A family of asymptotically good quantum codes based on code concatenation

A family of asym ptotically good q uantum codes based on code concaten ation Zhuo Li , ∗ Li-Juan Xing, and † Xin-Mei Wan g State Key Laboratory of Integrated Serv ice Networks, Xidian University, Xi’an, Shannxi 710071, China We explicitly construct an infinite fa mily of asymptotica lly good concatenated quantum stabilizer codes where the o uter co de uses CSS-type quantum Reed-Solomon code and the inner code uses a set of speci al quantum codes. In the field of quantum error-correcting codes, this is the first time that a fa mily of asymptotically good quantum codes is derived from bad codes. Its occurrence supplies a gap in quantum coding theory. PACS numbers: 03.67.Pp, 03.67.Hk, 03.67.Lx Quantum error correction is a bas ic technique for transmitting quantum inform ation reliably over a noisy quantum channel. Many explicit constructions of quantum error-correcting codes have been proposed so far [1-12]. Some of the best-known code constructions are the CSS code construction of Calderbank and Shor [1] and Steane [2] and the stabilizer code construc tion of Gottesm an [3] and Calderbank et al. [4, 5]. As in classical coding theory, we want to construct quantum codes with large minimum distance. More generally, we want to construct asymptotically good 1 quantum codes with both rate and di stance/length bounded away from zero. Ashikhmin et al. [13] and Chen et al. [ 14] constructed asymptotically good quantum codes based on algebraic geometry codes. Later, Matsumoto [15] improved the bound of Ashikhmin et al. [13]. In classical coding theory, code concatenation [16] is a basic method for constructing good error-correcting codes and most of the known asymptotically good binary codes are constructed by code concat enation [17]. In the quantum setting, code concatenation is also effectively used to construct good quantum error-correcting codes, although concatenation is mainly us ed for fault-tolerant quantum com putation [18]. Gottesman states code concatenation in his PhD thesis and gi ves the stabilizer of a quantum code constructed by concatenat ing the five-qubit c ode with itself. Calderbank et al. [5] also remark concat enated codes and Rains [19] proves the so-called product bound of concatenated codes. In this paper we derive an infinite fa mily of asymptotically good binary quantum stabilizer codes from quantum Reed-Solom on (RS) codes, which may be thought of as concatenated quantum codes where the out er code is CSS-type quantum RS code and the inner codes use distinct quantum codes. Firstl y we give the structures of the stabilizer and normalizer. Then we show that this fam ily of codes is asymptotically good. These codes are distingu ished by being the first fam ily of codes we have seen with the property that good quantum codes are obtai ned from bad codes. For this we first show that the long binary quantum codes obtained from RS codes are bad briefly. Let be an N C ,( 2 ) , 1 mN m N K K − +  binary quantum code obtained 2 from RS code [20]. If the rate (2 ) (2 ) mN K m N N K N =− = − is held fixed, the ratio distance le ngth ( 1 ) Km =+ N C approaches zero as . However, by a very clever construction it is possi ble to obtain an infinite family of good binary quantum codes from RS codes, as we now show. m →∞ Lemma 1 [12]. (CSS codes) Let be an [, weakly self-dual code over and let . Then there exists a quantum stabilizer code encoding C ] q nk () GF q min{ ( ) : \ } dw g t v v C ⊥ =∈ 2 nk − qudits into qudits with minimum distance , denoted by with stabilizer n d ,2 , q nn k d −  CC = × S and normalizer CC ⊥ ⊥ = × N . The starting point is a C SS-type quantum RS code 2 2 ,2 , 1 m NN K K − +  with stabilizer and normalizer RS =× SR R RS ⊥ ⊥ =× N RR where and are classical RS codes over with for 2 2 [, , 1 ] m NKN K =− + R 2 2 [, , 1 ] m NN KK ⊥ =− + R 2 (2 ) m GF ⊥ ⊆ RR 2 2 m N 1 = − and 2 KN ≤ ⎢ ⎥ ⎣ ⎦ . Let α be a primitive element of and let 2 (2 ) m GF 1 ,, m 2 β β … be a self-dual basis of over . Let , , be a typical codeword of with 2 (2 ) m GF (2 ) GF 01 2 1 (, , | , , ) NN N aa a a a −− = …… 2 (2 ) m i aG F ∈ j aa RS S 2 , 1 m ii j j β = = ∑ , (2 ) ij aG F ∈ ) [( ) ] mm i ij j N i i m N ij j ij m j jj ba s a s , . Let be the binary vector b 0, 1 0,4 2 1 , 1 1 ,4 2 0, 1 0,4 2 1 , 1 1 ,4 2 (, , ; ; , , | , , ; ; , , m N Nm m N Nm bb b b b c c c c + − −+ + − −+ = …… … …… … such that 22 ,, 1 , 1 1 , , 12 1 m j m β αβ β − ++ + − == =+ + + ∑∑ ,2 1 , 1 , 1 im i i ba s + =+ [( ) ] mm i i m jj N i m i m N i m jj i j m j jj ba t a t β = + ∑ , , 2 2 ,2 1 , 1 , 1 1 , , 12 1 m j m β αβ β − ++ + + + + + − == =+ + + ∑∑ ,4 2 , 1 , 1 im i m i ba t ++ =+ 1 11 () mm i ij j ij j i m m jj ca s βα β β ++ == =+ ∑∑ ,2 1 , 1 im i m cs β = + ∑ , , 2 ,, , 1 , 3 + + = , 2 ,2 1 , , 1 1 11 () mm i i m j j im j j im m jj ca t βα β β ++ + + + == =+ ∑∑ , ,4 2 , 1 im i m ct + + = , where for 01 , ,, ,( ij ij st G F ∈ 2 ) iN ≤≤ − 11 jm ≤ ≤+ . Let , , be a typical codeword of 01 2 1 (, , | , , ) NN N aa a a a ∗∗ ∗ ∗ ∗ −− = …… 2 (2 ) m i aG F ∗ ∈ RS N with 2 , 1 m ii j j j aa β ∗∗ = = ∑ , (2 ) ij aG F ∗ ∈ , . Let b ∗ be the binary vector 0 , 10 , 4 2 1 , 1 1 , 4 2 0 , 10 , 4 2 1 , 1 1 , 4 (, , ; ; , , | , , ; ; , , mN N m mN N m bb b b b c c c c ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ +− − + +− − = …… … …… … 2 ) + [( ) ] mm i ij j N i i m N ij j ij m j jj ba s a s such that 22 ,, 1 , 1 1 , , 12 1 m j m 4 β αβ β ∗− ∗ ∗ ∗ ∗ ++ + − == =+ + + ∑∑ ,2 1 , 1 , 1 im i i ba s β = + ∑ , ∗ ∗∗ + =+ [( ) ] mm i im j j N i m i m N i m j j i j m j jj ba t a t , 2 2 ,2 1 , 1 , 1 1 , , 12 1 m j m β αβ β ∗− ∗ ∗ ∗ ∗ ++ + + + + + − == =+ + + ∑∑ ,4 2 , 1 , 1 im i m i ba t β = + ∑ , ∗ ∗∗ ++ =+ 1 11 () mm i ij j ij j i m m jj ca s βα β β ∗∗ ∗ ++ == =+ ∑∑ ,2 1 , 1 im i m cs ∗∗ , 2 ,, , 1 , + = , + 2 ,2 1 , , 1 1 11 () mm i i m j j im j j im m jj ca t βα β β ∗∗ ∗ ++ + + + == =+ ∑∑ , ,4 2 , 1 im i m ct ∗∗ + + = , where for 01 , ,, ,( ij ij st G F ∗∗ ∈ 2 ) iN ≤≤ − 1 1 jm ≤ ≤+ . Let denote the code consisting of all such vectors and let L S b L N denote the code consisting of all such vectors . b ∗ Lemma 2. is dual to L S L N with respect to the symplectic inner product. Proof. Clearly and L S L N are binary linear codes. From definitio ns of and we have b b ∗ 22 2 2 ,, ,, , , , , 11 1 1 ( ) T r () () () () mm m m ij ij ij ij ij j ij j ij j ij j jj j j bc bc b c c b 2 1 m j β ββ ∗∗ ∗ ∗ == = = ⎛⎞ += + ⎜⎟ ⎝⎠ ∑∑ ∑ ∑ β = ∑ β + 2 ,1 , 1 1 , , , , 1 1 21 1 Tr [ ( ) ] ( ) mm m Ni i m Ni j j i j m j i j j i m m jj m j as a s a s ββ β β ∗∗ ++ + − + == + = ⎛ =+ ++ + ⎜ ⎝ ∑∑ ∑ 2 , 1 ,1 1 , , , ,1 1 21 1 [( ) ]( ) mm m Ni i m Ni j j i j m j i j j i m m jj m j as a s a s ββ β β β ∗∗ ∗ ∗ ++ + − + + == + = ⎞ ++ + + + ⎟ ⎠ ∑∑ ∑ , , , , ,1 , 1 ,1 , 1 ,1 , 1 ,1 , 1 1 () m N ij ij N ij ij i i m i i m i i m i i m j a a a a as as ss ss ∗∗ ∗ ∗ ∗ ∗ + ++ + + = = + +++ + ∑ + ) ∗ , 41 , , , , ,, ,, 22 1 () ( mm ij ij ij ij N i m j i m j N i m j i m j jm j bc bc a a a a + ∗∗ ∗ ∗ ++ + ++ + =+ = += + ∑∑ ,1 ,1 ,1 ,1 , 1 ,1 , 1 ,1 im im im im i im i im at at t t t t ∗∗ ∗ + ++ + + +++ + + ) ) ] 0 = . Thus the symplectic inner product 14 2 1 2 ,, ,, ,, ,, , 2 1 , 2 1 , 2 1 , 2 1 01 0 1 () [ () ( Nm N m ij ij ij ij ij ij ij ij i m i m i m i m ij i j bc bc bc bc b c b c −+ − ∗∗ ∗∗ ∗ ∗ ++ ++ == = = += + + + ∑∑ ∑ ∑ 41 ,, ,, , 4 2 , 4 2 , 4 2 , 4 2 22 () ( m ij ij ij ij i m i m i m i m jm b c b c bc bc + ∗∗ ∗ ∗ ++ ++ =+ ++ + + ∑ 12 ,, ,, 01 () Nm ij N ij ij N ij ij aa aa − ∗∗ ++ == =+ ∑∑ . T h e n t h e s t a t e m e n t f o l l o w s b y a d i m e n s i o n a r g u m e n t . Q . E . D . Clearly L N contains . Thus is weakly self-dual under symplectic inner product by Lemma 2. Then a quantum stab ilizer code can be derived from . L S L S L S Definition. For any and , define to be the quantum stabilizer code with stabilizer and normalizer N K , NK L L S L N which are obtained from the CSS-type quantum RS code . 2 2 ,2 , 1 m NN KK −+  Clearly is a binary quantum stabilizer code with param eters . In the other hand, may be thought of as concatenated quantum code where the outer code is CSS-type quantum RS code and the inner codes use distinct quantum codes. , NK L 2 2( 2 1 ) , 2 ( 2) Nm m N K +−  , NK L N 5 Before proving the first theorem we need som e lemmas. These involve the entropy function , defined by 4 () Hx 44 4 () l o g ( 1 ) l o g ( 1 ) 3 x Hx x x x =− − − − where 01 x ≤≤ . We shall also need the invers e function defined by 1 4 () Hy − 1 4 () x Hy − = iff 4 () y Hx = for 3 4 0 x ≤≤ . Lemma 3. Suppose n λ is an integer, where 3 4 0 λ < < . Then 4 () 0 34 n nH k k n k λ λ = ⎛⎞ ≤ ⎜⎟ ⎝⎠ ∑ . Proof. For any negative number we have r ) ⋅ } n 00 0 43 4 3 ( 3 4 ) ( 1 3 4 nn n r n k r k k rk rn kk k nn n kk k λλ λ == = ⎛⎞ ⎛⎞ ⎛⎞ ≤≤ ⋅ = + ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ⎝⎠ ∑∑ ∑ . Thus (1 ) 0 3{ 4 3 4 n kr r k n k λ λλ −− = ⎛⎞ ≤+ ⋅ ⎜⎟ ⎝⎠ ∑ . Choose 4 log ( 3( 1 ) ) r λ λ =− 4 . Then this sum is 4 () () 4( 1 ) 4 nH nH n λ λ λλ ≤− + = . Q . E . D . Lemma 4. If we are given M distinct nonzero quaternary (vectors of length L over ), where -tupl es L ( 4) GF (4 1 ) L M δ γ =− , 0 1 γ < < , 01 δ < < , then the sum of the weights of these is at least -tupl es L 1 4 (4 1 )( ( ) ( ) ) L LH o δ γδ − −− L . Proof. The number of these having weight -tupl es L L λ ≤ is at most 6 4 () 1 34 L LH k i L i λ λ = ⎛⎞ ≤ ⎜⎟ ⎝⎠ ∑ , by Lemma 3, for any 3 4 0 λ << . So the total weight is at least 44 () () (4 ) ( 1 4 LH LH LM L M M λλ λλ −= − ) . Choose 11 44 (1 l o g ) ( )( HL H λδ δ −− =− = − ) o L , with 3 4 λ < . Then the total weig ht is at least 1 4 ( 4 1 ) ( 1 () ) ( () () ) L Lo L H δ γδ − −− − o L L , 1 4 (4 1 )( ( ) ( )) L LH o δ γδ − =− − . Q . E . D . Theorem 5. Let R be fixed, 1 2 0 R < < . For each choose m 2 12 1 12 1 (1 ) (1 ) ( 2 1) 22 m mm KR N R mm ++ ⎢⎥ ⎢ =− =− − ⎢⎥ ⎢ ⎣⎦ ⎣ ⎥ ⎥ ⎦ . Then is a binary stabilizer code of length , NK L 2( 2 1 ) nN m = + with rate (2 ) (2 1 ) m mN K R R Nm − = ≥ + , and a lower bound on distance/length equal to 1 4 (1 4 ) (1 2 ) 4 H R − − as . m →∞ Proof. Let be any nonzero codeword of 0, 1 0, 4 2 1 , 1 1 , 4 2 0, 1 0,4 2 1 , 1 1 , 4 2 ( ,, ; ; ,, | ,, ; ; ,, ) mN N m mN N m cb b b b c c c c +− − + +− − = …… … …… … + L \ L N S . As we saw earlier, there must exists a nonzero codeword 01 2 1 (, , | , , ) NN N R aa a a a −− = S …… ∈ 2 , 1 m ii j j j aa N with β = = ∑ [( ) ] mm i ij j N i i m N ij j ij m j jj ba s a s , such that 22 ,, 1 , 1 1 , , 12 1 m j m β αβ β − ++ + − == =+ + + ∑∑ ,2 1 , 1 , 1 im i i ba s + =+ β = + ∑ , , 7 8 [( ) ] mm i im j j N i m i m N i m j j i j m j jj ba t a t 2 2 ,2 1 , 1 , 1 1 , , 12 1 m j m β αβ β − ++ + + + + + − == =+ + + ∑∑ ,4 2 , 1 , 1 im i m i ba t ++ =+ 1 11 () mm i ij j ij j i m m jj ca s βα β β ++ == =+ ∑∑ ,2 1 , 1 im i m cs β = + ∑ , , 2 ,, , 1 , + + = , 2 ,2 1 , , 1 1 11 () mm i i m j j im j j im m jj ca t βα β β ++ + + + == =+ ∑∑ , ,4 2 , 1 im i m ct + + = . From the definition of L N there are at least 1 K + nonzero pairs, saying , (| jj iN i aa + ) 0 j K ≤≤ , in the codeword . Thus either or is nonzero. Without losing generalization, we suppose , a ,1 , ,1 , (, , | , , jjj j ii m N i N i aa a a ++ …… ) m ) + ) m , 1 ,2 , 1 ,2 (, , | , , jj j j im i m N im Ni m aa a a ++ + …… ,1 , ,1 , (, , | , , jjj j ii m N i N i aa a a ++ …… 0 j K ≤≤ are nonzero. Then let us study vectors ,1 , 2 1 ,1 , 2 1 (, , | , , jjjj ii m i i m bb c c ) + + …… , in the codeword . From the definition of and , it is easy to prove that among these nonzero vectors each vector m ay occur as many as times. This is to say, contains at least 0 jK ≤≤ c , ij b , ij c 1 K + 2 m c (1 ) 2 m K + distinct nonzero binary . Now let ( 4 2)-tuples m + ω be a primitive elem ent of and let c be the quaternary vector ( 4) GF ′ 0 , 1 0 , 1 0 , 42 0 , 42 1 , 1 1 , 1 1 , 42 1 , 42 (, , ; ; , , mm N N N m N m cb c b c b c b c ) ω ωω ++ − − − + − + ′ =+ + + + …… … ω . Then contains at least c ′ (1 ) 2 m K + distinct nonzero quaternary . From the choice of , ( 2 1 ) -tupl es m + K 2 11 2 1 2 11 2 1 (1 ) (1 ) ( 2 1 ) 22 2 2 m m mm Km m RR mm ++ −+ ≥− ≥− − . We can now apply Lemma 4 with 21 Lm = + , (4 2 ) mm δ = + , 1 2 [1 ( 2 1 ) ] R mm γ = −+ , and deduce that 1 4 12 1 () 2 ( 2 1 ) ( 1 ) ( 2 1 ) ( ( ) ( ) ) 2 mm m wt c m R H o m m δ − + ′ ≥+ − − − , where the initial is because each 2 m ( 2 1 ) -tupl es m + occurs times (in the worst case). So 2 m 1 2 1 4 4 2 (1 4 ) distance 2 2 1 2 1 (1 ) ( ( ) ( ) ) (1 2 ) le ngth 2 1 4 4 2 4 mm m H mm R Ho m mm − − −+ ≥− − −+  R − m →∞ as . Q . E . D . To sum up, if let δ and R denote the lower bound to dist ance/length and rate of a family of quantum codes respectively as the length , we have found a family of asymptotically good concat enated quantum codes with n →∞ 1 2 0 R < < and 1 1 4 4 (1 4 ) (1 2 ) H δ − = R − . In fact in 1996 Calderbank and Shor [1] have proven the existence of good quantum codes. Then Calderbank et al. [5] proved the quantum Gilbert-Varshamov bound. But these proofs ar e not constructive. Later, Ashikhm in et al. [13] explicitly constructed asym ptotically good quantum c odes with 1 18 0 δ << , 11 10 3 1( 2 1 ) m R m δ −− =− − − and Chen et al. [14] with 0 t δ δ < < where 11 2 3 (2 3 ) (2 1 ) (2 1 ) tt t t δ −− =− + − for , 3 t ≥ 3 ( ) t Rt δ δ = − . Then Matsumoto [15] improved the bound of Ashikhmin et al. [13] with 11 1 2 0( 2 ) [ ( 2 1 ) m m δ ] − − <≤ − − , 1 10 3 12 ( 2 1 ) m R m δ − =− − − for . But all these quantum codes are derived from good classical codes directly. Compared with above codes, although the performance of our code is not very excellent, this is the first time that good quantum codes are explicitly constructed from bad codes. Its occurrence supplies a gap in quantum coding theory. 2 m ≥ ∗ Electronic ad dress: lizhuo@ xidian.edu.cn † Electronic address: l_zhuo@21cn.com [1] A. R. Cal derbank and P. W. S hor, Phys. Rev. A 54, 1098 (1996). 9 10 [2] A. M. Steane, Proc. R oy. Soc. Lond. A 452, 2551 ( 1996). 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