph states up to LC We have conducted an exhaustive search for the error set
equivalence, instead of all stabilizer states of length n, has {1} up to length n = 9, resulting in CWS codes correcting
E
dramatically reduced the search space, and exhaustive search a single amplitude damping error. As already mentioned, the
for single-error-correcting codes for the depolarizing channel codes found can not only correct a single error given by the
up to length n = 10 has been carried out. It turned out that KrausoperatorA1, atthe same timetheyalso correcta single
thebestCWScodewithlengthn=9hasdimensionK =12, errorA†.Inotherwords,thecodescorrectbothsingleX errors
1
beating the best stabilizer code of dimension 23 =8 [35]; for and single Y errors (and detect single Z errors as well). We
n = 10 the best CWS code has dimension K = 24, again summarize our results in Table I.
beating the best stabilizer code of dimension 24 =16 [20]. As we can see from the table, for the lengths 6,7,8,9, our
For the asymmetric channels as given in Eq. (7), however, codesoutperformthebestsingle-error-correctingcodesforthe
consideringonly all graphsstates as classified in [10] and the depolarizing channel—which also correct the error set {1},
E
error sets {j} (j = 1,2,3) is not sufficient to exhaustively i.e.,asingleamplitudedampingerror.Inparticular,forlengths
E
searchforallpossibleCWScodes.Thisisduetotheasymme- 8 and 9, the best CWS codes we have found (of dimensions
try between p and p , which implies that the error sets are 10 and 20 respectively) are nonadditive codes.
xy z
no longer invariant under LC operations. Therefore, in order For lengths 6,8,9, our codes also outperform the best
to exhaustively search for all possible CWS codes by using known CSS codes that are specifically designed to detect the
thestandardform,onewillneedtocheckallthepossibleerror error set {1}, based on a construction proposed in [14] (see
E
sets that are LC equivalent to a given {j}. also [31]). Therefore, for these lengths, we have found good
E
TABLEI
encodingasinglequbit,sincethecorrespondingclassicalcode
Dimension K of CWS codes ((n,K)) of length n detecting the error set
E{1} for different length n. The column d = 3 lists the largest dimension is trivially linear [9].
ocofluCmWnSEc{o1d}eslisthtsatthceorlraercgtesatsdiinmgelenseirornoroffoCrWthSe dceopdoelsarfiozuinngdcdheatencnteinl.gTthhee C One code has the stabilizer S1 generated by
error set E{1}. The column CSS lists the largest dimension of the known X I I I I I I I Z
Calderbank-Shor-Steane (CSS) codes [4], [32] that can correct the error set Z I I I X Z I Z X
E{1},basedonaconstruction proposedin[14].ThecolumnGF(3)lists the
I X I I I I I Z I
largestdimensionofcodescorrectingasingleamplitudedampingerrorbased
onaconstruction proposedin[31]. I Z I I X I Z X Z
I I X I I Z I I I
n d=3[4],[35] E{1} CSS[14] GF(3)[31] I I Z I Y Y Z Z I
5 2 2 2 2 I I I X I I Z I I
6 2 4 2 5
I I I Z Y Z Y I Z
7 2 8 8 8
8 8 10 8 16 The other code has the stabilizer S2 generated by
9 12 20 16 24
X I I I I I I I Z
Z I I I Z Z Z I X
codes that outperform all the previously known constructions I X I I I I I Z I
for detecting the error set {1}. I Z I Z Z Y I Y Z
E I I X I I I Z I I
Notice that the existence of a CWS code with dimension
I I Z Z Z X X Z I
K = 4, and hence a subcode of dimension 3, implies the
I I I X I Z I I I
existence of a stabilizer code with the same parameters [7,
I I I I X I Z Z Z
Theorem 7]. Hence the ((6,4)) codes we found, as listed in
Table I, include stabilizer codes encoding two qubits. As an It is straightforward to check that these codes detect the
example, one such code has stabilizer S generated by errorset {2}, since noelementsin {2} is also in C(S ) S
i i
E E \
(for i = 1,2). Furthermore, both codes are degenerate since
X X I I Z Z
some of the elements in {2} are indeed in S , for instance
X Z I Z I X E i
X1Z9.
Z I Y Z Y Z
These codes outperform the ((10,2)) code found in [11].
I I Z X I Z
RecallthatShor’snine-qubitcode,havingthesameparameters
It is straightforward to check that this code detects the error ((9,2)) as our codes, also corrects two amplitude damping
set {1}, sincenoelementsin {1} is alsoinC(S) S, where errors [14]. However, Shor’s code only corrects the Kraus
E E \
C(S) is the centralizer of the stabilizer S. operators A0 and A1, but does not detect the error set {2}.
E
However, with the exception of n = 7, the single-error- Therefore, for length n = 9, we have found good codes that
correcting codes constructed in [31] have larger dimensions outperformall the previousknownconstructionsfor detecting
than our codes. The codes constructed in [31] are specifically the error set {2}.
E
designedtocorrecttheKrausoperatorsA0andA1,thesecodes
C. Codes detecting a single amplitude damping error and
cannotdetecttheerrorset {1}.Asdetectionoftheerrors {1}
implies that a single errorEA† can be corrected as well,Eit is detecting multiple dephasing errors
1 For the error set {3}, we have performed an exhaustive
not a surprise that our codes have smaller dimensions.
E
searchfordifferentlengthsnandZ-errordetectingcapabilities
Notice that the codes constructed in [31] are also CWS
r up to n = 8, and a random search starting from randomly
codes, but errors are handled in a different way than the
Pauli error set {1}. It remains open how to generalize the selected graph states for n = 9 and different r. Our results
E are listed in Table II. We compare our results with the best
methodof[31]todealwithmorethanoneamplitudedamping
error, while the error set {1} can naturally be generalized, stabilizer codes that detect all errors up to weight r as given
e.g., to {2} for correctingEtwo amplitude damping errors, as in [4], and the codes detecting a single amplitude damping
E errors and Z errors up to weight r as found in [12].
demonstrated next.
As we can see from the table, for most lengths n and Z-
error weight r, the CWS codes found outperform the known
B. Codes correcting two amplitude damping errors
results. The entries for which we did not find improvements
We have performedan exhaustivesearch for codes correct- are n=6, r =1 and n=8, r =1. Codes with r =1 detect
ing two amplitude damping errors, i.e., detecting the error set single Pauli errors, i.e., they are codes of minimum distance
{2}, up to length n = 9. In fact, the resulting codes correct two. For even length, the corresponding stabilizer codes are
E
anycombinationof X andZ errorsupto weighttwo, as well knowntohavethe largestpossibledimensionforsingle-error-
as a single Z error. detectingcodes[29].Foroddlengthn=5,7,9,wefindcodes
No non-trivialCWS codes are found for length n 8. For with parameters matching those of the code family ((2m+
length n = 9, two LC-inequivalent codes encoding≤a single 1,3 22m−3,2)) given in [29]. Whenever the dimension is a
×
qubit have been found found. These are both stabilizer codes power of two, the codes we found include stabilizer codes.
TABLEII
Dimension K of CWS codes detecting the error set E{3} for different length n and parameter r. For each value of r, the first column lists the largest
dimensionofstabilizercodesthatdetectallerrorsuptoweightrasgivenin[4];thesecondcolumnliststhelargestdimensionofasymmetriccodesdetecting
asingleamplitude dampingerrorandphaseerrorsuptoweightr asfoundin[12];thethirdcolumnliststhelargestdimensionoftheCWScodes foundby
oursearch forcodes detecting the errorset E{3}.‘−’means that nonon-trivial codes exist based onthe construction. Thenumbers labeled with ∗ are the
bestparameters foundbyrandomsearch;otherwise themaximaldimensionisobtained byexhaustive search.
n/r 1 2 3 4 5 6 7
stab. [12] CWS stab. [12] CWS stab. [12] CWS stab. [12] CWS stab. [12] CWS stab. [12] CWS stab. [12] CWS
5 4 5 6 2 − 4 − − 2 − − 2 − − − − − − − − −
6 16 16 16 2 2 8 − − 4 − − 2 − − 2 − − − − − −
7 16 22 24 2 8 16 − − 8 − − 2 − − 2 − − 2 − − −
8 64 64 64 8 8 20 − 8 16 − − 4 − − 2 − − 2 − − 2
9 64 93 96∗ 8 16 40∗ − 8 20∗ − − 6∗ − − 4∗ − − 2∗ − − 2∗
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