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Is A Quantum Stabilizer Code Degenerate or Nondegenerate for Pauli Channel?

Fangying Xiao, Hanwu Chen

have to take the code space to orthogonal space. Abstract—Mapping an error syndrome to the error operator is the The decoding network of quantum code contains three parts, core of quantum decoding network and is also the key step of recovery. namely, error detection, error correction, decoding. Mapping error The definitions of the bit-flip error syndrome matrix and the syndromes to error operators is the core of quantum decoding network phase-flip error syndrome matrix were presented, and then the error and is also the key step to realize quantum error correction. In the third syndromes of quantum errors were expressed in terms of the columns section, we presented the definitions of the bit-flip error syndrome of the bit-flip error syndrome matrix and the phase-flip error matrix and the phase-flip error syndrome matrix, and then expressed syndrome matrix. It also showed that the error syndrome matrices of the error syndromes of all quantum errors in terms of the columns of a stabilizer code are determined by its check matrix, which is similar to the classical case. So, the error-detection and recovery techniques the bit-flip error syndrome matrix and the phase-flip error syndrome of classical linear codes can be applied to quantum stabilizer codes matrix. In the fourth section, it showed that the error syndrome after some modifications. Some necessary and/or sufficient conditions matrices of a stabilizer code are determined by its check matrix, which for the stabilizer code over GF(2) is degenerate or nondegenerate for is similar to the classical case. So, the error detection and correction Pauli channel based on the relationship between the error syndrome techniques of classical linear codes can be applied to quantum matrices and the check matrix was presented. A new way to find the stabilizer codes after some modifications. minimum distance of the quantum stabilizer codes based on their Until now, no technique has been developed to determine whether check matrices was presented, and followed from which we proved or not a stabilizer code is degenerate or nondegenerate other than that the performance of degenerate quantum code outperform (at exhaustive search. In the fifth section, based on the relationship least have the same performance) nondegenerate quantum code for Pauli channel . between the error syndrome matrices and the check matrix of quantum Index Terms—stabilizer codes, CSS codes, degenerate codes, Pauli stabilizer code some necessary and/or sufficient conditions for the channel stabilizer code over GF(2) is degenerate or nondegenerate were presented for Pauli channel. A new way to find the minimum distance I. INTRODUCTION of quantum stabilizer codes based on their check matrices was also presented, and followed from which we proved that the performance Quantum information can be protected by encoding it into a of degenerate quantum code are outperform (at least have the same quantum error-correcting code. Quantum error-correcting codes are performance as) nondegenerate quantum code for Pauli channel. We quite similar to classical codes in many respects, for example, an error hope that these results will helpful for design quantum degenerate is identified by measuring the error syndrome, and then corrected as codes [11],[12] with good performance. appropriate, just as in the classical case. However, there is an interesting class of quantum codes known as degenerate codes [1],[2],[3],[4] possessing a string property unknown in classical codes. II. PRELIMINARY For classical error-correcting codes errors on different bits necessarily The commutator and anti-commutator[1] of the operator A and the lead to different corrupted codewords. But for a degenerate quantum operator B are [A,B] =AB-BA and {A, B} =AB+BA respectively. If [A, code, the error syndrome is not unique, and error syndromes are only B] =0, then the operator A and the operator B are said to commute. If repeated when EF† belongs to the stabilizer S, implying that E and F {A, B} =0, then A and B are said to anticommute. act the same way on the codewords. So they can sometimes be used to Four extremely useful matrices called Pauli matrices with their correct more errors than they can identify. The phenomenon of corresponding notations are described following: degenerate quantum codes is a sort of good news-bad news situation ⎡10⎤⎡⎤⎡⎤⎡⎤ 01 1 0 0−i for quantum codes. The bad news is that some of the proof techniques IXZ===,, , Y = ⎢01⎥⎢⎥⎢⎥⎢⎥ 10 0− 1i 0 used classically to prove bounds on error-correction fall down because ⎣ ⎦⎣⎦⎣⎦⎣⎦ they can’t be applied to degenerate quantum codes, for example, the from which we see that [X, Y] =2iZ, [Y, Z] =2iX, [Z, X] =2iY. quantum Hamming bound. The good news is that degenerate quantum The Pauli group G1 on 1 qubit is the matrix group consisting all of codes seem to be among the most interesting quantum codes. They are the Pauli matrices I, X, Y, Z, together with multiplicative known to outperform all nondegenerate quantum codes for very noisy quantum channel (for example, Pauli channel)[5],[6],[7] and have factors ±1, ±i , i.e., important applications in purifying quantum states[8] and proving the G=±{, I ± iI , ± X , ± iX , ± Y , ± iY , ± Z , ± iZ } security of quantum communication protocols[9],]10]. It is possible 1 that they are able to store quantum information more efficiently than The Pauli group Gn on n-qubit is the group generated by the any nondegenerate code, because distinct errors do not necessarily operators described above applied to each of n qubits in the tensor 2n product Hilbert space C , Xiao Fangying and Chen Hanwu are with the School of Computer Science and Engineering, Southeast University, Nanjing, Jiangsu 210096, China 2

c III. ERROR SYNDROME ⎪⎪⎧⎫gigg=⊗⊗⊗12" gn , Ggn = ⎨⎬, ⎪⎪gIXZYcj ∈∈{}{}, , , , 0,1,2,3 ⎩⎭Suppose SMMM= 12,,," nk− is the stabilizer of a where i =−1 . quantum stabilizer code CS(). Let the encoded state φ was It follows from the definition of the Pauli group that if AB, ∈ Gn ， then [,]AB = 0or {,}AB = 0. Since Y= iZX, so any element of suffered from some inference denoted E ∈Gn during transmitting

n on a noisy channel which transmitted the encoded state to ab G can be denoted as g =⊗iXZc jj ⋅ , where a ,bF. n ()ij∈ 2 ′ (say). The error detection can be done by simply j=1 φ = E φ A homomorphism from G onto F 2n defined as measure the generators of the stabilizer, as shown in Figure 1 [1]. It n 2 will give us a list of eigenvalues, the error syndrome, which will tell us 2n whether the error E commutes or anticommutes with each of the ϕ : GFn 6 2 (1) generators. that is,ϕαβϕϕggg=  , where () ()()()XZ() From figure 1 we see that if [EM,0i ] = , then si = 0 ; if n c ab jj , ， EM,0= , then s =1 . g =⊗iXZG() ⋅ ∈n c ∈{0,1, 2,3} { i } i j=1 12 nk−− nk n The vector ssssEEEE=∈,,," F2 represents the α = (,aa12 ," , an )and β =∈(,bb12 ," , bn ) F 2. ( ) It follows from the above definition that the Pauli matrices I, X, Y, Z can be mapped to the following binary vectors: 0 s1 ϕ()IXZY==== (0|0),ϕϕϕ ( ) (1|0), ( ) (0|1), ( ) (1|1) . 0 s2 2n An isomorphism from F onto G n define as: 2 # # # } 2n Gn 0 snk− ϕ′ :,FGG6 nn= (2) 2 {}±±1, i φ′ " φ′ n n ab that is, ϕαβ′()()=⊗()XZjj ⋅ . Fig. 1. Syndrome measurement. j=1 error syndrome of the error E ∈G , where if ⎡⎤, n ⎣⎦EM,0j = For example, ϕ(iXIZYXZY )= (1001101 0011011) and then j ; if , then j where, . sE = 0 {EM,0j } = sE = 1 j =−1, 2," , nk ϕ′(1001101 0011011) = XIZYXZY . ⊗(1)ini−⊗− ( ) The operator IXI⊗⊗ denotes the bit-flip error on Let g ∈ Gn , such that ⊗−(1)ini ⊗ ( − ) the i-th qubit, written X i , i.e., XIi =⊗⊗ XI , then ϕϕϕ()ggg= () () | () , then the quantum weight of g, X Z 12 nk− nk− the error syndrome of X is ssss=∈,,," F. i XXXXiiii()2 written wgQ (), is the number of components that is not I, that is Set the rows of s to be ss,,," s , then X XX12 Xn wgwgQXZXZ()=+−⋅ (())ϕϕϕϕ wg (()) wg (() ()) g where w(u) is the Hamming weight of u and“.”is the inner product. the nnk×()− matrix 12 nk− The symplectic weight of a vector vvv= ()12| , where ⎡⎤s ⎡⎤ss" s X1 XX11 X 1 n ⎢⎥⎢⎥ vv, ∈ F, written wv(), is s ss12" snk− 12 2 s ⎢⎥X 2 ⎢⎥XX X nnk×()− 22 2 sFX ==⎢⎥ ∈2 w() v=+−⋅ wv ( ) wv ( ) wv ( v ). ⎢⎥# ##"# s 1212 ⎢⎥⎢⎥ 12 nk− As a result, wgQs()= wϕ () g . ⎢⎥sX ⎢⎥ss" s () ⎣⎦n ⎣⎦XXnn X n Let MM,,," M are the generators of the stabilizer S, is called the bit-flip error syndrome matrix(BSM). { 12 nk− } More generally, if the bit-flip errors happen to multiple qubits, where M in∈G (1≤≤ink − ) , H is an ()2nk−× n matrix n c a written X v , where , . over GF(2) whose rows contains the vectors EXv= iX∏ cX ∈{0,1,2,3} aFv ∈ 2 , that is, v=1 ϕϕ()()MM12,,," ϕ ()M nk− n Suppose aaaa=∈[ 12,,," n ] F 2, if av =1, then a bit-flip error ⎡⎤ϕ ()M1 ⎢⎥ happens to the v-th qubit; if , then no error happens to the v-th HHH= # (3) av = 0 ⎢⎥[]X Z ⎢⎥ qubit. Since ⎣⎦ϕ ()M nk− H is called the check matrix[1] of S. 3

n the error syndrome s of Z (1,2,,)ln= " . cX av Zl l EMXj⋅= i X v ⋅ M j ∏ ⊗−(1)mnm ⊗− ( ) v=1 Suppose YIm =⊗⊗ YI (1≤≤mn) which is j n−1 asn ⋅ the case of that two kinds of errors (bit-flip and phase-flip) happened ()Xn cX aavn =− ( 1) iXMX∏ vjn ⋅ ⋅ contemporarily on the m-th qubit. Since Y=iZX, so v=1 (5) n YiZXmmm= ⋅ as⋅ j n ∑()u Xu uv=1 cX a It follows from Eq (5) that an arbitrary error EPn∈G can be =− ( 1) Mijv ⋅ ∏ X v=1 expressed as n nn as⋅ j cP abvl ∑()u Xu E =⋅iX Z u=1 Pvl∏ ∏ =− ( 1) MEjX ⋅ vl==11 T j jj j n where abvl,∈∈ Fc2 , P { 0,1, 2,3} . Let aaaa= [ 12,,," n ] Let ssssXXXX=∈⎡⎤,,," F2 , then ⎣⎦12 n n n and bbbb=∈[ 12,," ,n ] F 2, such that j jj sasasEuXX=⋅=⋅ nn Xu∑() c ab u=1 P vl EMPj⋅= i∏∏ X v ⋅ Z l ⋅ M j n vl==11 cX av Therefore, the error syndrome of E = iXis nn X ∏ v j v=1 bs⋅ ZPc abvl =− ( 1) iXMZ∏∏vj ⋅ ⋅ l 12 nk− sasasasas=⋅⎡⎤,,, ⋅" ⋅ =⋅ (4) vl==11 EXXXX X ⎣⎦ nn bs⋅+⋅jj as ⎛⎞ It follows from Eq (4) that the error syndrome s of any =− ( 1) ZXM ⋅iXc P abvl ⋅ Z EX jvl⎜⎟∏∏ n ⎝⎠vl==11 c a X v bs⋅+⋅jj as error EXv= iX can be expressed as a linear combination of ZX ∏ =− ( 1) MEjP ⋅ v=1 Then the error syndrome s of X (1,2,,)vn= " . Xv v j jj s = as⋅+⋅ bs ⊗−(1)ini ⊗ ( − ) EXZP The operator Z IZIin(1 ) denotes i =⊗⊗≤≤ nn c ab the phase-flip error on the i-th qubit. Similarly, the matrix Therefore，the error syndrome of P vl is EPvl=⋅iX∏∏ Z ⎡⎤ss12" snk− vl==11 ZZ11 Z 1 ⎢⎥ s= asbsasbsasbs ⋅1122 +⋅,,, ⋅ +⋅" ⋅nk−− +⋅ nk 12 nk− EXZXZXZP ( ) ⎢⎥ssZZ" s Z nnk×−() 22 2 12nk−− 12 nk sFZ =∈2 =⋅as,,, as ⋅"" as ⋅ +⋅ bs ,,, bs ⋅ bs ⋅ (6) ⎢⎥##"# ()()XX X ZZ Z ⎢⎥ =⋅as +⋅ bs ⎢⎥ss12" snk− XZ ⎣⎦ZZnn Z n It follows from Eq (6) that the error syndrome sE of any j P where if ⎡⎤ZMij,0= then sZ = 0 and if {ZMij,0} = ⎣⎦ i error EPn∈G can be expressed as a linear combination of the error j then sZ =1 , in=1, 2," , , j =−1, 2," , nk, is called the syndrome s of Z (1,2,,)ln= " and the error syndrome s of i Zl l Xv phase-flip error syndrome matrix (PSM). " . If the phase-flip errors happen to multiple qubits, that X v (1,2,,)vn= n Remarkably, suitable error syndrome measurements would c b is Z l , where . collapse an arbitrary error (including coherent superpositions of EZ = iZ∏ l bFclZ∈∈2 ,0,1,2,3{ } l=1 bit-flip and phase-flip errors) into the discrete set of only bit-flip n and/or phase-flip errors, because it can be expressed as a superposition Let bbbb=∈[ 12,," ,n ] F 2, Similarly, the error syndrome of of basis operations—the error basis (which is here given by the Pauli n matrices). And these discrete Pauli errors can be easily reversed to c b Z l is recover the original state. So, it only has to find out the error EZ = iZ∏ l l=1 syndromes of the bit-flip error X v (1≤≤vn ) and the phase-flip 12 nk− sbsbsbsbs=⋅⎡⎤,,, ⋅" ⋅ =⋅ EZZZZ ⎣⎦ Z error Zl (1≤ ln≤ ) on 1-qubit, and then the error syndrome of an arbitrary error can be expressed in terms of the columns of the bit-flip From which we see that the error syndrome sE of any Z error syndrome matrix and the phase-flip error syndrome matrix n c b according to Eq (6). error Z l can be expressed as a linear combination of EZ = iZ∏ l From previous analysis we knew that the errors in the l=1 4

set {±±Es , iEs} can be mapped to the same error Therefore, the error syndrome of an arbitrary error E ∈Gn can be nk− expressed in terms of the columns of the check matrix, i.e., syndrome sF∈ 2 , since from an observational point of view the T quantum states corrupted by the errors with different global phase sEHHEZX=Λ⎡⎤ϕ()( ⎣⎦) factor are identical. For this reason we may ignore the global phase T factors as being irrelevant to the observed properties of the physical = ⎡⎤⎡⎤ϕϕ (EEHH )XZZX ( ) () system and having no affect to the recovery. In the rest of the article ⎣ ⎦⎣ ⎦ TT we will only consider the errors in Gn . =+ϕϕ (EH )XZ ( EH ) ZX

⎡ 0 In ⎤ where and . IV. ANALYZE THE ERROR SYNDROME OF STABILIZER CODES Λ=⎢ ⎥ ϕϕϕ()EEE=⎡⎣⎦ ()XZ () ⎤ ⎣In 0 ⎦ Suppose CS() is a stabilizer code with It has come to light that the error-detection and recovery of the stabilizer SMMM= 12,,," nk− whose check matrix classical codes are totally dependent on their parity check matrices. Theorem 1 makes a connection between the error syndrome matrices is . Whether a bit-flip error commutes (BSM and PSM) and the check matrix of the stabilizer code. Thereby, H()nk−×2 n=⎡⎣⎦HHXZ ⎤ X i expressing the error syndrome as a linearly combination of the with or anticommutes with M j dependent on whether or not the i-th columns of BSM and PSM is transformed into expressing it as a linearly combination of the columns of the check matrix. So, the component of M j is Z or Y. If M ji = Z or Y, then{XMij,0} = , error-detection and recovery of the quantum stabilizer codes are also totally dependent on their check matrices, which is similar to the otherwise ⎡⎤. It follows from Eq (3) that the syndrome ⎣⎦XMij,0= classical case. Therefore, the error-detection and recovery schemes and techniques of the classical codes can be applied to quantum of the bit-flip error in the i-th qubit, X i (1≤≤in ) , corresponds to stabilizer codes easily. the n+i-th column of H which is also the i-th column of the V. IS A STABILIZER CODE DEGENERATE OR NONDEGENERATE? sub-matrix H Z . Similarly, the syndrome of the phase-flip error in the In this section we describe some necessary and/or sufficient i-th qubit, Z (1≤≤in ) , corresponds to the i-th column of H which i conditions for a stabilizer code over GF(2) is degenerate or nondegenerate based on its check matrix for Pauli channel which can is also the i-th column of the sub-matrix H X . That is, also be applied to depolarization channel. ϕ ()MHjni++ jni, XMi j=−(1) MX ji =− (1) MX ji (7) Strictly speaking, degeneracy is not a property of a quantum code alone, but a property of a code together with a family of errors it is ϕ ()MHji ji ZiMMZMZ j=−(1) ji =− (1) ji (8) designed to correct [5]. Here we will show a definition of degenerate and nondegenerate stabilizer codes for Pauli channel. whereϕ M is the i-th component of the 2n-dimentional binary ()j i Let , consider a mapping GgGwgtnt| = { ∈≤ n | Q () } vectorϕ ()M j and H ji is the element at row j, column i of matrix 2n from G nt| to F2 : H,1,12≤≤−j nk ≤≤ i n. It follows from (7) and (8) that, for a nk− f : GFnt| 6 2 stabilizer code CS()the error syndromes of X i and Zi are nk− that is, for ∀∈g Gnt| , f ()gvvv=∈()12 ,," ,nk− F 2 . If sHH= ,,," H Xnininknii ()1,++ 2, −+ , [gM,0i ] = then vi = 0 , if {gM,0i } = then vi =1 . If f is sHHH= ,,," Zi ()12ii nki− , nonsingular ， we say that the stabilizer codes with parameters j j where sH= and s = H . nk,,21 t+ and nk,,22 t+ are nondegenerate, otherwise, Xi jn, + i Zi ji aba b Let they are degenerate. 12nn 12 In the rest of this section we only discuss the stabilizer codes with H ==()HHXZ|() HH XX "" HHH XZZ | H Z parameters nk,,21 t+ , and all the conclusions can also be i ab where H X (1≤≤in ) is the i-th column of H X and applied to stabilizer codes with parameters . i abnk,,21 t+ H Z (1≤≤in ) is the i-th column of H Z . Then Lemma 1. If the set of vectors vaaa= { ,,," } is linearly i T i T 12 2m sH= , sH= XZi ()ZXi () independent, then the sum of any set of m vectors of v are different The following theorem now follows easily after the discussion from each other. above. Proof: suppose the vectors aa,,," a and the ii2 im Theorem 1. If the check matrix of a stabilizer code CS() is 1 vectors aa,,," a , where T T j12jjm ()H X | H Z , then sHX = Z and sHZ = X . 5

then, and . {ii12,,"" , imm}∩<{ j 1 , j 2 , , j} m, satisfy 0(),()≤ wu12 wu≤ t 1(≤⋅≤wu12 u ) t mm Therefore, wQ () E= wu ()1212+−⋅≤ wu ( ) wu ( u ) t. ca= ca′ , with c =1and cc,0,1′ ∈ ∑∑kikl l j p kl { } It follows from Eq (2) that, if we pick different columns from kl==11 mm [H X | H Z ] which corresponds to different sets {ii12,," , im} Then, we have ca+= cb′ 0 . Thus v is fails to be ∑∑kikl lj kl==11 and { jj12,,," jp } , then they will correspond to different linearly independent. Therefore, we produce a contradiction. □ error E ∈Gnt| . Finally, combining (i) and (ii) we get that the stabilizer code C(S) is Theorem 2. Let HH| be a check matrix such that any [ XZ]nnk×−2( ) nondegenerate. □ 4t columns linearly independent, then the stabilizer code Note that the reverse of theorem 2 is generally not true. That is, there would exist a set of 4t linearly dependent columns of C(S)=[[n,k,2t+1]] with check matrix[H X | H Z ] is nondegenerate. HH| even if the stabilizer code with parameters Proof: Since C(S) is nondegenerate if (and only if) for [ XZ]nnk×2(− ) [[n,k,2t+1]] is nondegenerate. Because if the 4t linearly dependent arbitrary E , EG∈ nt| and E ≠ E , they satisfy ss≠ . We 12 12 EE12 columns satisfy the following conditions: will perform two steps in order to prove the theorem. (i) Let us assume as before that 0,4≤ mp≤ t, {}ii12,,"" , imp+={ jj 1 , 2 , , j} 4t and 12nn 12 ()H XZ|HHHHHHH= () XX "" XZZ| Z , ti< {}11,,"" imp∩<{ jj ,,} 2t . assume further that vmp, is the sum of the ii12th, th," , im th Then, at least one of the two corresponding errors that have the same error syndrome has quantum weight lager than t, which does not columns of H and the j th,jj th," , th columns of H . X 12 p Z violate the definition of nondegenerate code. m p We have the following corollary. ijnk− That is ab. Since any vHHFmp,2=+∈∑∑ X Z 4t Corollary 1. if the stabilizer code C(S)=[[n,k,2t+1]] is nondegenerate, ab==11 then any 2t columns of its check matrix [H X | H Z ] are linearly columns of are linearly independent, it follows from [H X | H Z ] independent. lemma 1 that if m and p satisfy12≤+≤mp t, then we will get Proof: Suppose the ii12th, th," , im th columns of H X and the different v with respect to different mp, jj12th, th," , jp th columns of H Z are linearly dependent, sets{ii12,," , im} and{ j12,,,jj" p } . where ii12,," , im and j12,,,jj""p ∈{ 1,2,, n} and they (ii) Let satisfy the conditions 0,2≤≤mp t and 2n uuu==()12(){, jj 12 ,"" , jpm }|{,, iii 12 ,} ∈ F 2 , {}ii12,,"" , imp+ { jj 1 , 2 , , j} = 2t . Then n where u1 and uF22∈ , {ii12,," , im} and j12,,,jj" p m p { } ij ab (9) vHHmp, = ∑∑ X+= Z 0 are supports of u1 and u2 respectively. Therefore ab==11 m p T T ijab Pick a subset ii,,," i from {ii,," , i} with a elements vHHuHH=+ = { kk12 ka } 12 m mp, ∑∑ X Z( () Z X ) ab==11 and a subset jj,,," j from jj,,," j with b T { ll12 l} { 12 p} =+uHTT uH ∈ F()nk− b ()12ZX 2 elements, where a+b=t. Let It follows from Eq (2) that, the operator corresponding to u is ba p nn m Eab, =⋅XZ j i uu ∏ lkcd∏ E ==ϕ′()uZ21kl ⋅ X = ZX ⋅ cd==11 ∏∏kl ∏∏ ijab kl==11 ab == 11 E =⋅XZ T mapb−−, ∏ c∏ d cjj∈−{}12,,,"" jplll{} jj , ,, j dii ∈−{} 12 ,,, "" i mkkk{} ii , ,, i So, the error syndrome of E , sE , is ()vmp, . 12ba 12 Then, wE≤ t, wE≤ t If the sets {ii12,," , im} and { jj12,,," jn} satisfy the Qab( , ) Qmapb( −−, ) following conditions: It follows from Eq (9) that, Eab, and Emapb−−, possess the same 0,≤≤≤+≤mp t ,1 m p 2 tand error syndrome. Therefore, we produce a contradiction. □ ii,,"" , i ∪≤jj , , , j t {}12mp{ 1 2 } The theorem 2 and corollary 1 are the sufficient condition and necessary condition for stabilizer codes to be nondegenerate 6 respectively. We have the following theorem that describes a t ⎛−⎞⎛⎞nnjtj− ⎛22 ⎞ necessary and sufficient condition for a stabilizer code to be in A is ∑∑⎜⎟⎜⎟⋅ ⎜ ⎟, then for any element vA∈ , nondegenerate. jl==01⎝⎠⎝⎠jl ⎝ ⎠ Theorem 3. A stabilizer code C=[[n,k,2t+1]] is nondegenernate if and only if the number of elements in the set we can find a unique pair of ii12,," , im and jj12,,," jp such

⎧⎫T mpij 0,0≤≤mt ≤≤ pt that vHH=+ab. So the error mapped to the ⎪⎪∑ ab==11X ∑ Z ⎪⎪mp HHiiijjjnijab+∈,,"" , ,, , , 1,2,, " ⎨⎬()∑∑ab==11XZ12 m 1 2 p{} error syndrome v is Ejjjiii= ϕ′ ,,,"" ,,, . ⎪⎪(({ 12pm} {} 12 )) ⎪⎪0,,,<∪{}ii12"" imp{} j 1 ,,, j 2 j ≤ t ⎩⎭Because the set A contains all the vectors that corresponding to pairs t t tj− ⎛⎞n i ⎛−⎞⎛⎞nnj ⎛22 ⎞ of ii,," , i and jj,,," j which satisfy is ∑⎜⎟3 or ∑∑⎜⎟⎜⎟⋅ ⎜ ⎟. 12 m 12 p i=1 ⎝⎠i jl==01⎝⎠⎝⎠jl ⎝ ⎠ 1,,,,,,≤ wjjjsmm({}12"") ({} iii 12 ) ≤ t. Thereby, the Proof: (a) For Pauli channel, a stabilizer code C is nondegenerate if ( ) (and only if) the different errors in its correctible error set collection of the errors E that corresponding to pairs of ii12,," , im GInt| −{ } are mapped to different error syndromes. Since the and jj12,,," jp is GInt| −{ } . Finally, C is a nondegenerate code. t ⎛⎞n i □ number of elements in GInt| −{ } is GInt| −={} ⎜⎟3 . By theorem 2, we have the following corollary. ∑ i i=1 ⎝⎠ Corollary 2. Let H be a check matrix such that any 4t columns are linearly independent, but there exists a set of 4t linearly dependent Let E is an arbitrary operator of GInt| −{ } and suppose columns, then the stabilizer code C(S) defined by H has the minimum

ϕ()E ==()(uv | uu12"" unn | vv 12 v ) with distance dt≥+21. If C(S) is nondegenerate, then its minimum distance satisfies . uu====" u1 and vv====" v1 , 21tdt+ ≤≤ 41 + ii12 im jj12 jp Proof: The minimum distance of C(S) is the quantum weight of the where iijj,,,,,""∈{ 1,2,, " n} . Then, ii,," , i element in N(S)-S with the minimum quantum weight, where N(S) is 11mp 12 m the normalizer of S. Without lost of generality, suppose the and j ,,,jj" satisfy the following conditions 12 p ii12th, th," , im th columns of H X and the columns of are linearly dependent, 0,≤≤mp t , 0,,,<∪{}ii12"" imp{ j 1 ,,, j 2 j} ≤ t. jj12th, th," , jp th H Z m p Thus ij i.e., ab. Then, we have 0,0≤≤mt ≤≤ pt vHHmp, = ∑∑ X+= Z 0 mp,0≥ ⎧⎫ab==11 ⎪⎪iijj,,,,,""∈ 1,2,, " n ⎪⎪11mp{}and mp+ ≥+41 t . Let us assume as before ′ GInt| −={}⎨⎬ϕ (|) ( uv ) ()uv|,,,,= (){} j11"" jpm{} i i ⎪⎪that Ejjjiii= ϕ′ { 12,,,""pm} {} 12 ,,, , ⎪⎪(( )) ⎩⎭1|≤≤wuvts ()() then E ∈ NS()and TT The error syndrome of E is suHvHEZX=+. Then, the error wE= w jj,,,"" j ii ,,, i≥+ 2 t 1 Qs() ({ 12 p} {} 12 m) syndromes of the errors in GInt| −{ } are ( ) Let us consider the case with m + p = 4t+1: ⎧⎫0,0≤≤mt ≤≤ pt T Case (i): If , then ⎪⎪mpij j12,,,jjiii""pm∩{} 12 ,,,= 2 t AHHiiijjjn=+ab,,"" ,, , , , ∈ 1,2,, " { } ⎨⎬()∑∑ab==11XZ12 m 1 2 p{} ⎪⎪ 0,,,<∪ii"" i j ,,, j j ≤ t wEQ ( ) = 21 t+ ; ⎩⎭{}12mp{} 1 2 The number of elements in A satisfies if { j12,,,jjiiit""pm}∩{} 12 ,,,< 2, then t ⎛−⎞⎛⎞nnjtj− ⎛22 ⎞ n ≤⋅ (10) A ∑∑⎜⎟⎜⎟ ⎜ ⎟ wEQ ( ) >+21 t . jl==01⎝⎠⎝⎠jl ⎝ ⎠ Case (ii): If {}ii,,"" , i∩ j , j , , j = 0, then If the different errors in GInt| −{ } are mapped to different error 12mp{ 1 2 } syndromes, then wEQ ( ) = 41 t+ ; t ⎛−⎞⎛⎞nnjtj− ⎛22 ⎞ if ii,,"" , i∩ j , j , , j > 0, then GInnt| −=={} A ∑∑⎜⎟⎜⎟ ⋅ ⎜ ⎟ {}12mp{ 1 2 } jl==01⎝⎠⎝⎠jl ⎝ ⎠ (b) Conversely, it follows from Eq (10) that, if the number of elements wEQ ( ) < 41 t+ . 7

distance dt≥+21. If C is nondegenerate, then its minimum It follows from case (i) that wEQ ()>+21 t . Then we distance is d = 2t +1. have dt≥+21. Proof: Our corollary follows from theorem 2 and corollary 3. If C(S) is nondegenerate, then □ minwee ( ) |∈−≤ NS ( ) S min we (′ ) | e ∈ S. The above assertion is similar to the classical case that the { QQ} { } minimum distance of a classical linear code is determined by its parity From which we see that dt≤+41. Combining case (i) and case (ii), check matrix, since that the CSS codes detect and correct quantum errors by making use of the error-correcting properties of the classical we get 21tdt+≤ ≤ 41 +. □ codes. It is easy to see that some sequence of elementary row We know from corollary 2 and corollary 4 that the degenerate transformations and column permutation will transform the check quantum stabilizer codes outperform the nondegenerate quantum matrix H into the following standard form [3]: stabilizer codes which had proved in reference [5]. rnkrkrnkrk - - - -

r {⎡⎤ I A12 A B 0 C VI. CONCLUSION R ==⎡⎤⎢⎥⎣⎦RRXZ nrk−−{0⎣⎦ 0 0D I E We have presented some necessary and/or sufficient conditions for a stabilizer code over GF(2) is degenerate or nondegenerate for Pauli From which we see that the stabilizer code C with check matrix R R channel. It would be interesting to investigate the necessary and/or sufficient conditions for a stabilizer code over GF(q) is degenerate or is isomorphic to the stabilizer code CnktH =+a ,,21bwith check nondegenerate. It would be also interesting to investigate the matrix H. Therefore, CH is nondegenerate if (and only if) CR is necessary and/or sufficient conditions for a stabilizer code over GF(q) is degenerate or nondegenerate for other quantum channels. It already nondegenerate. Thus, to demonstrate that is nondegenerate it CR proved that all CSS codes with alphabet size q>4, where q is a prime only has to show that C is nondegenerate. power, must obey the Hamming bound [13]. Some special cases of H interest are stabilizer codes and CSS codes over a nonprime power Theorem 4. If r = n - k and there exists a column of B with Hamming alphabet and small alphabet q <5. We hope our work will helpful for weight less than or equal to t-1, then CR is degenerate. solving this problem and for design quantum degenerate codes with Proof: Follows from theorem 2. □ good performance. CSS codes are a special kind of stabilizer codes with the check matrix of the following form: REFERENCES ⎡⎤A 0 [1] M.A.Nielsenand1.L.Chuang, Quantum Computation and Quantum Information. Cambridge, Britain: Cambridge University Press, 2000. HHH=⎡XZ ⎤=⎢⎥ ⎣⎦0 B [2] D. Gottesman, “Class of Quantum Error-correcting Codes Saturating the ⎣⎦ Quantum Hamming Bound,” Phys. Rev. A, vol. 54, no.3, pp.1862~1868, It is easy to see that, none of the columns of H can be expressed Sept., 1996. X [3] D. Gottesman, “Stabilizer Codes and Quantum Error Correction,” Ph.D. as a linear combination of the columns of H . Therefore, by theorem dissertation, California Institute of Technology, Pasadena, CA, 1997. Z [4] Calderbank A R, Rains E M, Shor P W, et al. “ Quantum error correction 2, we have the following corollary. via codes over GF(4),” IEEE Trans. Inf. Theory, vol. 44, no.4, Corollary 3. The CSS code C= [[n,k,2t+1]] is nondegenerate if and pp.1369-1387, Jul., 1998 [5] G. Smith, J. A. Smolin, “Degenerate Quantum Codes for Pauli only if any 2t columns of H X and H Z are linearly independent Channels ,” Phys. Rev. Lett., vol. 98, no.3, p. 030501, Jan., 2007. respectively. [6] N. J. Cerf, “Pauli Cloning of a Quantum Bit ,” Phys. Rev. Lett., vol. 84, Proof: The condition is clearly sufficient. Then we will prove that it no.19, pp. 4497-4500, May, 2000 [7] D. P. DiVincenzo, P. W. Shor, J. A. Smolin, “Quantum-channel capacity is necessary. Without lost of generality, suppose the of very noisy channels,” Phys. Rev. A, vol. 57, no.2, pp.830-839, Feb., 1998. ii12th, th,"" , itt th, j 1 th, j 2 th, , j th columns of H X are [8] K. H. Ho, H. F. Chau, “Purifying Greenberger-Horne-Zeilinger states ttij linearly dependent, then we have HHab+=0 . using degenerate quantum codes,” Phys. Rev. A, vol. 78, no.4, p.042329, ∑∑ab==11XX Oct., 2008. [9] Hoi-Kwong Lo, “Proof of unconditional security of six-state quantum key Let the errors E1 and E2 be define as distribution scheme,” Quantum Information and Computation, vol. 1, t t no.2, pp.81-94, Apr., 2001. E = Z and E = Z respectively. [10] P. W. Shor, J. Preskill, “Simple Proof of Security of the BB84 Quantum 1 ∏ ik 2 ∏ jk Key Distribution Protocol,” Phys. Rev. Lett., vol.85, no.2, pp.441-444, k =1 k =1 Jul., 2000. Thus, wE== wE t. It is easy to see that [11] R. Daskalov, P. Hristov, “New Quasi-Twisted Degenerate Ternary Linear QQ()12 () Codes,” IEEE Trans. Inf. Theory, vol. 49, no.7, pp.2259-2263, Sept., T t i T t j 2003. s = H a and s = H b . Thus, E and E have EX1 ∑ a=1 EX2 ∑b=1 1 2 [12] S. A. Aly, A. Klappenecker, P. K. Sarvepalli, “Remarkable Degenerate Quantum Stabilizer Codes Derived from Duadic Codes,” quant- the same error syndrome. Therefore, we produce a contradiction. ph/o601117, Jan., 2006. □ [13] P. Sarvepalli, A. Klappenecker, “Degenerate quantum codes and the Corollary 4. Let H be a check matrix such that any 2t columns are quantum Hamming bound,” Phys. Rev. A, vol. 81. no.3, p.032318, Mar., linearly independent, but there exist a set of 2t linearly dependent 2010. columns, then the CSS code C defined by H has the minimum