A Characterization Of Quantum Codes And Constructions

Transcription

1 A Characterization Of Quantum Codes And Constructions Chaoping Xing Department of Mathematics, National University of Singapore Singapore , Republic of Singapore ( Abstract We give a characterization of quantum codes in the paper. This characterization converts the algebraic structure of quantum codes into a combinatorial structure and makes it possible to employ classical codes and boolean functions to construct good quantum codes. In addition, we make use of classical algebraic geometry codes to get the best known asymptotic bound on quantum codes through this characterization. 1 Introduction Quantum information has received much attention for the past few years. Since the pioneering work in [4, 5, 19, 20, 21], the theory of quantum error-correcting codes has been rapidly developed. As in classical coding theory, one of the central tasks in quantum coding theory is to construct good quantum codes. The first systematic mathematical construction is given by Calderbank et al [4] in the binary case and then generalized by Rains [15], Ashikhmin and Knill [3] and Matsumoto and Uyematsu [14] to the non-binary case. The p-ary quantum codes by this construction are called stabilizer quantum codes and derived from classical codes over finite fields F p or F p 2 through various techniques. Another construction is presented by Partially supported by the MOE-ARF research grant R

2 Schlingemann and Werner [18] using combinatorial properties of matrices over finite fields. Many good quantum codes with various parameters have been constructed in these ways, but all p-ary quantum codes produced by these constructions have dimensions being powers of p. On the other hand, quantum codes with general dimension have been studied by Knill [10, 11] and Rains [16], etc. In this paper we present a characterization of (binary and non-binary) quantum codes. Based on this characterization, we are able to derive a method to construct q-ary quantum codes with dimensions not necessarily equal to powers of q. With this method, the construction of quantum codes is reduced to finding functions from F n q to F q satisfying a system of quadratic relationships. Furthermore, we show that certain classical codes can be used to construct good quantum codes. As the first consequence of the construction, a binary ((7, 4, 3))-quantum code is found. As the second consequence of this construction, we are able to construct quantum codes of minimum distance 2. In particular, we produce a class of binary quantum ((n, 2 n 2 ( 1 2 (n 1)/2), 2))-codes for odd length n 5. By a simple propagation rule, Rains [16] gives a family of binary quantum ((n, 3 2 n 4, 2))-codes for odd n 5. For n 11, our codes are better than those by Rains [16] in the sense that our codes have bigger size. Moreover, our binary ( quantum ((n, 2 n (n 1)/2), 2))-codes of odd length achieve the quantum Singleton bound asymptotically. Other advantages of our characterization of quantum codes include formulation of propagation rules for quantum codes. Classical coding theory has a much longer history and various constructions and propagation rules have been proposed. Based on our characterization of quantum codes, we can induce some propagation rules for quantum codes similar to those in classical coding theory. In particular, some new binary quantum codes are found based on our propagation rules. At the last, our characterization makes it possible to convert asymptotic lower bound from classical algebraic geometry codes quantum codes. The paper is organized as follows. In Section 2, we recall the definitions and basic facts on quantum codes and show our characterization of quantum codes and derive some consequences including several propagation rules by making use of this characterization. In Section 3, we first discuss the asymptotic behavior of quantum codes and then apply classical algebraic geometry codes to derive the asymptotic quantum algebraic geometry bound using the construction in Section 3 (or the stabilizer method). Finally, we derive one of the main results in this paper, i.e., we employ the construction in Section 2 to derive an asymptotic lower bound improving 2

3 the asymptotic quantum algebraic geometry bound (the same improvement has been done recently in classical coding theory [22]). It seems that this improvement cannot be obtained from additive quantum codes. 2 A Characterization of Quantum Codes Binary quantum codes have been generalized to the q-ary quantum codes with q being a power of a prime [3]. Firstly we recall the definition of q-ary quantum codes. Throughout this section, we assume that q = p m for a prime p and an integer m 1. Let C q be a complex vector space of dimension q and { a>: a F q } be a normal orthogonal basis of C q with respect to the hermitian inner product <, > H (or < >). A q-ary quantum state v>is a non-zero vector in C q : 0 v>= α c c> (α c C), c Fq where F q denotes the finite field with q elements. For n 1, the n-th tensor product (C q ) n = C qn has a basis { } c >= c1 > c 2 > c n >: c =(c 1,,c n ) F n q. A q-ary n-system state v > is a non-zero vector in (C q ) n : 0 v >= α c c >, c F n q where α c C. Let ζ be the p-th primitive root of unity: ζ = e 2πi p C. Let δ ij be the Kronecker symbol and define two p p matrices T =(t ij ), R =(r ij ) with t ij = δ i,j 1 mod p and r ij = ζ i δ ij. Fix a basis {α 1,...,α m } of F q as a vector space over F p. For two elements a = m i=1 a iα i and b = m i=1 b iα i, consider the error operator T a R b = T a 1 R b 1 T a 2 R b 2 T am R bm. Then we have (T a R b )(T c R d )=ζ <b,c> T a+c R b+d, 3

4 where <, > stands for the usual inner product of F q F m p over F p with respect to the basis {α 1,...,α m }, and (T a R b )(T c R d )=ζ <a,d> <b,c> (T c R d )(T a R b ). The action of T a R b on a basis element c >of C q with c F q is as follows T a R b c>= ζ <b,c> c + a>. Now for two vectors a =(a (1),a (2),...,a (n) ) and b =(b (1),b (2),...,b (n) )inf n q, we define the error operator on (C q ) n by E ab = T a (1)R b (1) T a (2)R b (2) T a (n)r b (n). The set of all quantum errors E := { ζ λ E ab : a, b F n q,λ F } p forms a non-abelian group of order p 2mn+1 under the group law E ab E uv = ζ <b,u> E a+u E b+v, where < b, u >= n i=1 <b(i),u (i) >. It is easy to verify that the center C(E) ofe is { } ζ λ I : λ F p. The quantum weight of E ab is defined by w Q (E ab )=# { i : 1 i n, (a (i),b (i) ) (0, 0) F 2 q}. Definition 2.1. A complex subspace Q {0} of ( C q) n = C qn of length n. is called a q-ary quantum code We denote by K = dim C Q the dimension of Q over C and put k = log q K. Then k is a real number and 0 k n. The minimum distance of a quantum code Q is defined to be the largest positive integer d satisfying the following condition: for any u > and v > in Q with < u v >= 0 and e Ewith quantum weight d 1, we have < u e v >= 0. 4

5 A q-ary quantum code Q with length n, dimension K ( k = log q K) and minimum distance d is denoted by ((n,k,d)) q or [[n,k,d]] q. A quantum ((n,k,d)) q -code Q is called pure if < u e v >= 0 for all e Ewith 1 w Q (e) d 1 and u, v Q (note that we do not require < u v >= 0 here). Now we present a characterization on a quantum ((n,k,d)) q -code Q. It gives an alternative way to construct quantum codes. An n-system state v >= α c F c c > can be identified with a mapping ϕ: F n n q q C defined by ϕ(c) =α c for all c F n q. For a map ϕ: F n q C and a partition {1, 2,,n} = A B, we denote ϕ(c) byϕ(c A, c B ), where c A and c B are the subvectors of c coordinated by A and B, respectively. For two maps ϕ, ψ : F n q C, we define their hermitian inner product by (ϕ, ψ) = ϕ(c)ψ(c) C, c F n q where ϕ(c) stands for the conjugate of the complex number ϕ(c). Theorem 2.2. There exists a quantum ((n,k,d)) q -code with K 2 if and only if there exist K non-zero mappings ϕ i : F n q C (1 i K) (2.1) satisfying the following condition: for each partition{1, 2,,n} = A B with A = d 1 and B = n d +1, and any c A, c A Fd 1 q, 1 i, j n, { ϕ i (c A, c B )ϕ j (c A, c 0 if 1 i j K B)= (2.2) f if 1 i = j K, c B F n d+1 q where f = f(c A, c A ) is independent of i. For the detailed proof of the above theorem, see [8, 9]. Using the above characterization, we give a binary ((7, 4, 3))-quantum code due to Sze Ling Yeo. This is a new code compared with the Table III of [4] as the code of length 7 and minimum distance 3 in the Table III of [4] has dimension only 2. 5

6 Example 2.3. We consider the set {f λ (x)} 4 λ=1 with x =(x 1,...,x 7 ) of quadratic boolean functions, where f λ (x) =x 1 x 2 + x 1 x 3 + x 1 x 3 + x 1 x 4 + x 1 x 7 + x 2 x 6 + x 3 x 5 + x 4 x 7 + x 5 x 7 + x 6 x 7 + u λ x T with u 1 = ( ), u 2 = ( ), u 3 = ( ) and u 4 = ( ). Define the mappings ϕ λ : F 7 2 C, u ( 1)f λ(u). Then we can check that (2.2) in Theorem 2.2 is satisfied for p =2and(n,K,d)=(7, 4, 3). Hence, the set {ϕ λ (x)} 4 λ=1 gives a binary ((7, 4, 3))-quantum code. Using the above characterization of quantum codes, we are now able to give a construction of quantum codes from classical codes. Proposition 2.4. Let C be a q-ary classical linear code of length n and {v i } K i=1 be a set of K distinct vectors in F n q. Put d v := min{wt(v i v j + c) : 1 i j K and c C} and d = min{d v,d(c )}, where wt denotes the Hamming weight. If d>0, then we have a q-ary ((n,k,d))-quantum code. Corollary 2.5. Assume that we have two q-ary classical linear codes [n, k i ] (i =1, 2) such that C 1 C 2. Then there exists a q-ary ((n,k,d))-quantum code with K = q k 2 k 1 and d = min{d(c 2 ),d(c1 )}. Remark 2.6. The result of Corollary 2.5 can be derived from stabilizer codes [4] as well. However, we will see that Proposition 2.4 can give better quantum codes than the stabilizer codes by employing classical nonlinear codes. Example 2.7. Let C = {0, 1} be the binary [n, 1,n]-code with n 4. Then its dual code is a binary [n, n 1, 2]-code. Define V to be the subset of F n 2 as follows: Case 1. n is even. V := {v F n 2 : wt(v) =(n 2)/2 2i for all 0 i (n 2)/4}. 6

7 Then V =2 n 2. Case 2. n is odd. V := {v F n 2 : wt(v) =(n 3)/2 2i for all 0 i (n 3)/4}. ( Then V =2 n (n 1)/2). It is clear that wt(u v + c) 2 for any two distinct u, v V and c C. By Proposition 2.4, we get a binary ((n, V, 2))-quantum code, i.e., we have a binary ((n, 2 n 2, 2))-quantum ( code for any even n 4 and a binary ((n, 2 n (n 1)/2), 2))-quantum code for any odd n 5. Remark 2.8. (i) For even n 4, binary quantum ((n, 2 n 2, 2))-codes have been constructed in [4, 16]. For odd length n, a pure binary quantum ((5, 6, 2))-code was obtained in [17]. This code is optimal in the sense that binary quantum ((5,K,2))-codes do not exist for K 7. By a simple propagation rule, Rains [16] obtains a family of binary quantum ( ((n, 3 2 n 4, 2))-codes for all odd n 5. Our binary quantum ((n, 2 n (n 1)/2), 2))- codes obtained in Example 2.7 are better than Rains result for 2 n 11. For instance, we have binary quantum codes ((11, 386, 2)) and ((13, 1586, 2)), while the previous result only indicates existence of ((11, 384, 2)) and ((13, 1536, 2)). (ii) Rains [16] defines an asymptotic quantity K 2 = lim m K 0(2m +1)/2 2m 2, where K 0 (2m + 1) denotes the maximal dimension K of binary pure quantum ((2m + 1,K,2))-codes. By the quantum Singleton bound we know that K 2 2. Rains result gives only K 2 3/2. Now it is easy to see from the above example that K 2 =2. In [8], we give some propagation rules. These propagation rules play a key role on determining asymptotic behavior of quantum codes. We state these propagation rules without detailed proof. Proposition 2.9. Suppose there is a q-ary ((n,k,d))-quantum code. Then (i) (subcode) there exists a q-ary ((n, K 1, d))-quantum code; (ii) (lengthening) there exists a q-ary ((n +1,K, d))-quantum code; 7

8 (iii) (puncturing) there exists a q-ary ((n 1,K, d 1))-quantum code; (iv) there exists a q-ary ((n,k,d 1))-quantum code. Example In this example, we can see that some good quantum codes can be obtained by the first propagation rule in Proposition 2.9, while the second and third propagation rules provide some new codes. (i) By the Table III of [4], we know the existence of a binary ((12, 16, 4))-quantum code. Hence, from the first propagation rule in Proposition 2.9, we get binary ((12,K,4))- quantum codes for all K 16. By the Table III of [4], we know that ((12, 8, 4)) and ((12, 4, 4))-quantum codes are the best known in the sense that give length n and dimension K, the minimum distance is the best known. One can find many such examples in this way by using the Table III of [4]. (ii) By the Table III of [4], we know the existence of pure binary ((6, 1, 4)), ((12, 1, 6)) and ((18, 1, 8))-quantum codes. Hence, from the second propagation rule in Proposition 2.9, we get pure binary ((7, 1, 4)), ((13, 1, 6)) and ((19, 1, 8))-quantum codes, respectively. These three codes are new compared with the Table III of [4]. Furthermore, we can get many best known codes from this propagation rule. For instance, a binary ((16, 4, 6))- quantum code gives a binary ((17, 4, 6))-quantum code. (iii) By the Table III of [4], we know the existence of a binary ((28, 2 8, 6))-quantum code. Hence, from the third propagation rule in Proposition 2.9, we get a binary ((27, 2 8, 5))- quantum code. This is a new code compared with the Table III of [4] as the code of length 27 and dimension 2 8 in the Table III of [4] has minimum distance only 4. Furthermore, we can get many best known codes from this propagation rule. For instance, a binary ((16, 4, 6))-quantum code gives a binary ((15, 4, 5))-quantum code. 3 Asymptotic bounds from AG-codes As in classical coding theory, one wonders how a good family of quantum codes performs when the length tends to infinity. By linear programming methods, some asymptotic upper bounds have previously been derived (see [1]). 8

9 The first asymptotic lower bound is given in [2] using algebraic geometry codes. This bound was improved later in [6, 13]. All these three bounds are concerned with binary quantum codes and they are far away from some reasonable upper bounds. In [3], the asymptotic quantum Gilbert-Varshamov bound is given (the finite version of the quantum Gilbert-Varshamov bound is derived in [7]). This asymptotic Gilbert-Varshamov bound for binary quantum codes is much better than those from algebraic geometry codes. However, as in classical coding theory, the quantum Gilbert-Varshamov bound can be improved by algebraic geometry codes using the stabilizer methods or other constructions (see Theorem 3.6 and 3.7). For a q-ary quantum code Q, we denote by n(q),k(q), and d(q) the length, the dimension over C, and the minimum distance of Q, respectively. Let Uq Q be the set of ordered pairs (δ, R) R 2 for which there exists a family {Q i } i=1 of q-ary codes with n(q i ) and δ = lim i d(q i ) n(q i ), R = lim i log q K(Q i ), n(q i ) where log q denotes the logarithm to the base q. One of the central asymptotic problems for quantum codes is to determine the domain Uq Q. As in classical coding, it is a hard problem to determine Uq Q completely. Instead, we are satisfied with some bounds on Uq Q. First, we give a general description on the domain Uq Q. Proposition 3.1. There exists a function αq Q (δ),δ [0, 1], such that Uq Q domain {(δ, R) R 2 : 0 R<αq Q (δ), 0 δ 1} is the union of the with some points on the boundary αq Q(δ). Moreover, αq q (0) = 1,αQ q (δ) =0for δ [1/2, 1], and (δ) decreases on the interval [0, 1]. α Q q Remark 3.2. (i) From Proposition 3.1, determination of the domain Uq Q is almost equivalent to determining the function αq Q(δ). (ii) In classical coding, the boundary α q (δ) is in the domain U q and the function α q (δ) is continuous (see [23]). However, we do not know whether the same result holds for quantum codes. This is due to the lack of one propagation rule for quantum codes, namely, the existence of a q-ary ((n,k,d))-quantum code does not ensure the existence of a q-ary ((n 1, K/q, d))-quantum code (however, it is true that, in classical coding theory, the existence of a q-ary (n,k,d)-code implies the existence of a q-ary (n 1, K/q,d)-code). 9

10 The first lower bound on α Q 2 (δ) was derived by [2] using algebraic geometry codes and later on this bound was improved by Chen-Ling-Xing [6] and Matsumoto[13]. A very good existence lower bound for p-ary quantum codes was introduced by Ashikhmin and Knill [3]. It is called the quantum Gilbert-Varshamov bound. The finite version of the q-ary quantum Gilbert-Varshamov bound was given by Feng and Ma in [7], and the asymptotic version of q-ary quantum codes can be derived easily from this finite version. As in classical coding theory, the quantum Gilbert-Varshamov bound is a benchmark for the function αq Q (δ). For 0 <δ<1, define the q-ary entropy function H q (δ) :=δ log q (q 1) δ log q δ (1 δ) log q (1 δ), and put R GV (q, δ) :=1 δ log q (q +1) H q (δ). Then the Gilbert-Varshamov bound says that αq Q (δ) R GV (q, δ) for all δ (0, 1 ). (3.1) 2 For q = 2, the Gilbert-Varshamov bound is much better than the constructive bounds given in [2, 6, 13] from algebraic geometry codes. However, the situation is quite similar to the one in classical coding theory, i.e., for large q, the q-ary quantum Gilbert-Varshamov bound can be improved by the algebraic geometry bound as shown below. Before proceeding to the algebraic geometry bounds, we recall some background on classical algebraic geometry codes. Let X /F q be an algebraic curve of genus g. We denote by F q (X ) the function field of X. An element of F q (X ) is called a function. We write ν P for the normalized discrete valuation corresponding to the point P of X /F q. For a divisor G, we form the vector space L(G) ={x F q (X )\{0} : div(x)+g 0} {0}. Then L(G) is a finite-dimensional vector space over F q, and we denote its dimension by l(g). By the Riemann-Roch theorem we have l(g) deg(g)+1 g, 10

11 and equality holds if deg(g) 2g 1. Let P be a subset of X (F q ) and label the points in P as fellow P = {P 1,P 2,...,P n }. Choose a divisor G such that Supp(G) P =. Then ν Pi (f) 0 for all 1 i n and any f L(G). Consider the map φ : L(G) F n q, f (f(p 1 ),f(p 2 ),...,f(p n )). Then the image of φ forms a subspace of F n q that was defined as an algebraic geometry code by Goppa. The image of φ is denoted by C L (G; P). If n is bigger than the degree of G, then φ is an embedding and the dimension k of C L (G; P) is equal to l(g). The Riemann-Roch theorem makes it possible to estimate the parameters of the code C L (G; P). Proposition 3.3 (see Theorem of [23]). Let X /F q be an algebraic curve of genus g and let P be a set of n points on X. Choose a divisor G with g deg(g) <nand Supp(G) P =. Then C L (G; P) is an [n, k, d]-linear code over F q with k deg(g) g +1, d n deg(g). Moreover, the dimension k is equal to deg(g) g +1 if deg(g) 2g 1. Furthermore, the minimum distance d(cl (G; P)) of its dual code is at least deg(g) 2g +2. Proposition 3.4. If there is an algebraic curve X /F q with at least n +1points and genus g, then one has a q-ary ((n, q n 2m+2g 2,m 2g + 2))-quantum code for any 2g 2 <m<n. Proof. Let P 0,P 1,...,P n be n + 1 distinct rational points of X. Putting P = {P 1,...,P n }, C 1 = C L (mp 0, P) and C 2 = C L ((n m +2g 2)P 0, P) and applying Corollary 2.5, we obtain the desired result. Remark 3.5. If X /F q is the projective line, i.e., g = 0, then we get q-ary ((n, q n 2m 2,m+2))- quantum codes for any 1 <m<n 1. This class of quantum codes achieves the quantum Singleton bound. 11

12 Let N(X ) denote the number of F q -rational points of a curve X /F q of genus g(x ). According to the Weil bound: N(X ) q +1+2g(X ) q, the following two definitions make sense. For any prime power q and any integer g 0, put N q (g) := max N(X ), where the maximum is extended over all curves X /F q with g(x )=g. We also define the following asymptotic quantity A(q) := lim sup g N q (g). g We know from [23] that A(q) = q 1ifq is a square. The asymptotic version of Proposition 3.4 is the following result. Theorem 3.6. For a prime power q, one has α Q q (δ) 1 2δ 2 A(q). (3.2) Remark 3.7. The bound (3.2) is better than the quantum Gilbert-Varshamov bound (3.1) in an interval for some large q. In particular, when q is a square prime power bigger than or equal to 19 2, the bound (3.2) improves on the bound (3.1) in an interval around (q 2 1)/(2q 2 1). For instance, for q =19 2, the bound (3.2) improves on the bound (3.1) in the interval (0.3515, ). Recently, algebraic geometry bound was improved by nonlinear codes in classical coding [22]. Now we can employ the construction in Section 2 to derive an asymptotic lower bound improving the asymptotic quantum algebraic geometry bound given in Theorem 3.6. It seems that this improvement cannot be obtained from additive quantum codes. Theorem 3.8. For a prime power q, one has α Q q (δ) 1 2δ 2 A(q) + log q (1+ 1q 3 ). (3.3) 12

13 References [1] A. Ashikhmin and S. Litsyn, Upper bounds on the size of quantum codes, Sep., 1997, quant-ph/ [2] A. Ashikhmin, S. Litsyn and M. A. Tsfasman, Asymptotically good quantum codes, Phys. Rev. A, Vol. 63, no. 3, p , Mar [3] A. Ashikhmin and E. Knill, Non-binary quantum stabilizer codes, IEEE Trans. IT-47 (2001), [4] A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, Quantum error correction via codes over GF (4), IEEE Trans. IT-44 (1998), [5] A. R. Calderbank and P. W. Shor, Good quantum error-correcting codes exist, Phys. Rev. A, 54 (1996), [6] H. Chen, S. Ling and C. P. Xing, Asymptotically good quantum codes exceeding the Ashikhmin-Litsyn-Tsfasman bound, IEEE Trans. Inform. Theory, Vol. 47(2001), [7] K. Q. Feng and Z. Ma, A finite Gilbert-Varshamov bound for pure stabilizer quantum codes IEEE Trans. Inform. Theory, Vol. 50(2004), [8] K. Q. Feng and C. P. Xing, A construction of non-additive quantum codes, preprints, [9] K. Q. Feng, S. Ling and C. P. Xing, Asymptotic Bounds on Quantum Codes from Algebraic Geometry Codes, preprint, [10] E. Knill, Non-binary unitary error bases and quantum codes, Aug., 1996, quantph/ [11] E. Knill, Group representations, error bases and quantum codes, Aug., 1996, quantph/ [12] E. Knill and R. Laflamme, A theory of quantum error correcting codes, Phys. Rev. A, 55 (1997),

14 [13] R. Matsumoto, Improvement of the Ashikhmin-Litsyn-Tsfasman bound for quantum codes, IEEE Trans. Inform. Theory, Vol. 48(2002), [14] R. Matsumoto and T. Uyematsu, Constructing quantum error-correcting codes for p m - state systems from classical error-correcting codes, IEICE Trans. Fundamentals, E83-A(10) (2000), [15] E. M. Rains, Non-binary quantum codes, IEEE Trans. IT-45 (1999), [16] E. M. Rains, Quantum codes of minimal distance two, IEEE Trans. IT-45 (1999), [17] E. M. Rains, R. H. Hardin, P. W. Shor and N. J. A. Sloane, A nonadditive quantum code, Phys. Rev. Lett., 79 (1997), [18] D. Schlingemann and R. F. Werner, Quantum error-correcting codes associated with graphs, Phys. Rev. A, 65 (2002), [19] P. W. Shor, Scheme for reducing decoherence in quantum memory, Phys. Rev. A, 52 (1995), [20] A. M. Steane, Simple quantum error correcting codes, Phys. Rev. Lett., 77 (1996), [21] A. M. Steane, Multiple particle interference and quantum error correction, Proc. Roy. Soc. London A, 452 (1996), [22] H. Stichtenoth and C. P. Xing, Excellent non-linear codes from algebraic function fields, preprints, [23] M. A. Tsfasman and S. G. Vlǎduţ, Algebraic-Geometric codes, Kluwer, Holland,