Table Of Content

Twoinfinitefamiliesofnonadditivequantumerror-correctingcodes Sixia Yu1,2, Qing Chen1,2, and C. H. Oh2 1HefeiNationalLaboratoryforPhysicalSciencesatMicroscaleandDepartmentofModernPhysics& DepartmentofModernPhysicsUniversityofScienceandTechnologyofChina,Hefei230026,P.R.China 2PhysicsDepartment, NationalUniversityofSingapore, 2ScienceDrive3, Singapore117542 We construct explicitly two infinite families of genuine nonadditive 1-error correcting quantum codes and provethattheircodingsubspacesare50%largerthanthoseoftheoptimalstabilizercodesofthesameparam- etersviathelinearprogrammingbound. Allthesenonadditivecodescanbecharacterizedbyastabilizer-like structureandthustheirencodingcircuitscanbedesignedinastraightforwardmanner. One major family of quantum error-correcting codes whenthelengthtendstoinfinitythatoutperformsallthesta- (QECCs) [1, 2, 3, 4], which are powerful tools to fight the bilizercodesofthesameparameters. 9 quantum noises in various quantum informational processes, InthisLetterweshallconstructtwoinfinitefamiliesofgen- 0 0 are called as additive or stabilizer codes [5, 6, 7, 8]. The uine nonadditive 1-error-correcting codes with coding sub- 2 coding subspace of a stabilizer code is specified by the joint spaces being 50% larger then the corresponding optimal 1- +1eigenspaceagroupofcommutingmultilocal(directprod- error-correcting stabilizer codes of the same parameters to n a uctof)Paulioperators. Usually[[n,k,d]]denotesastabilizer show that the nonadditive error-correcting codes outperform J codeoflengthn,thenumberofphysicalqubits,anddistance the stabilizer codes even when the length n tends to infinity. 14 dic,ail.eq.u,cboitrsre(2ctki-ndgimupentosi(cid:98)odn−a2l1s(cid:99)u-qbuspbaitceer)r.ors,thatencodesklog- Astrllutchtuerneoannadddthiteivreefcooredethseareencchoadriancgt-edreizceoddibnygacsirtacbuiiltiszcear-nlibkee ] The first example of nonadditive codes, codes that cannot designedinastraightforwardmanner. h be described within the framework of stabilizer, was an infi- Two families of nonadditive 1-error correcting codes that p nitefamilyof1-error-detectingcodes[9,10],e.g.,((5,6,2)), weshallconstructhavethefollowingparameters - nt wbiiltihzecrocdoidnegssoufbtshpeacsaemsbeepinagra5m0e%terlsa.rgReerctehnatnlythaenootphteimrfaalmsitlay- D(m,a) =((Nma,322Nma−2m−6,3)) (1) a qu ohfas1-beereronrcdoentesctrtuincgtecdodines[1w1i]thasntdillsllairgghetrlyenimcopdrionvgesdubinsp[a1c2e]s. where Nma = 22m+35−5 +a with a = 0,1 and m ≥ 1. To ensure that they are genuine nonadditive we shall prove that [ Here we have denoted by ((n,K,d)) a nonadditive code of the corresponding optimal 1-error-correcting stabilizer codes 1 lengthnanddistancedthatencodesaK-dimensionallogical ofthesamelengthhaveparameters v subspace(aboutlog K logicalqubits). 2 35 The first example of nonadditive code [13], namely [[Nma,Nma −2m−6,3]] (2) ((9,12,3)), that outperforms all the stabilizer codes of the 9 byworkingoutanalyticallythelinearprogrammingboundfor 1 samelengthwhilecapableofcorrectingarbitrarysinglequbit the lengths Na. Notice that the quantum Hamming bound . errorshasrecentlybeenconstructedviaagraphicalapproach m 1 permits exactly one more logical qubit, i.e., log (3Na + 0 based on graph states. Later on an optimal 10-qubit code (cid:100) 2 m 1) =2m+5. Thefirstnonadditivecodesofthesetwofami- 9 ((10,24,3)) has been found via a comprehensive computer (cid:101) liesare((41,3 232,3))and((42,3 233,3))respectivelywhile 0 search [12]. Recently a family of codes of distance 8 that · · : the optimal stabilizer codes have the parameters [[41,33,3]] v encode 3 more logical qubits than the best known stabilizer and[[42,34,3]]. i codes have been constructed based on nonlinear classical X Ourconstructionisbasedonafamilyofstabilizercodesof codes [14]. However the possibility of being equivalent to ar somesubcodeofanoptimalstabilizercodeorevenastabilizer lengths{22r+3}mr=1[5,8]andtwononadditivecodesoflength 9and10discoveredrecently[12,13]andisakindofpasting code of the same parameters under local unitary transforma- stabilizer codes with nonadditive codes that generalizes the tionshasnotyetbeenexcluded. pastingofstabilizercodesinRef.[16]. Wedenoteby , v v Generally, being without a stabilizer structure, the nonad- X Z three Pauli operators acting nontrivially only on some qubit ditive codes promise larger coding subspaces while they are labeledbyv andby theidentityoperator. Furthermorefor hardertoconstructandidentifythanthestabilizercodes. On I agivenindexsetU wedenote = andsimilarly onehandthereisnosystematicconstructionsofarandallthe forotherPaulioperators. XU v∈UX goodcodesarefoundviacomputersearch[12, 15], whichis Letuslookattheoptimalstabilizero(cid:81)flength22r+3atfirst. impossibleforarelativelylargelength(e.g. n 11). Onthe AccordingtoRef.[5]thestabilizerofthecodehas2r+5gen- ≥ otherhandanobviouscriterionforagenuinenonadditivecode eratorswithtwoofthembeing and wherewehave is to check whether or not its coding subspace is larger than labeled22r+3physicalqubitswiXthUUr = Z1U,r2,3,...,22r+3 . r allthestabilizercodesofthesameparameters. Howeverthe { } Theremaining2r+3generatorsaregivenby exact bound for stabilizer codes is generally unknown. As a resultitisofinteresttofindnonadditiveerror-correctingcode r = hk hk−1+h1+h2r+3 k U . (3) Sk X Z | ∈ r (cid:8) (cid:9) 2 HHeerreehh ddeennootteessaa2222rr++33--ddiimmvveeccttoorrtthhaattiisstthheekk--tthhrrooww wweehhaavveeddeennootteedd ==11((11++ ++ ))..IIffwweeddeennoottee kk VVaabb 22 XXaa XXbb−−XXaaXXbb ooffaa((22rr++33)) 2222rr++33mmaattrriixxHH == [[cc,,cc,,......,,cc ]] == wweehhaavveeTTrr ==2288.. IIttsshhoouullddbbeennootteedd wwhhoosseekk--tthhccooll×u×ummnncc bbeeiinnggtthheebrbriinnaarryyrr0e0epprr1e1esseennttaattii2o2o2n2nrr++oo33f−f−k1k1,, AttAhha0a0tttthheeAAcco1o1AdAdee22iAiAss3n3noonnddeeggeenneerraattVeeV,0,0AwwAh0h0iicchhccaannbbeeeeaassiillyysseeeennffrroomm kk ee..gg..,,ccTT == ((00,,00,,......,,11))aannddcc == ((11,,11,,......,,11))aanndd iittsswweeiigghhttddiissttrriibbuuttiioonnss.. 11 2222rr++33 11 hh00 == 00iisstthheezzeerroovveeccttoorr.. AAnnddffoo−r−raavveeccttoorrhhwwiitthhccoomm-- TThheennoonnaaddddiittiivveeooppttiimmaallccooddee((((1100,,2244,,33))))≡≡DD((00,,11))hhaassaa appanonodndneennththss={={hhvv||vv∈∈UUhhrrvv}}..wweehhaavveeddeennootteeddXXhh== vv∈∈UUrrXXvhvhvv gogornrnaa1p1p0h0h-v-vssetetararttttieieccebebsasasasaisissssschchoooorwrwrrenensspipinnooFnnFdidigigin.n.11ggbbttaoaonntdtdhhtetehhgeegrcracaopoprrhrhreeGsGspp11oon=n=ddii(nn(VgVg1g1g,,rrEaEap1p1h)h) DDZeZessppiitteetthhveev∈i∈irrUUrnrnoZoZnnvvaaddddiittiivveenneesssstthheeccooddeess((((99,,Q(cid:81)1122,,33))))aanndd ssttaatteeiiss GG11.. OObbvviioouussllyytthheeggrraapphhiissuunncchhaannggeedduunnddeerrtthhee ((1100,,2244,,33))))Q(cid:81)aaddmmiittaassttaabbiilliizzeerr--lliikkeessttrruuccttuurreeaannddccaannbbeemmoosstt ffoolllloowwiinn|g|gttwwi(cid:105)ooppeerrmmuuttaattiioonnss ccoonnvveenniieennttllyyffoorrmmuullaatteeddbbyyuussiinnggtthheeggrraapphhssttaatteess[[2200,,2211]].. ππ==((1144))((2233))((6699))((7788)),, ττ==((1122))((3344))((6677))((8899)).. ((66)) WWeeddeennootteebbyyGG==((VV,,EE))aassiimmpplleeuunnddiirreecctteeddggrraapphhwwiitthhaa sseettVV ooffvveerrttiicceessaannddaasseettEEooffeeddggeess..TTwwoovveerrttiicceessaarreeccoonn-- tthhaattaaccttnnaattuurraallllyyoonnVV.. TTwwoosswwaappppiinnggooppeerraattoorrss aanndd ππ onnoneenccttVeVed0d0ww==iitthh99aanananneeddddggVeVe1i1iffff={={aa,1,1b0b0}}vv∈∈eerrEtEtiicc..eeTsTswwaaororeeggsrsrhahapopowhhwssnnGGiinana(F(Faaiigg==..11.0.0,B,B11y)y) MaMarrbbτiτittraraaarrrereyyddCCeefifinneeVdVd.v.viFiFaao,o,rerel.l.gagat.t.e,e,rMrMuussπeπeZZwwCCee||+d+deei(cid:105)fiVxfiVxn1n1ee==ttwwZZooππc(c(oCoCn)n)t|t|r+r+ooil(cid:105)lMlVxlMVxee11ddf--fossowrwraaaanppn llaabbe|e|lliinn|g|g VV qquubbiit|ts|sbby|y|VVwweeccaannddeefifinneetthheeggrraapphhssttaatteeccoorrrree-- ooppeerraattiioonnssww⊆⊆iitthhqquubbiittss00aanndd55aassssoouurrcceessrreessppeeccttiivveellyyaass || || ssppoonnddiinnggttootthheeggrraapphhGGaa GG == ++VVwwhheerree || i(cid:105) UUGG|| i(cid:105)xx 11 == 11++ ++((11 )) ,, ((77aa)) == 11++ZZaa++ZZbb−−ZZaaZZbb,, ((44)) TTππ 22 XX00 −−XX00MMππ UUGG 22 11(cid:0)(cid:0) (cid:1)(cid:1) {{aa,Y,(cid:89)bb}}∈∈EE TTττ == 22 11++XX55++((11−−XX55))MMττ ,, ((77bb)) aanndd||++i(cid:105)VxVx ddeennootteesstthheejjooiinntt++11eeiiggeennssttaatteeooffXXvvffoorrvv∈∈VV.. wwhhiicchhccaannbbeerreeaaddiillyy(cid:0)i(cid:0)immpplleemmeenntteeddvviiaaTTooffflfliiaa(cid:1)n(cid:1)nddccoonnttrrooll--nnoott OObbvviioouussllyyUUG2G2==11aannddtthheeggrraapphhssttaattee||GGi(cid:105)iissaallssootthhee++11jjooiinntt ggaatteess.. AAccccoorrddiinnggttooRReeff..[[1122]],,tthhoossee2244ggrraapphh--ssttaatteebbaasseessooff eeiiggFFeernrnoossmtmtaatttetehhoeoeffgtgthrhraeaeppfhfhoolsllsltotoaawwtteeiinngGgGnnsasattaabbbbaaiislsliiiizszseefrfrososrrGGtvthvhe=e=wwUUhhGGooXlXleevvUsUsyGyGss.t.teemmaa 1tt1hh,,e2e2,c,c.o.o.d.d..ee,,6D6Da(a(0n0n,,d1d1))µµaa,r,rνeνeg=g=iivv0e0e,n,n11bbwwyyhh{{ee|r|CrCeeµiµ2i2νν44i(cid:105)ss=u=ubbsZsZeeCtCtssµiµioνoν|ff|GGVV11i(cid:105)a}a}rrewewiitthhii== || i(cid:105) 11 bbaassiissoofftthheewwhhoolleessyysstteemmccaannbbuuiilltt GG CC VV ..FFoorr CC ttrriivviiaallggrraapphhwwiitthhnnooeeddggeetthheeggrraapphh--ss{t{taZaZtteeb|b|aassii(cid:105)iss||rreeaadd⊆s⊆sZZCC}|}|++i(cid:105)VxVx.. CCµiµiνν==ππµµ◦◦ττνν((CCii))△(cid:52)ννBB△(cid:52)µµττνν((AA)) ((88)) AAnnyyggiivveennccoolllleeccttiioonnooffvveerrtteexxssuubbsseettssooffVVwwiillllddeefifinneeaabbaassiiss wwiitthhAA BB==AA BB AA BBbbeeiinnggtthheessyymmmmeettrriiccddiiffffeerreennccee tthhaattssppaannssaassuubbssppaacceewwhhiicchhiissrreeffeerrrreeddttooaasstthheeggrraapphh--ssttaattee △(cid:52) ∪∪ −− ∩∩ bbeettwweeeennttwwoosseettssAAaannddBBaanndd bbaassiiss.. WWiitthhaaggrraapphhaannddaaccoolllleeccttiioonnooffvveerrtteexxssuubbsseettsswwiillll ddeefifinneeaassuubbssppaaccee.. AA== 00,,22,,33 ,,BB== 55,,11,,22 ,,CC == ,, 11 {{ }} {{ }} ∅∅ FFoorrtthheennoonnaaddddiittiivveeccooddee((((99,,1122,,33))))≡≡DD((00,,00))wweeccoonnssiiddeerr CC22=={{11,,22,,33,,99}},,CC33=={{11,,22,,77,,88}},,CC44=={{11,,22,,66,,77,,99}},,((99)) tthheellooooppggrraapphhGG ==((VV,,EE))oonn99vveerrttiicceessaasssshhoowwnniinnFFiigg..11aa CC == 11,,33,,77,,88,,99 ,,CC == 11,,33,,44,,66,,77,,99 .. 00 00 00 55 66 {{ }} {{ }} aannddccoorrrreessppoonnddiinnggggrraapphhssttaattee GG ..TThheejjooiinntt++11eeiiggeennssppaaccee 00 oofftthheeffoolllloowwiinngg66oobbsseerrvvaabblleess|| i(cid:105) IIttttuurrnnssoouutttthhaatttthheennoonnaaddddiittiivveeccooddeeDD((00,,11))aaddmmiittssaallssooaa ssttaabbiilliizzeerr--lliikkeessttrruuccttuurree.. WWiitthhtthheehheellppoofftthheeffoolllloowwiinnggeenn-- αα == ,, ((55aa)) ccooddiinnggooppeerraattoorr 11 UUGG00XX{{33,,88}}UUGG00 αα == ,, ((55bb)) 22 UUGG00XX{{66,,22}}UUGG00 UUeenncc==ZZ22UUGG11TTττTTππZZ22 ((1100)) αα == ,, ((55cc)) 33 UUGG00XX{{99,,55}}UUGG00 iittccaannbbeerreeaaddiillyycchheecckkeeddtthhaatttthheeffoolllloowwiinngg66mmuuttuuaallllyyccoomm-- == (( )) ,, ((55dd)) AA11 UUGG00XX{{44,,77,,33,,66,,99}}VV6699UUGG00 mmuuttiinnggoobbsseerrvvaabblleess == (( )) ,, ((55ee)) iisseexxaaccttllyytthhee1AA1AA2223-23-ddi=i=mmcUUcUUooGGdGGd00i00in(n(gXXgXXs{{s{{u11u11,,b,,b7474s,,s,,33p33p,,a,,69a69c}}c}}eVVeVVo33o3396f96f)t)tUUhUUheGGeGGc00c00o,o,ddeeDD((00,,00))..HH((e5e5rfrfe)e) ββββ2112 ==== UUUUeeeennnnccccXXXX{{{{2626,,,,7337,,,,7878}}}}UUUUee††ee††nnnncccc,,,, ((((11111111baba)))) ββ33 == UUeennccXX{{33,,44,,66,,99}}UUe†e†nncc,, ((1111cc)) ! 9 ! ! BB00 == UUeenncc((XX66VV6677))UUe†e†nncc,, ((1111dd)) !2 1 !9 7 5 8 ! ! BB11 == UUeenncc XX{{11,,22}}VV3377 UUe†e†nncc,, ((1111ee)) !3 !8 6 ! ! BB22 == UUeenncc(cid:0)(((cid:0)XX44VV3366))UUe†e†(cid:1)n(cid:1)ncc ((1111ff)) 4 ! ! !4 !7 2 ssttaabbiilliizzeetthheeccooddiinnggssuubbssppaacceeooffDD((00,,11)),,ii..ee..,,oonneeoonneehhaanndd (a) !5 !6 (b) 1 0 3 ! oβoβnkn !k||tCtChhµeiµeiννoio(cid:105)tth=h=eerrBBhhlal|a|CnCnddµiµiνtνthih(cid:105)ee==jjoo|i|iCnCntµtiµi+ν+νi1(cid:105)1fefoeoiirgrgeaeanlnllslsppppaoaocscsesesiibobolflfeetthkhke,e,slsle,e,ii6,6,µµoo,,bbννsseaearnrvnvd-d- aabblleess,,wwhhoosseepprroojjeeccttoorriissggiivveennbbyy FFIIGG..11:: TTwwooggrraapphhssGGaa == ((VVaa,,EEaa))aarreesshhoowwnnwwiitthhVV00 == 33 22 {b{ba1a1s,s,e2e2s,s,f.f.o.o.r.r.,t,tw9w9}o}onananonodndnaaVdVd1d1di=it=tiivve{e{0c0c,o,o1d1d,e,e.s.s..(.(.((,9,999,,}1}1.2.2T,T,3h3h)e)e)y)yaapnpnrdrodov(v(i(i(d1d1e0e0,t,th2h2e4e4,g,g3r3ra)a)p)p).h.h--ssttaattee PP11== 11++22ββii 11++22BBii,, ((1122)) ii==11 ii==00 Y(cid:89) Y(cid:89) 3 TABLEI:ThestabilizingobservablesofthenonadditivecodesDam wnoetihcainvgeTrOi(a) =0fori≤2m+5andTrO2(am)+6 =2Nma−1 whosephysicalqubitsarelabeledwithUm ... U1 Va(a=0,1). ∪ ∪ ∪ TheblankentriesstandforsuitableidentityoperatorsIUk orIVa. TrPma = Tr(12+2mO+2(6am)+6) = 232Nma−2m−6. (14) Um Um−1 U2 U1 V0orV1 Thusweobtainthe1-errorcorrectingcodeofparametersex- ··· (0,1) O1 XUm actly as given in Eq.(1). Now we shall demonstrate that its (0,1) coding subspace is 50% larger than the corresponding opti- O2 ZUm (0,1) m malstabilizercodessothatourcodesaregenuinenonadditive O3(0,1) S1m XUm−1 codesthatareneitherequivalenttosomestabilizercodesun- O4 S2 ZUm−1 derlocalunitarytransformationsnorsubcodesofsomelarger (0,1) m m−1 O5 S3 S1 1-errorcorrectingstabilizercodesofthesamelength. ... ... ... ... ThequantumHammingboundfora1-errorcorrectingsta- O2(0m,1−)4 S2mm−6 S2mm−−18 ··· b[[inli,zke,r3c]o],deb,eei.ngg.,inn−trokdu≥ce(cid:100)dloign2i(ti3anlly+1fo)r(cid:101)fthoeransotnab-dileigzeenrecroadtee (0,1) m m−1 O2m−3 S2m−5 S2m−7 ··· XU2 codes,isvalidforbothdegenerateandgeneratecodesofdis- (0,1) m m−1 O2m−2 S2m−4 S2m−6 ··· ZU2 tance 3 and 5 [6] and of a large enough length [19]. In the O2(0m,1−)1 S2mm−3 S2mm−−15 ··· S12 XU1 case of n = Nma we have the quantum Hamming bound (0,1) m m−1 2 n k 2m + 5. This is not enough to prove the nonad- O2m S2m−2 S2m−4 ··· S2 ZU1 − ≥ (0,1) m m−1 2 1 α orβ ditivenessofourcodes. Howeverbyworkingoutanalytically O2m+1 S2m−1 S2m−3 ··· S3 S1 1 1 (0,1) m m−1 2 1 α orβ thelinearprogrammingboundwehave O2m+2 S2m S2m−2 ··· S4 S2 2 2 Theorem If there exists a stabilizer code [[Na,k,3]], de- (0,1) m m−1 2 1 α orβ m O2(0m,1+)3 S2mm+1 S2mm−−11 ··· S52 S31 3 or 3 generate or non-degenerate, with Nma = 22m+35−5 + a and O2m+4 S2m+2 S2m ··· S6 S4 A1 B1 m 0anda=0,1thenNa k 2m+6. (0,1) m m−1 2 1 or ≥ m− ≥ O2m+5 S2m+3 S2m+1 ··· S7 S5 A2 B2 Proof. Givenastabilizercode[[n,k,d]]itsweightdistribu- O2(0m,1+)6 A0orB0 tions{Ai}ni=0isdefinedby 1 A = Tr(P )2 (i=0,1,...,n), (15) i 22k | Eω | hasexactlydimension24, i.e., TrV1P = 24sinceTrV1B0 = |(cid:88)ω|=i 29. An encoding circuit can therefore be designed in a simi- wherethesummationisoverallerrorssupportedon iqubits larmannerasthatof((9,12,3))[13]. Wenotealsothatthis andP istheprojectorontothecodingsubspace. Itisobvious nonadditivecodeisnon-degenerate. thatA 0,A =1,and A =2ssothat A /2s canbe Nowwearereadytopresentourconstruction. Weconsider i ≥ 0 i i { i } regardedasaprobabilitydistributionwiths = n k. Foran NUma qVubiwtsitahndUlabe=lt2h2ekm+3byanddisjVoint=se9t+Uma∪wiUthmk−1∪m..a.n∪d arbitraryfunctionf(x)we(cid:80)denoteitsaverage − 1 a k a ∪ | | | | ≤ a=0,1.Weclaimthatthejoint+1eigenspaceofthose2m+ d 1 6observables{Oi(a)}2i=m1+6witha=0or1asdefinedinTable (cid:104)f(x)(cid:105)≡ 2s f(i)Ai. (16) IisthecodeD(m,a)inEq.(1)withthefollowingprojectoronto (cid:88)i=0 thecodingsubspace Inthefollowingweshallformulateasubsetofthelinearpro- 2m+61+ (a) grammingboundfor1-errorcorrectingcode,whichservesour a = Oi . (13) purposeperfectly. Foracompletesetoflinearprogramming Pm 2 boundseeRef.[8,17]. i=1 (cid:89) Linear Programming bound (Restricted) If there exists a InTableIobservables r aredefinedinEq.(3)and α , , Si { i Aj} stabilizer code [[n,k,3]] then the following conditions hold β , are defined via Eq.(5) and Eq.(11) respectively. { i Bj} true Blank entries represent suitable identity operators. By jux- taposition of some operators in the same row we mean their A = 3n 4x , (17a) 1 (cid:104) − (cid:105) directproduct. 1 Firstofall,these2m+6stabilizingobservablesdetectall A2 = (4x 3n+1)2 3n 1 , (17b) 2(cid:104) − − − (cid:105) 2-qubiterrorsbecausefirstlyallerrorshappenedonU-blocks n orV blockscanbedetectedbecauseallthesubcodesarepure (cid:98)2(cid:99)A2i 2s−1. (17c) 1-errorcorrectingcodesandsecondlyalltwoerrorshappened ≥ i=0 on different qubit blocks can always be detected by the sta- (cid:88) bilizercontaining and forsomek. Thusweobtain Inthecaseofa=0,i.e.,n=N0 withm 0weintroduce XUk ZUk m ≥ a pure 1-error correcting codes of length Na. Secondly, by anonnegativefunctionf(x) = (3n+1 4x)2 anditiseasy m − 4 tocheckthataslongasn 5 happenonsomeU-blockornot. Ifyesweusethedecodings ≥ forGottesmanscodesandifnotthenwehaveonlytodecod- f(0) = (3n+1)2 >(3n+5)(3n 7)+16, (18a) − ingD(0,a) withthedetailedcircuitinthecaseofa = 0being f(1) = (3n 3)2 >4(3n+5), (18b) givenin[12]. − f(2) = (3n 7)2 >2(3n+5)+16. (18c) We acknowledge the financial support of NNSF of China − (Grant No. 90303023, 10675107, and 10705025) and the If there exists a stabilizer code [[n,k,3]] then Eqs.(17a-17c) A*STARgrantR-144-000-189-305. musthold. Asaresult Noteadded. Onfinishingthepaperanotherinfinitefamily ofgenuinenonadditivecodeshasbeenreportedin[22]. f(x) =3n+1+4A +2A , (19a) 1 2 (cid:104) (cid:105) n n (cid:98)2(cid:99) (cid:98)2(cid:99) 16+16A + f(2i)A 16 A 8.2s,(19b) 2 2i 2i ≥ ≥ i=2 i=0 (cid:88) (cid:88) [1] P.W.Shor,Phys.Rev.A,2,2493(1995). wherewehaveusedthatfactthatf(2i) 16since 3n+1,the ≥ 4 [2] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, andW. uniquezerooff(x),isanoddinteger. Puttingallthesepieces K.Wootters,Phys.Rev.A54,3824(1996). togetherweobtain [3] A.Steane,Phys.Rev.Lett.77,793(1996). [4] E.KnillandR.Laflamme,Phys.Rev.A55,900(1997). n [5] D.Gottesman,Phys.Rev.A541862(1996). 2s f(x) = f(i)A (cid:104) (cid:105) i [6] D.Gottesman,arXive:quant-ph/9705052. (cid:88)i=0 [7] A. Calderbank, E. Rains, P. Shor, and N. Sloane, Phys. Rev. n (cid:98)2(cid:99) Lett.76,405(1997). f(0)+f(1)A1+f(2)A2+ f(2i)A2i [8] A.Calderbank,E.Rains,P.Shor,andN.Sloane,IEEETrans. ≥ i=2 Inform.Theory,44,1369(1998). f(0) 16+f(1)A1+(f(2)(cid:88)16)A2+8.2s [9] E.M.Rains,R.H.Hardin,P.W.Shor,andN.J.A.Sloane,Phys. ≥ − − > (3n+5)(3n 7+4A +2A )+8.2s Rev.Lett.79,953(1997). 1 2 − = (3n+5) f(x) 8 +8.2s, (20) [10] E.M.Rains,IEEETrans.Inf.Theory45,266(1999). (cid:104) − (cid:105) [11] J.A. Smolin, G. Smith and S. Wehner, Phys. Rev. Lett. 99, inwhichthestrictinequalitycomesfromthef(0)term. Tak- 130505(2007). ingintoaccountof f(x) > 8weobtain2s > 3n+5, i.e., [12] S.Yu,Q.Chen,andC.H.Oh,arXiv:0709.1780v1[quant-ph] n k 2m+6. (cid:104) (cid:105) [13] S.Yu,Q.Chen,C.H.Lai,andC.H.Oh,Phys.Rev.Lett.101, − ≥ 090501(2008) Inthecaseofa = 1,i.e.,n = N1 withm 0wedefine m ≥ [14] M.GrasslandM.Roetteler,Proc.2008IEEEInt.Symp.onInf. g(x) = (3n+2 4x)(3n 2 4x) which is nonnegative − − − Theory(ISIT2008),300(Toronto,Canada,July2008). on integers because 3n4+2 is an integer. It is obvious that as [15] A.Cross, G.Smith, J.Smolin, andB.Zeng, IEEETrans.Inf. long as n 5 we have g(i) > 2(3n+2) for i = 1,2 and Theory55,433(2009). ≥ mostimportantlyg(0) > (3n+2)(3n 4). Ifthereexistsa [16] D.Gottesman,arXive:quant-ph/9607027. stabilizercode[[n,k,3]]thenEqs.(17a-1−7c)musthold,which [17] E. Rains, IEEE Trans. Inform. Theory 44, 1388 (1998); ibid, leadsto g(x) =3n 4+2A +2A . Asaresultwehave IEEETrans.Inform.Theory45,2361(1999). 1 2 (cid:104) (cid:105) − [18] P.ShorandR.Laflamme,Phys.Rev.Lett.78,1600(1997). 2s g(x) g(0)+g(1)A +g(2)A [19] A.AshikhminandS.Litsyn, IEEETrans.Inform.Theory45, 1 2 (cid:104) (cid:105) ≥ 1206(1999). > (3n+2)(3n 4+2A +2A ) − 1 2 [20] D. Schlingemann and R.F. Werner, Phys. Rev. A 65, 012308 = (3n+2) g(x) , (21) (cid:104) (cid:105) (2001). [21] M. Hein, J. Eisert, and H.J. Briegel, Phys. Rev. A 69. in which the strict inequality sign is due to the g(0) term. 062311(2004). Since g(x) >0wehave2s >3n+2,i.e.,n k 2m+6. (cid:3) (cid:104) (cid:105) − ≥ [22] M. Grassl, P. Shor, G. Smith, J. Smolin, and B. Zeng, arXiv: 0901.1319[quant-ph]. It should be noted that the optimal stabilizer codes of pa- rametersasgiveninEq.(2)existandconstructisalreadygiven by the stabilizers in Table I with the stabilizers acting on qubits V or V being replaced by 6 stabilizers of the pure 0 1 optimalstabilizercodes[[9,3,3]]or[[10,4,3]]. Thestabilizer-likestructuresofourcodessimplifysignifi- cantlytheencodinganddecodingprocedures. Letussuppose we have already the encoding and decoding circuits for the codesD(0,a) andfortheGottesman’scodes. Withsomeaddi- tionalcontrolled-notgatesinfrontoftheencodingcircuitsof theseindividualcodesweobtaintheencodingsofourcodes. Todecodewehaveonlytocheckatfirstthefirst2mgenera- tors in Table I, from which we can be sure wether the errors