Nonbinary Quantum Error-Correcting Codes from Algebraic Curves

Transcription

1 Nonbnary Quantum Error-Correctng Codes from Algebrac Curves Jon-Lark Km and Judy Walker Department of Mathematcs Unversty of Nebraska-Lncoln, Lncoln, NE USA e-mal: {jlkm, November 15, 2004 Abstract We gve a generalzed CSS constructon for nonbnary quantum error-correctng codes. Usng ths we construct nonbnary quantum stablzer codes from algebrac curves. We also gve asymptotcally good nonbnary quantum codes from a Garca- Stchtenoth tower of functon felds whch are constructble n polynomal tme. keywords Algebrac geometrc codes, nonbnary quantum codes. 1 Introducton Bnary quantum error-correctng codes have been constructed n several ways. One nterestng constructon uses algebrac-geometry codes [2], [6], [7], [12], wth the man dea beng to apply the bnary CSS constructon [4], [5], [16] to the asymptotcally good algebrac-geometry codes arsng from the Garca-Stchtenoth [11] tower of functon felds over F q 2 (where q s a power of 2 attanng the Drnfeld-Vladut bound [17]. It s natural to consder nonbnary quantum codes. Beyond the smple fact that nonbnary error-correctng codes are nterestng n the classcal case, Rans [14] ponts out that there are ndeed applcatons n whch nonbnary quantum codes would be more approprate than bnary quantum codes. Though nonbnary quantum codes have been consdered n [1], [3], [9], [14], the majorty of attenton has been gven thus far to the bnary case. In partcular, the queston of asymptotcally good nonbnary quantum codes has not been studed untl now.

2 In ths paper we gve a nonbnary verson of the generalzed bnary CSS constructon based on two bnary lnear codes gven by Calderbank et al. [4]. We then apply ths constructon to the tower of functon felds defned n [11] wth concatenaton to Reed-Solomon codes to obtan asymptotcally good nonbnary quantum codes whch are constructble n polynomal tme. 2 Prelmnares In ths secton we gve some defntons and basc facts about quantum codes. Frst, we recall the generalzed bnary CSS constructon of quantum stablzer codes. In Secton 3 we wll generalze ths constructon to the nonbnary case. Theorem 2.1. [4, Theorem 9] Suppose C 1 C 2 F n 2 are bnary lnear codes wth dmensons k 1 and k 2, respectvely. Then there exsts a bnary [[n, k 2 k 1, d]] quantum code, where d = mn{d(c 2 \ C 1, d(c 1 \ C 2 }. Here, and n the remander of ths paper, the notaton d(a \ B means the mnmum weght of any vector n A but not n B. Let q = p m, where p s an odd prme throughout the paper. We call C F n q an F p -lnear code f C s lnear over F p. Ths generalzes the noton of addtve F 4 -codes, snce beng an addtve subgroup of F n 4 s equvalent to beng an F 2 -vector space contaned n F n 4. Addtve F 4 -codes whch are self-orthogonal under the trace nner product were used to construct stablzer quantum codes n [4]. Ths dea was generalzed n [1] to the relatonshp between self-orthogonal codes over F q 2 and q-ary quantum codes for any odd prme power q. An explct error bass for p m -ary quantum codes s descrbed as follows [1]. Let T and R be the lnear operators actng on the p-dmensonal complex space C p defned by T,j = δ,j 1 (mod p and R,j = ξ δ,j, where ξ = e 2π 1/p, the ndces range from 0 to p 1, and δ,j = 1 f = j and 0 otherwse. The set of operators T R j forms an orthogonal bass under the nner product defned by A, B = Tr(A B, where A s the Hermtan transpose of A [1], [15]. Fx a bass {γ 1, γ 2,..., γ m } for F p m over F p. For a, b F p m we can wrte unquely Defne a = a 1 γ 1 + a 2 γ a m γ m, b = b 1 γ 1 + b 2 γ b m γ m. T a R b = (T a 1 T a 2 T am (R b 1 R b 2 R bm. The set of operators T a R b forms an orthogonal bass of untary operators actng on the p m -dmensonal complex vector space C pm [1]. 2

3 Let a = (a (1,..., a (n, b = (b (1,..., b (n F n q. As seen above, t s enough to consder the error operators gven by The set of operators E a,b = T a (1R b (1 T a (2R b (2 T a (nr b (n. E = {ξ E a,b a, b F n q and 0 p 1} form an error group of order p 2mn+1. Quantum stablzer codes are defned as jont egenspaces of the operators of a commutatve subgroup S of E [1]. See also the appendx of [3]. 3 A q-ary CSS Constructon In ths secton, we explore CSS constructons for nonbnary quantum codes. We begn wth a constructon gven n [1] that s analogous to the frst constructon presented n [4], and then follow the lead of [4] to derve other constructons. We note that our q-ary CSS constructon generalzes the p-ary CSS constructon [13, Theorem 5] as the latter constructon uses only self-orthogonal codes over F p 2 where p s a prme. The man result s Theorem 3.4, whch wll be used n Secton 4 to construct asymptotcally good sequences of nonbnary quantum codes. As n the prevous secton, we wrte q = p m, where p s an odd prme. For a = (a (1,..., a (n, b = (b (1,..., b (n F n q, let a b = a ( b ( be the usual nner product on F n q. For (a b, (a b F 2n q, set (a b (a b = Tr(a b a b, where Tr : F q F p s the trace map. We see that f q = p then (a b (a b = (a b a b whch was studed n [13], [14]. Proposton 3.1. [1, pp. 3069] Suppose C F 2n q s an F p -lnear code of length 2n havng p r codewords. Let C be the dual of C wth respect to (. If C C, then there s a q-ary [[n, n r, d]] quantum code wth d = m d(c \ C. For x, y F n q 2, defne x y = (x y q xq y. Ths map s F q -blnear and generalzes the nner product of [13, pp. 1879]. Note that for any γ 0 F q, there exsts γ F q 2 \ F q satsfyng γ q = γ 0 γ; ndeed snce the trace map Tr : F q 2 F q s onto and F q -lnear, we may pck γ F q 2 \ F q wth Tr(γ = γ 0. Further, for any such γ, {1, γ} s a bass for F q 2 over F q snce γ F q. Lemma 3.2. Suppose D F n q s an F q-lnear code satsfyng D D 2, where D s the dual of D wth respect to (. Fx γ 0 F q and choose γ F q 2 \ F q satsfyng γ q = γ 0 γ. Defne f : F n q 2 F2n q by f(x 1,..., x n = (x (1 1,..., x (1 n x (2 1,..., x (2 n, where x = x (1 + γx (2 for = 1,..., n. Then f(d f(d = (f(d, where (f(d s the dual of f(d wth respect to (. 3

4 Proof. Clearly, f(d f(d snce D D. It remans to show that f(d = (f(d. To do ths, let x D, y D. Then 0 = x y = (x y q xq y = ( (x (1 + γx (2 (y (1 + γy (2 (x (1 + γx (2 q (y (1 + γy (2 = ( (x (1 + γx (2 (y (1 + γ q y (2 = ( q (x (1 + γ q x (2 (y (1 + γy (2 x (1 + γ q x (1 + γx (2 ( x (1 + γx (1 + γ q x (2 = (γ q γ ( x (1 x (2 = (γ 0 2γ ( x (1 x (2 + γ q+1 x (2 + γ q+1 x (2. But γ 0 2γ F q 2 \ F q, and so ( x (1 x (2 = 0. Therefore ( ( f(x f(y = Tr x (1 x (2 = 0. Ths shows f(d (f(d. Snce these two codes have the same number of codewords, they must be equal. Proposton 3.3. Let C 1 C 2 F n q be F q -lnear codes, so that C2 C1, where C s the dual of C under the usual nner product. Let ω be a prmtve element of F q 2 and wrte ω = ω q. Set D = ωc 1 + ωc 2 F n q 2. Then the dual D of D s gven by D = ωc 1 + ωc 2. Hence D D and d(d \ D = mn{d(c 2 \ C 1, d(c 1 \ C 2 }. Proof. Note frst that D = q k 1+n k 2, and so D = q 2n (n+k 1 k 2 = q n k 1+k 2 = ωc 1 + ωc 2. 4

5 Now pck x C 1, y C 2, a C 1, and b C 2. Then (ωx + ωy ( ωa + ωb = ((ωx + ωy (ωa + ωb ( ωx + ωy ( ωa + ωb = (ω 2 ω 2 ( x a y b = 0, snce x a = y b = 0. The last sentence of the proposton follows snce C 1 C 2, C 2 C 1, and ωc 1 ωc 2 = ωc 1 ωc 2 = {0}. Next, we gve a constructon whch produces a q-ary quantum code from any two F q -lnear codes C 1 C 2 F n q. Ths s a q-ary verson of the bnary CSS constructon [4, Theorem 9] as t s also based on two lnear codes over F q, and so t s a generalzaton of [13, Theorem 5] whch s based on self-orthogonal codes. Theorem 3.4. Let q = p m, where p s an odd prme and m 1 s an nteger. Suppose C 1 C 2 F n q are F q -lnear codes wth dmensons k 1 and k 2, respectvely. Then there exsts a q-ary [[n, k 2 k 1, d]] quantum code, where d = mn{d(c 2 \ C 1, d(c 1 \ C 2 }. Proof. Set D = ωc 1 + ωc 2, as n Proposton 3.3. Then f(d (f(d by Proposton 3.3 and Lemma 3.2. Note that f(d s an F q -lnear code n F 2n q, hence an F p -lnear code wth p r elements, where r = m(k 1 + n k 2. Our clam now follows by applyng Proposton 3.1 by lettng C = f(d. Example 3.5. Let C 2 be the ternary Golay [11, 6, 5] code and let C 1 be the subcode of C 2 consstng of codewords whose weght s dvsble by 3. Then C 1 s a ternary [11, 5, 6] code and n fact s equal to C 2. By Theorem 3.4, we obtan a ternary double-error correctng quantum [[11, 1, 5]] 3 code. 4 Good Sequences of q-ary Quantum AG Codes We assume the results from Secton II of [7] and use the deas of Secton III of that paper. Fx t 1 and set q = p 2t = (p t 2. The authors [7] only used a trval bnary MDS code n the concatenaton whle we use Reed-Solomon codes over F p. Let X be a curve over F q of genus g. Fx an F q -ratonal pont P = P (X X(F q, let D = D(X be the sum of the F q -ratonal ponts on X other than P, and set N = N(X = deg D = #X(F q 1. Pck ntegers m 1 = m 1 (X and m 2 = m 2 (X wth 2g 2 < m 1 < m 2 < N, and set T j = T j (X = C(m j P, D, the (functonal algebrac geometrc code defned on 5

6 X from the dvsors m j P and D. Then the code T j s an [N, m j g + 1, N m j ] code over F q, T 1 T 2, and the dual Tj of T j s an [N, N m j + g 1, m j 2g + 2] code over F q. As n [7], we use concatenaton [10] to obtan F p -lnear codes C 1 and C 2 from T 1 and T 2. More precsely, for any nteger 0 r p + 1 2t, let π : F q = F p 2t F 2t+r p be an F p -lnear njectve map such that the mage C of π s a [2t + r, 2t, r + 1] Reed-Solomon code over F p. Defne π : F N q F N(2t+r p by π((x 1,..., x N = (π (x 1,..., π (x N. Then we have C 1 := π(t 1 π(t 2 =: C 2. Thus C j, (j = 1, 2 s an F p -lnear [(2t + rn, 2t(m j g + 1, (r + 1(N m j ] code (see [10] or [8]. The dual of C j (j = 1, 2 s Cj = S (π (Tj, where S s the drect sum of N copes of C and π s the F q -lnear njectve dual bass map, as descrbed n [7]. For any vector x C1 \C2, we have wt(x m 1 2g + 2, just as n the bnary case (see [7], proof of Theorem 1.2. The followng proposton follows from Theorem 3.4. Proposton 4.1. Wth notaton as above, we get a p-ary quantum code B = B(X wth parameters [[(2t + rn, 2t(m 2 m 1, mn{(r + 1(N m 2, m 1 2g + 2}]] p. Example 4.2. Consder the Hermtan curve defned by y pt + y = x pt +1 over F p 2t; ths s the base level of the Garca-Stchtenoth tower [11]. In ths case, the codes T j, j = 1, 2, are [p 3t, m j pt (p t 1 + 1, p 3t m 2 j ] lnear codes over F p 2t. For any ntegers r, m 1, and m 2 wth 0 r p + 1 2t and p t (p t 1 2 < m 1 < m 2 < p 3t, we get a p-ary quantum code B wth parameters [[(2t + rp 3t, 2t(m 2 m 1, mn{(r + 1(p 3t m 2, m 1 p t (p t 1 + 2}]] p. Next, we consder the asymptotc behavor of our quantum codes. Let X = {X} be a Garca-Stchtenoth tower [11] of polynomally constructble curves over F q havng ncreasng genus g = g(x and attanng the Drnfeld-Vladut bound,.e., satsfyng lm sup X X #X(F q g = p t 1. Note that f k = k(x = m 2 m 1, then 0 < k N 2g. Conversely, gven any nteger k wth 0 < k N 2g, set (r + 1N + 2g + k 2 m 2 =. r + 2 6

7 and Then we have the followng. Therefore, B has parameters (r + 1(N m 2 = (r + 1N (r + 2m 2 + m 2 (r + 1N ((r + 1N + 2g + k 2 + m 2 = 2g k m 2 = m 1 m 2 2g m 2 = m 1 2g + 2 m 1 2g + 2 = m 2 k 2g + 2 (r + 1N + 2g + k 2 = r + 2 k 2g + 2 (r + 1N + 2g + k 2 r + 2 r + 1 r + 2 k 2g + 2 = r + 1 (N 2g k + 1. r + 2 [[(2t + rn, 2tk, d := r + 1 r + 2 (N 2g k + 1]] p. For any sequence of ntegers {k = k(x X X} wth 0 < k < N 2g for each X, we k have 0 < lm sup 1 X X 2 N q 1 by the Drnfeld-Vladut bound. Indeed, by choosng the k values of k approprately, we can have lm sup X X N We put R := lm sup X X = 2t 2t + r λ, 2tk (2t + rn d δ := lm sup X X (2t + rn r + 1 = (r + 2(2t + r (1 2 p t 1 λ. 7 = λ for any λ wth 0 < λ 1 2 q 1.

8 To get an expresson for R n terms of δ, we solve for λ n terms of δ and substtute, yeldng R p (δ := R = 2t ( 1 2 2t(r + 2 2t + r p t 1 r + 1 δ. In order to have R > 0, we need δ < δ(p, r, t, where We have proved the followng. δ(p, r, t = (r + 1(p t 3 (r + 2(2t + r(p t 1. Theorem 4.3. Let p be any odd prme number. Suppose that t 1 and r 0 are ntegers satsfyng 2t+r p+1. Let δ(p, r, t be as above. Then for any δ wth 0 < δ < δ(p, r, t, there exst polynomally constructble famles of p-ary quantum codes wth n and asymptotc parameters (δ, R p (δ, where R p (δ = (2t(r + 2 (δ(p, r, t δ. r + 1 Remark 4.4. The case when p = 2 was dscussed n [7]. In ths case we requre that t 3 s an nteger and r = 0 or 1. Then pluggng n p = 2 and r = 1 nto δ(p, r, t n Theorem 4.3 gves whch s Theorem 1.2 of [7]. δ(2, 1, t = δ t = 2 2 t 3 3 (2t + 1(2 t 1, R 2 (δ = 3t(δ t δ, Usng the same deas, we can construct p t -ary quantum codes. Theorem 4.5. Let p be an odd prme, and let t 1 and r 1 be ntegers wth r p t 1. Set δ(p, r, t = (r + 1(pt 3 (r (p t 1. Then for any δ wth 0 < δ < δ(p, r, t, there exst polynomally constructble famles of p t -ary quantum codes wth n and asymptotc parameters (δ, R p t(δ, where R p t(δ = 2(r + 2 (δ(p, r, t δ. r + 1 8

9 Proof. We proceed as n the proof of Theorem 4.3. For any nteger r wth 1 r p t 1, we have a [2 + r, 2, r + 1] Reed-Solomon code C over F p t. Let π : F p 2t F 2+r p be an F t p t-lnear njectve map wth π (F p 2t = C. The code C j wll be an F p t-lnear [(2+rN, 2(m j g+1, (r + 1(N m j ] code wth Cj = S + π (Tj, where S s the drect sum of N copes of C and π s the dual bass map correspondng to π. Applyng the CSS constructon, we get a p t -ary quantum code B = B(X wth parameters [[(2 + rn, 2k, d := r + 1 (N 2g k + 1]]. r + 2 Now set and wrte R p t R p t := R = lm sup X X δ = lm sup X X 2k (2 + rn = r λ, d (2 + rn = r + 1 (r n terms of δ to obtan the result. ( 1 2 p t 1 λ References [1] A. Ashkhmn and E. Knll, Nonbnary quantum stablzer codes, IEEE Transactons on Informaton Theory 47 (2001, [2] A. Ashkhmn, S. Ltsyn, and M.A. Tsfasman, Asymptotcally good quantum codes, quant-ph/ [3] H. Barnum, C. Crépeau, D. Gottesman, A. Smth, and A. Tapp, Authentcaton of quantum messages, quant-ph/ [4] A.R. Calderbank, E.M. Rans, P.W. Shor, and N.J.A. Sloane, Quantum error correcton va codes over GF (4, IEEE Transactons on Informaton Theory 44 (1998, [5] A.R. Calderbank and P.W. Shor, Good quantum error-correctng codes exst, Phys. Rev. A 54 ( [6] H. Chen, Some good quantum error-correctng codes from algebrac geometry codes, IEEE Transactons on Informaton Theory 47 ( [7] H. Chen, S. Lng, and C. Xng, Asymptotcally good quantum codes exceedng the Ashkhmn-Ltsyn-Tsfasman bound, IEEE Transactons on Informaton Theory 47 (2001,

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