aa r X i v : . [ qu a n t - ph ] J a n Quantum Goethals-Preparata Codes
Markus Grassl
Institute for Quantum Optics and Quantum InformationAustrian Academy of SciencesTechnikerstraße 21a, 6020 Innsbruck, AustriaEmail: [email protected]
Martin R¨otteler
NEC Laboratories America, Inc.4 Independence Way, Suite 200Princeton, NJ 08540, USAEmail: [email protected]
Abstract —We present a family of non-additive quantumcodes based on Goethals and Preparata codes with parameters ((2 m , m − m +1 , . The dimension of these codes is eight timeshigher than the dimension of the best known additive quantumcodes of equal length and minimum distance. Index Terms —Non-additive quantum code, Goethals code,Preparata code
I. I
NTRODUCTION
Most of the known quantum error-correcting codes(QECCs) are based on the so-called stabilizer formalism whichrelates quantum codes to certain additive codes over GF (4) (see, e. g., [3], [7]). It is known that non-additive QECCs canhave a higher dimension compared to additive QECCs with thesame length and minimum distance [5], [14], [17], [18]. Allthese examples of non-additive QECCs are examples of so-called codeword stabilized quantum codes which are obtainedas the complex span of some so-called stabilizer states, whichcorrespond to self-dual additive codes. In [9] we have extendedthe framework of stabilizer codes to the union of stabilizercodes (see [8]). This allows to construct non-additive codesfrom any stabilizer code. In general, these non-additive QECCscorrespond to non-additive codes over GF (4) which can bedecomposed into cosets of an additive code which containsits dual. Using a construction similar to that of so-called CSScodes (see [4], [15]), families of non-additive quantum codesbased on the binary Goethals and Preparata codes were derivedin [9]. Here we present a new family of non-additives quantumcodes which have a dimension that is eight times higher thanthe dimension of the best known additive quantum codes.II. U NION S TABILIZER C ODES
A. Stabilizer codes
We start with a brief review of the stabilizer formalism forquantum error-correcting codes and the connection to additivecodes over GF (4) (see, e. g., [3], [7]). A stabilizer codeencoding k qubits into n qubits having minimum distance d ,denoted by C = [[ n, k, d ]] , is a subspace of dimension k ofthe complex Hilbert space ( C ) ⊗ n of dimension n . The codeis the joint eigenspace of a set of n − k commuting operators S , . . . , S n − k which are tensor products of the Pauli matrices σ x = (cid:18) (cid:19) , σ y = (cid:18) − ii (cid:19) , σ z = (cid:18) − (cid:19) , or identity. The operators S i generate an Abelian group S with n − k elements, called the stabilizer of the code. It isa subgroup of the n -qubit Pauli group P n which itself isgenerated by the tensor product of n Pauli matrices andidentity. We further require that S does not contain any non-trivial multiple of identity. The normalizer of S in P n , denotedby N , acts on the code C = [[ n, k, d ]] . It is possible to identify k logical operators X , . . . , X k and Z , . . . , Z k such thatthese operators commute with any element in the stabilizer S ,and such that together with S they generate the normalizer N of the code. The operators X i mutually commute, and sodo the operators Z j . The operator X i anti-commutes with theoperator Z j if i = j and otherwise commutes with it.It has been shown that the n -qubit Pauli group correspondsto a symplectic geometry, and that one can reduce the problemof constructing stabilizer codes to finding additive codes over GF (4) that are self-orthogonal with respect to a symplecticinner product [2], [3]. Up to a scalar multiple, the elementsof P can be expressed as σ ax σ bz where ( a, b ) ∈ F is abinary vector. Choosing the basis { , ω } of GF (4) , where ω is a primitive element of GF (4) with ω + ω + 1 = 0 , weget the following correspondence between the Pauli matrices,elements of GF (4) , and binary vectors of length two:operator GF (4) F I σ x σ y ω (11) σ z ω (01) This mapping extends naturally to tensor products of n Paulimatrices being mapped to vectors of length n over GF (4) orbinary vectors of length n . We rearrange the latter in such away that the first n coordinates correspond to the exponentsof the operators σ x and write the vector as ( a | b ) , i. e., g = σ a x σ b z ⊗ . . . ⊗ σ a n x σ b n z ˆ= ( a | b ) = ( g X | g Z ) . (1)Two operators corresponding to the binary vectors ( a | b ) and ( c | d ) commute if and only if the symplectic inner product a · d − b · c = 0 . In terms of the binary representation,the stabilizer corresponds to a binary code C which is self-orthogonal with respect to this symplectic inner product, andthe normalizer corresponds to the symplectic dual code C ∗ .In terms of the correspondence to vectors over GF (4) , thestabilizer and normalizer correspond to an additive code over F (4) and its dual with respect to an symplectic innerproduct, respectively, which we will also denote by C and C ∗ .The term additive quantum code refers to this correspondence.The minimum distance d of the quantum code is given as theminimum weight in the set C ∗ \ C ⊂ GF (4) n which is lowerbounded by the minimum distance d ∗ of the additive code C ∗ .If d = d ∗ , the code is said to be pure , and for d ≥ d ∗ , thecode is said to be pure up to d ∗ .Fixing the logical operators X i and Z j , there is a canonicalbasis for the additive quantum code C . The stabilizer group S of the quantum code together with the logical operators Z j generate an Abelian group of order n which correspondsto a self-dual additive code. The joint +1 -eigenspace is one-dimensional, hence there is a unique quantum state | . . . i ∈C stabilized by all elements of S . An orthonormal basis of thecode C is given by the states | i i . . . i k i = X i · · · X i k k | . . . i , (2)where ( i i . . . i k ) ∈ F k . B. Union stabilizer codes
The stabilizer group S gives rise to an orthogonal decompo-sition of the space ( C ) ⊗ n into common eigenspaces of equaldimension. The stabilizer code C is the joint +1 -eigenspaceof dimension k . In general, the joint eigenspaces of S canbe labeled by the eigenvalues of a set of n − k generators of S . Moreover, the n -qubit Pauli group P n operates transitivelyon the eigenspaces. Hence one can identify a set T ⊂ P n of n − k operators such that ( C ) ⊗ n = M t ∈T t C . (3)Note that each of the spaces t C is a quantum error-correctingcode with the same parameters as the code C and stabilizergroup t S t − . The decomposition (3) corresponds to the de-composition of the n -qubit Pauli group P n into cosets withrespect to the normalizer N of the code C and likewise to thedecomposition of the full vector space GF (4) n into cosets ofthe additive code C ∗ .The main idea of union stabilizer codes is to find a subset T of the translations T such that the space L t ∈T t C is agood quantum code (see [8], [9]). Definition 1 (union stabilizer code):
Let C = [[ n, k ]] be astabilizer code and let T = { t , . . . , t K } be a subset of thecoset representatives of the normalizer N of the code C in P n . Then the union stabilizer code is defined as C = M t ∈T t C . Without loss of generality we assume that T contains identity.The dimension of C is K k , and we will use the notation C = (( n, K k , d )) .Similar to (2) a canonical basis of the union stabilizer code C is given by | j ; i i . . . i k i = t j X i · · · X i k k | . . . i , (4) where j = 1 , . . . , K , ( i i . . . i k ) ∈ F k , and X i are logicaloperators of the stabilizer code C .In order to compute the minimum distance of this code, wefirst consider the distance between two spaces t C and t C .As for a fixed stabilizer code C two spaces t C and t C areeither identical or orthogonal, we can define the distance ofthem as follows: dist( t C , t C ) := min { wgt( p ) : p ∈ P n | pt C = t C } . (5)Here wgt( p ) is the number of tensor factors in the n -qubitPauli operator p that are different from identity. Clearly, dist( t C , t C ) = dist( t − t C , C ) . The two spaces areidentical if and only if t − t is an element of the normalizergroup N , or equivalently, if the cosets C ∗ + t and C ∗ + t ofthe additive normalizer code C ∗ are identical. (Note that wedenote both an n -qubit Pauli operator and the correspondingvector over GF (4) by t i .) Hence the distance (5) can also beexpressed in terms of the associated vectors over GF (4) . Lemma 2:
The distance of the spaces t C and t C equalsthe minimum weight in the coset C ∗ + t − t . Proof:
Direct computation shows dist( t C , t C ) = dist( C ∗ + t , C ∗ + t )= dist( C ∗ + ( t − t ) , C ∗ )= min { wgt( c + t − t ) : c ∈ C ∗ } = min { wgt( v ) : v ∈ C ∗ + t − t } . While the distance between the cosets C ∗ + t j is an upperbound on the minimum distance of the union code C , the trueminimum distance can be derived from the following codeover GF (4) . Definition 3 (union normalizer code):
With the union sta-bilizer code C we associate the (in general non-additive) unionnormalizer code given by C ∗ = [ t ∈T C ∗ + t = { c + t j : c ∈ C ∗ , j = 1 , . . . , K } , where C ∗ denotes the additive code associated with thenormalizer N of the stabilizer code C . We will refer toboth, the vectors t i and the corresponding unitary operators,as translations . Theorem 4:
The minimum distance of a union stabilizercode with union normalizer code C ∗ is given by d = min { wgt( v ) : v ∈ ( C ∗ − C ∗ ) \ e C }≥ d min ( C ∗ )= min { dist( c + t i , c ′ + t i ′ ) : t i , t i ′ ∈ T , c, c ′ ∈ C ∗ c + t i = c ′ + t i ′ } , where C ∗ − C ∗ := { a − b : a, b ∈ C ∗ } denotes the set of alldifferences of vectors in C ∗ , and e C ≤ C is the additive codethat corresponds to all elements of the stabilizer group S thatcommute with all t j ∈ T . Proof:
Let E ∈ P n be an n -qubit Pauli error of weight < wgt( E ) < d . For two canonical basis states | ψ a i and | ψ b i s given in (4) we consider the inner product h ψ a | E | ψ b i = h j ; i i . . . i k | E | j ′ ; i ′ i ′ . . . i ′ k i = h . . . | X i · · · X i k k t j Et j ′ X i ′ · · · X i ′ k k | . . . i = ± h . . . | X i + i ′ · · · X i k + i ′ k k t j t j ′ E | . . . i . If E ∈ S commutes with all t j ∈ T , then h ψ a | E | ψ b i = δ ab .Otherwise, E / ∈ C ∗ − C ∗ since < wgt( E ) < d , and hencethe inner product vanishes.III. T HE B INARY G OETHALS AND P REPARATA C ODES
In this section we recall some properties of the binaryGoethals codes [6] and the Preparata codes [13]. It has beenshown that variations of these codes have a simple descriptionas Z -linear codes [10], but in our context the description interms of cosets of linear binary codes is used.In the following m is an even integer ( m ≥ ) and n = 2 m − − . Let α be a primitive element of the finite field GF (2 m − ) . By µ i ( z ) we denote the minimal polynomial of α i over GF (2) , i. e., the polynomial with roots α j for j = i k .The idempotent θ i ( z ) is the unique polynomial satisfying θ i ( α i ) = 1 and θ i ( α j ) = 0 for j = i k .Codewords of a cyclic code can be represented by polynomials f ( z ) , and we use ( f ( z ); f (1)) to denote the codeword of theextended cyclic code obtained by adding an overall paritycheck. Similar, we use ( f ( z ); f (1); g ( z ); g (1)) to denote thejuxtaposition of codewords of two extended cyclic codes. Definition 5 (Goethals code [6]):
The Goethals code G ( m ) of length m is the union of m − cosets of the linearbinary code C G = [2 m , m − m + 2 , . The code C G isobtained via the | u | u + v | construction applied to the extendedcyclic codes C and C . The cyclic code C is a single-errorcorrecting code with generator polynomial µ ( z ) , and C isgenerated by µ ( z ) µ r ( z ) µ s ( z ) where r = 1 + 2 m/ − and s = 1 + 2 m/ − . The non-zero coset representatives are givenby ( z i ; 1; z i θ ( z ); 0) for i = 1 , . . . , n − .An alternative description of Goethals codes has been givenin [1]. The codewords are described by pairs ( X, Y ) of subsetsof GF (2 m − ) . The corresponding codeword is given by thejuxtaposition of the characteristic functions χ X and χ Y of thetwo set X and Y , i. e. ( X, Y ) ˆ= (1 X ( α i ); 1 X (0); 1 Y ( α i ); 1 Y (0)) , where X ( α i ) is a short-hand for the vector X ( α i ) = (1 X ( α ) , X ( α ) , . . . , X ( α n − )) and S ( x ) = ( if x ∈ S, if x / ∈ S. The non-zero elements of X and Y give rise to the polyno-mials f X ( z ) and f Y ( z ) given by f S ( z ) = n − X i =0 S ( α i ) z i . (6) Definition 6 (Goethals code [1]):
The Goethals code G ( m ) of length m consists of the codewords described by all pairs ( X, Y ) satisfying:a) | X | is even, | Y | is even,b) X x ∈ X x = X y ∈ Y y ,c) X x ∈ X x r + X x ∈ X x ! r = X y ∈ Y y r ,d) X x ∈ X x s + X x ∈ X x ! s = X y ∈ Y y s .In order to relate the two definitions, we distinguish threecases.1) X = Y : Conditions c) and d) imply that P x ∈ X x = 0 .This is true for all codewords of the cyclic code gener-ated by µ ( z ) . Adding an overall parity check impliesCondition a).2) X = ∅ : The left hand side of Conditions b), c),and d) vanishes, so the solutions for Y correspondto an extended cyclic code with generator polynomial µ ( z ) µ r ( z ) µ s ( z ) .3) X = { , x = α i } : From (6) it follows that f Y ( α ) = P y ∈ Y y . So Condition b) holds for the set Y corre-sponding to f Y ( z ) = z i θ ( z ) . The left hand side ofConditions c) and d) vanishes, so the solutions for Y are elements of the extended cyclic code with generatorpolynomial µ r ( z ) µ s ( z ) . As neither r nor s is a power oftwo, the polynomial θ ( z ) and hence f Y ( z ) = z i θ ( z ) vanishes for α r and α s , i. e., Conditions c) and d) hold.Finally, all codewords of the Goethals code as given inDefinition 5 are the juxtaposition of two binary vectors ofeven weight, i. e., Condition a) holds. Hence any codewordgiven by Definition 5 fulfills the conditions of Definition 6.The equivalence of the definitions follows from the fact thatthe codes have equal size.Next we consider the definition of Preparata codes similarto Definition 6 given in [1]. Definition 7 (Preparata code [1]):
The extended Preparatacode P ( m ) of length m and parameter σ consists of thecodewords described by all pairs ( X, Y ) satisfying:a) | X | is even, | Y | is even,b) X x ∈ X x = X y ∈ Y y ,c) X x ∈ X x σ +1 + X x ∈ X x ! σ +1 = X y ∈ Y y σ +1 ,Here σ is a power of two and gcd( σ ± , n ) = 1 .For σ = 2 m/ − and n = 2 m − − we compute (cid:0) m − − (cid:1) − (cid:16) m/ − ± (cid:17) (cid:16) m/ ∓ (cid:17) = 1 , showing that gcd( σ ± , n ) = 1 . Hence for this particularchoice of σ , the Preparata code of Definition 7 containsthe Goethals code. What is even more, we can describe thePreparata code similar to Definition 5 as the union of cosetsf the linear binary code C P which contains the linear binarycode C G . Definition 8:
The extended Preparata code P ( m ) of length m is the union of m − cosets of the linear binary code C P = [2 m , m − m + 1 , . The code C P is obtained viathe | u | u + v | construction applied to the extended cyclic codes C and C . The cyclic code C is a single-error correctingcode with generator polynomial µ ( z ) , and C is generatedby µ ( z ) µ s ( z ) where s = 1 + 2 m/ − . The non-zero cosetrepresentatives are given by ( z i ; 1; z i θ ( z ); 0) .Comparing Definitions 5 and 8 we see that we can use the verysame coset representatives to construct the Goethals and thePreparata code as union of cosets of the linear binary codes C G and C P , respectively. Moreover, all codes lie between codesthat are equivalent to the Reed-Muller codes RM ( m − , m ) and RM ( m − , m ) = [2 m , m − m − , (see [11]). This isillustrated by the following diagram: RM( m − , m )[2 m , m − m + 2 ,
8] = C G [2 m , m − m + 1 ,
6] = C P ✘✘✘✘✘✘ G ( m ) = S i C G + t i P ( m ) = S i C P + t i [2 m , m − m − ,
4] = RM( m − , m ) The components of the codes are summarized as follows: C : cyclic code generated by µ ( z ) C : cyclic code generated by µ ( z ) µ s ( z ) C : cyclic code generated by µ ( z ) µ r ( z ) µ s ( z ) r = 1 + 2 m/ − , s = 1 + 2 m/ − C G : | u | u + v | construction applied to the extended cycliccodes C and C C P : | u | u + v | construction applied to the extended cycliccodes C and C t i : n + 1 coset representatives with t i = ( ( z i ; 1; z i θ ( z ); 0) for i = 0 , . . . , n − , (0 , . . . , for i = n. IV. T HE Q UANTUM G OETHALS -P REPARATA C ODES
Before presenting the new family of non-additive quantumcodes, we recall Steane’s construction to enlarge the dimensionof CSS codes.
Theorem 9 (see [16]):
Let C = [ n, k, d ] and C ′ = [ n, k ′ >k + 1 , d ′ ] be linear binary codes with C ⊥ ≤ C < C ′ . Thenthere exists an additive quantum code C = [[ n, k + k ′ − n, ≥ min( d, d ′ / . Given a generator matrix G of the code C and a generator matrix D of the complement of C in C ′ , the normalizer of the code C is generated by G GD AD , where A is a fixed-point free linear transformation.As the code C G contains a code that is isomorphic to theReed-Muller code RM ( m − , m ) it follows that C ⊥G ≤ C G .Hence we can apply Steane’s construction [16] to the chain C ⊥G ≤ C G < C P of linear binary codes and obtain an additivequantum code with parameters C = [[2 m , m − m + 3 , .In a second step we use the K = 2 m − coset represen-tatives t i of the decomposition of both the Goethals andthe Preparata code. This yields a non-additive code withdimension K m − m +3 = 2 ℓ where ℓ = 2 m − m + 1 . G GD AD t t ... ... t t K ... ... t K t ... ... t K t K Fig. 1. Structure of the non-additive union normalizer code of the quantumGoethals-Preparata codes.
Theorem 10:
Let C = [[2 m , m − m +3 , be the additivequantum code obtained from the chain of linear binary codes C ⊥G ≤ C G ≤ C P using Steane’s enlargement construction.Furthermore, let T = { ( t i | t j ) : i, j = 0 , . . . , m − − } where t i are the coset representatives used to obtain the Goethals andPreparata code. Then the quantum Goethals-Preparata code is a union stabilizer code given by C and T . The minimumdistance of the quantum Goethals-Preparata code is eight. Proof:
Let G denote a generator matrix of the code C G and let D be such that ( GD ) generates C P . The structure of thenon-additive union-normalizer code of the quantum Goethals-Preparata codes is illustrated in Fig. 1. A generator matrix ofthe normalizer of the additive quantum code C is given abovethe horizontal line, while the set of translations is listed belowthe horizontal line. Every codeword of the non-additive unionnormalizer code is of the form g = ( g X | g Z ) = ( c + v + t i | c + w + t j ) , where c , c ∈ C G = [2 m , m − m +2 , and v, w ∈ C P /C G .For g, g ′ ∈ C ∗ , g = g ′ we compute dist( g, g ′ ) = dist(( c + v + t i | c + w + t j ) , ( c ′ + v ′ + t ′ i | c ′ + w ′ + t ′ j ))= wgt(( c ′′ + v ′′ + t i − t ′ i | c ′′ + w ′′ + t j − t ′ j )) , here c ′′ = c − c ′ and c ′′ = c − c ′ are codewords of C G ,and v ′′ = v − v ′ , w ′′ = w − w ′ are codewords of C P /C G . Ingeneral, the weight of g = ( g X | g Z ) is given by wgt(( g X | g Z )) = 12 (wgt( g X ) + wgt( g Z ) + wgt( g X + g Z )) . Hence we get dist( g, g ′ ) =12 wgt( c ′′ + v ′′ + t i − t ′ i ) (7a) + 12 wgt( c ′′ + w ′′ + t j − t ′ j ) (7b) + 12 wgt( c ′′ + c ′′ + v ′′ + w ′′ + t i − t ′ i + t j − t ′ j ) . (7c)By Steane’s construction the vectors v ′′ and w ′′ are eitherboth zero, or both are non-zero and they are different. For v ′′ = w ′′ = 0 , we can assume without loss of generality thatthe vectors in (7a) and (7b) are both non-zero. The weightof these vectors equals the distance between two codewordsof the Goethals code, so it is at least 8. For v ′′ = 0 = w ′′ the terms (7a) and (7b) equal the distance of two codewordsof the Preparata code, so they are lower bounded by 6. Wewill show that for v ′′ = w ′′ , the vector in (7c) is a non-zerocodeword of the linear code isomorphic to the Reed-Mullercode RM ( m − , m ) , hence its weight is at least 4. For this,consider the vectors a = ( a ; a ) = c ′′ + c ′′ + v ′′ + w ′′ = 0 and b = ( b ; b ) = t i − t ′ i + t j − t ′ j . The coset representativesare of the form t i = ( z i ; 1; z i θ ( z ); 0) , so the second half b of b is a codeword of the extended cyclic code generatedby θ ( z ) , while a is a codeword of the extended cyclic codegenerated by µ ( z ) . The intersection of the two codes is trivial,so a = b only if a = b = 0 . Then wgt( b ) ≤ while wgt( a ) ≥ since = a ∈ C P . Hence a = b .To our best knowledge, the best additive quantum code withthe same length and minimum distance has dimension m − m − . Codes with these parameters can, e. g., be obtainedby applying Steane’s construction to extended primitive BCHcodes [2 m , m − m − , and [2 m , m − m − , (see [16]).In the following table we give the parameters of the first codesin these families. Additionally, we give the parameters of thenon-additive quantum codes derived from Goethals codes in[9]. Goethals enlarged BCH Goethals-Preparata ((64 , , , , , , , , , , , , , , , , , , V. C
ONCLUSIONS
We have constructed some new non-additive quantum codesfrom nested non-linear binary codes which can be decomposedinto cosets of linear codes which contain their dual. It isinteresting to find more good non-linear binary or quaternarycodes with this property. Recently, Ling and Sol´e have constructed some non-additivequantum codes from Z -linear codes using a CSS-like con-struction [12]. So far it is not clear whether the non-additivecodes presented here can also be put into the framework of Z -linear codes. A CKNOWLEDGMENTS
We acknowledge fruitful discussions with Vaneet Aggarwaland Robert Calderbank. Markus Grassl would like to thankNEC Labs., Princeton for the hospitality during his visit aswell as Tero Laihonen, Kalle Ranto, and Sanna Ranto fordiscussions on variations of Goethals codes. This work waspartially supported by the FWF (project P17838).R
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