We present some results that show that bounds from classical coding theory
still work in many cases of quantum coding theory

Ashikhmin, Alexei

English

arXiv

We present some results that show that bounds from classical coding theory still work in many cases of quantum coding theory

Ashikhmin, A.

UNT Digital Library

I Los Alamos National Laboratory is operated by the University of California for the United States Department of Energy under contract W-7405-ENG-36 TITLE: AUTHOR(S): SUBMITTED TO: REMARKS ON BOUNDS FOR QUANTUM CODES Alexei Ashikhmin Fifth Quantum on Computation Workshop DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsi- bility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. 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Remarks on Bounds for Quantum codes Alexei Ashikhmin Los Alamos National Laboratory Abstract We present some results that show that bounds from classical cod- ing theory still work in many cases of quantum coding theory. 1 Nondegenerate Quantum Codes Let & be an ((n,  li, d ) )  nondegenerate quantum code with projection oper- ator P. Let A; and B; be the quantum enumerators of &, [1],[2] A; = Tr(EP)Tr(EP), wt(E)=i B; = Tr(EPEP),  where E is an error operator. (See [l],[2] for definition of error operators and their weights.) wt(E)=i It is shown in [1],[2] that 1 "  At = - B;Pt(i, n),  2" 2=0 . where Pt(i, n) = ~f ,o( - l ) j3 t - j  (1) ("-!) is the Krawtchouk polynomial. Re- t -3  place A; by A; f K 2  and B; by B;/K.  Then we get . n, 1 .-  At = -E B;Pt(i, n) and A0 = Bo = 1. i=O 2°K Now we can use well known techniques to  get upper bounds for the quantity 2°K (see for example [3]).We describe this technique here for completeness. 1 c Let f(z) = Cr=oua;zi be a polynomial . Consider its expansion by Krawtchouk polynomials Then from (1) and (2) we lhave Suppose that fo > 0, fi 2 0, if i = 1,. . .,n, andf(0) > 0, f(i) 5 0, if i = d , .  . .,n. (3) Then taking into account that for nondegenerate codes Ai = Bi = 0 for i = 1,. . ., d - 1, we have From classical coding theory we know many useful polynomials that  satisfy conditions (3). Here we list some of them. Let f(z) = 44-d+1 ny=,i(l - 5 )  [4]. Then > 0. (3f i  = -(d?,) - So we get a quantum Singleton bound [5][6]: K 5 2n-2d+2. Note that fi = 0 for i 2 n - d + 2 and hence if a quantum code meets the Singleton bound it should be nondegenerate up to  n - d + 2 (a quantum code is called nondegenerate up to  t if A; = 0 for i = 1,. . . , t -  1 [6].)(In [4] this proposition is proven for all quantum codes.) Let d = 2e + 1 and ) 2  * ( EL0 (9) P, ( i  - 1, n - 1) f; = Then [4] f ( i )  = 0 for i = 11,. . ., n and f ( 0 )  = 4"/ c;f=o 3 i ( l ) ,  and from (4) we get the quantum analog of the Hamming bound: K 5 2n 3i(9).. 2 Analogously, from [3] it is straightforwardly follows that where and d k ( n )  is the smallest root of Pk(z,n).  Using the asymptotic version of this bound [3], as n tends to  infinity log, K 2 2 H 4 ( s )  - n/2 and d < nr4(z), where ( 5 )  Y q ( 4  = l/d(q - 1 - ( Q  - 2)s  - q/GGF-G),  and If&) = -zlog,z - (1 - z)logq(l  - z) + "log& - 1). (See figure 1,B.) Note that as tends to  zero $ M 0.316 which is bet- ter then the value n/3 from 191. However the latter bound is valid for all quantum codes. 2 Stabilizer Codes 2.1 Nondegenerate Stabilizer Codes In the case of nondegenerate stabilizer codes just all bounds for classical codes are valid. For example we can use the best known asymptotic bound for nonbinary codes from Ell]. (See figure 1, A). Note that as tends to  zero $ M 0.308 2.2 General Case Let Q be a quantum stabilizer [[n, IC]] code (degenerate or nondegenerate). In [7] it was shown that if the minimum distance of Q is d then there exist a classical binary code [n - 1, k, d ] .  Using this observation and a classical lin- ear programming bound, an upper bound for quantum stabilizer codes was 3 obtained [7] (see figure 1,C). Arguments used in [7] seem to  be complicated. We show that a simpler approach yield somewhat stronger bound. Let C and CI ,  C E C', be additive [n, n - k] and [n, n + I C ]  codes over GF(4) associated with Q .  '[f an additive code has cardinality 2t we will say that t is its dimension. So the dimensions of C and C' are n - k and n + IC respectively. It is known that Q has minimum distance d iff all code words from the set C' \ C have weights greater than or equal t o  d (see [lo] for details). We can consider codes C and C' as linear binary codes C and C' of length 2n. C' is dual to  C with respect to the symplectic inner product, i.e. if (a, b) E C and (a', b') E C' then (a, b) 0 (a', b') =< a, b' > @ < a', b >= 0. In [7]  it was shown that a generator matrix G of C can be written in the form 0 0 0 D1 D2 I, 1 ' Is AI A2 BI B2 0 G =  [ where Is and B1 are s x s matrices, A1 and B1 are s x k matrices, A2 and B2 are s x T matrices, D1 is T x s matrix, and D2 is T x k matrix. It is not difficult to check that all r'ows of the matrix G'- 0 0  O A T I k O '  O Ik D l  B,T o O i Is A1 A2 B1 B2 0 0 0 0 D1 0 2  IT are linearly independent and orthogonal (with respect to  the symplectic inner product) to rows of the matrix G. So the matrix G I  is a generator matrix for the code CI. It is obvious that if the minimum distance of codewords from CL \ C is d then the minimum dilstance of the binary code, say S, with generator is not less than d. The code S is an [n + k , 2 k ]  binary code. Applying now any classical bound for this code we get a bound for code Q. For example we can use the second McEliece-Rodemich-Rumsey- Welch bound [8]. The resulting quantum bound is better than the bound from 171 (see figure 1, D). In fact in [7] only a code with the generator matrix 1 0  0 0 AT I k  0 1  4 was taken into account. A generator matrix of C can be written in the form [lo] We can get a stronger bound if we consider codes C and CL over GF(4). Iko wA1 G =  [ W I k o  4" 7 (7) where Bj is a binary matrix, Aj is an arbitrary matrix, and w is an element of GF(4) of order 3. We will say that G defines a code of type 4'OOk1. If in some I coordinates an additive code, say S ,  of length n contains only 0-s and CY-s, a E GF(4),a # 0, then we will say that S is a mixed code of lengths 1 and n - 1. Note that if C is a code of type 4k02k1 of length n then corresponding quantum code Q is an [[n, k ] ]  = [[n, 2n  - 2k0 - k l ] ]  code. Proposition 1 Let d be a minimum distance of Q .  Then  d is  not grater than the minimum distance of the best additive mixed code of lengths k1 and n - Eo - E l  and dimension 2n - 4ko - 2kl .  I n  particular i) i f  kl = 0 then d is not greater than the minimum distance of the best [n - ko, 2 n  - 4ko] = [(n + k ) / 2 , 2 k ]  additive code. ii) i f  kl < 2n;4kQ = 2k then d is not greater than the minimum distance of the best [n - ko - k l ,  2 n  - 4ko - 3k1] = [ T ,  n f k - k l  2k - IC11 additive code. Proof Let G' be a generator matrix of a complementary code of C to  C I ,  i.e. [,!?,I is a generator matrix of the code CI .  Due to the structure (7) of matrix G we can make elements of G' on the first Eo positions equal 0 and elements on the next k1 positions equal to  0 or w.  So G' will have the form[0 0 1  0 2 1  where D1 is an 2 n  - 4ko - 2k1 x k1 matrix consisting from 0-s and w-s and 0 2  is is an arbitrary 2 n  - 4ko - 2k1 x n - ko - kl matrix. Since the minimum distance of a complementary code has to  be not less than minimum distance of the quantum code Q we get the assertion. i) follows from the previous. ii) In the case k1 < 2n<44k0 the matrix [0 D1 0 2 1  can be transformed to the form [: ::], where A1 is an kl x kl matrix consisting form 0-s and w-s, B1 and B2 are arbitrary k1 x n - LO - kl and 2 n  - 4k0 - 3k1 x n - ko - kl matrices. Since the 5 subcode with the generator matrix [O 0 B2] has length n-ko  - k l ,  dimension 2n - 4ko - 3kl and its minimum distance has to  be not less than minimum distance of Q ,  the assertion follows. 0 Let S be a mixed code of lengths I and n - I and dimension k . Using usual standard w e  can get the Plotkin an d hamming bounds for the code S. 1 Lemma 1 d 5 s G ( Z 2 k - 1  + 3(n  - 92"' Lemma 2 Combining Lemma 1 and Proposition 1 we get a Plotkin type bound for degenerate codes. Proposition 2 n + IC 3 4"' IC1 4k--1 d <  -~+-- 2 4 k - 1  2 4'"-1'  Combining Lemma 1 and Proposition 1 we get a Hamming type bound for degenerate codes. Proposition 3 Consider the case kl = 0. Then we have Asymptotically this becomes 6 In the case kl = 0 we can also apply asymptotic bounds for complemen- tary codes [(n + k)/2,2k] from Proposition 1. For example we can use an asymptotic bound from I113 (see fig. 1,E). Note that as tends to  zero d = 0.375. n Let IC1 .> 0. Then according to  Proposition 1 we can estimate a minimum distance of Q for n - kl < k < k1/2 as minimum distance of an additive ,2k - kl] code. For example using the asymptotic bound from [ll] and fixing k1 = 0, kl = 0.2,kl = 0.5 we get bounds A,B,C, figure 2. If k > k1/2 we have to  apply bounds for a mixed code. Generalizations of asymptotic bounds for mixed codes are not known to the author. Note that we can still use asymptotic bounds based on representation of complimentary code in the binary form (6) or the bound from [7] (bounds C and D on fig.1) for an arbitrary quantum stabilizer code. References [l] P.W. Shor and R. Laflamme, “Quantum analog of the MacWilliams identities in classical coding theory,” LANL e-print quant-ph/9610040. [2] E.M. Rains, “Quantum weight enumerators,” LANL e-print quant- ph/9612015. [3] V.I. Levenshtein, “Krawtchouk Polynomials and Universal Bounds for Codes and Designs in Hamming Spaces,” IEEE Trans. on Inf. Theory, 41, no. 5, 1995, pp. 1303-1321. [4] P. Delsarte, “Bounds for unrestricted codes, by linear programming,” Philips Res. Reports, 27 , 1972, pp. 272-289. [5] E. Knill and R. Laflamme, “A theory of quantum error correcting codes,” LANL e-print quant-ph/9604034. [6] E.M. Rains, “Nonbinary quantum codes,” LANL e-print quant-ph/97?. [7] R. Cleve, “Quantum stabilizer codes and classical linear codes,” LANL e-print quant-ph/9612048. 7 t [8] R.J. McEliece, E.R. Rodemich, H. Rumsey Jr., and L.R. Welch, “New upper bounds on the rate of a code via the Delsarte-MacWilliams in- equalities,” IEEE Trans. on Inf. Theory, vol. IT-23, 1977, pp. 157-166. [9] E.M. Rains, “Quantum shadow enumerators,” LANL e-print quant- ph/961101. [lo] A.R. Calderbank, E.M. Rains, P.W. Shor and N.J.A. Sloane, “Quan- tum errors correction via codes over GF(4),” LANL e-print quant- ph/9608006. [ll] M. J.  Aaltonen, “A new bound on nonbinary block codes,”Discr. Math., v01. 83, pp. 139-160, 1990. 8 

Remarks on Bounds for Quantum Codes

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