Non-Additive Quantum Codes from Goethals and Preparata Codes

Non-Additi v e Quantum Codes from Goethals and Preparata Codes Markus Grassl Institute for Quantum Optics and Quantum Information Austrian Academy of Sciences T echnikerstraße 21a, 6020 Innsbruck, Austria Email: markus.grassl@oeaw .ac.at Martin R ¨ otteler NEC Laboratories America, Inc. 4 Independence W ay , Suite 200 Princeton, NJ 08540, USA Email: mroetteler@nec-labs.com Abstract — W e extend the stabilizer formalism to a class of non- additive quantum codes which are constructed fr om non-linear classical codes. As an example, we present infinite families of non- additive codes which are derived from Goethals and Preparata codes. I . I N T RO D U C T I O N Recently , se veral ne w non-additi ve quantum error -correcting codes (QECCs) hav e been constructed that hav e higher dimen- sion than additi ve QECCs with the same length and minimum distance [5], [16], [17]. The first example of such a code is the code ((5 , 6 , 2)) of [11] which has been found via numerical optimization. Afterwards, the code has been identified as the span of a particular state and its image under fiv e unitary transformations (see also [7]). The recently discovered codes ((9 , 12 , 3)) and ((10 , 24 , 3)) (see [16], [17]) start with a so- called graph state which corresponds to a stabilizer state, i. e., a stabilizer code with parameters [[ n, 0 , d ]] (see [8], [13]). A basis of the quantum code is obtained by this initial state together with its image under some tensor products of Pauli matrices and identity (Pauli operators). The distance between any pair of these states can be defined as the minimal weight of a Pauli operator transforming one state into the other (see below). The problem of finding a code of high dimension can be stated as finding a maximal clique in a sear ch graph whose vertices are all images of the initial state. There is an edge between two states if their distance is at least the prescribed minimum distance. Using the formalism of graph states, Cross et al. show in [5] that it is sufficient to consider Z -only operators in order to define the search graph, but this has the disadv antage that the distance between two of these states is not necessarily equal to the number of Z operators in the tensor product. Moreover , constructing a code (( n, K , d )) requires to find a clique of size K in a search graph with 2 n vertices. Fixing one basis state, the graph can be slightly simplified. Howe ver , for the code ((10 , 24 , 3)) the simplified graph still has 678 vertices and 149.178 edges. A different approach for constructing non-additiv e QECCs based on Boolean functions and projection operators has been presented in [1]. Ev aluating the Boolean function at the projection operators, the code is giv en as the sum of the image of products of the projections. Finally , we mention the non- additiv e QECCs obtained by the method given in [12]. Those codes ha ve the same parameters as the additiv e CSS codes that can be obtained using the same underlying binary codes. In this paper , we extend the approach of combining unitary images of stabilizer states to arbitrary stabilizer codes as starting point. For a stabilizer code [[ n, k , d ]] , the search graph has only 2 n − k vertices, and a clique of size K yields a quantum code of dimension K × 2 k . What is more, we present some infinite families of non-additiv e quantum codes which are constructed using non-linear binary codes, av oiding the NP-hard problem of finding a maximal clique in the search graph. Finally , we show how to obtain encoding circuits for the resulting non-additi ve codes using encoding circuits for the underlying stabilizer codes. I I . A B R I E F R E V I E W O F T H E S TA B I L I Z E R F O R M A L I S M W e start with a brief revie w of the stabilizer formalism for quantum error-correcting codes and the connection to additiv e codes over GF (4) (see, e. g., [4], [6]). A stabilizer code encoding k qubits into n qubits having minimum distance d , denoted by C = [[ n, k , d ]] , is a subspace of dimension 2 k of the complex Hilbert space ( C 2 ) ⊗ n of dimension 2 n . The code is the joint eigenspace of a set of n − k commuting operators S 1 , . . . , S n − k which are tensor products of the Pauli matrices σ x =  0 1 1 0  , σ y =  0 − i i 0  , σ z =  1 0 0 − 1  , or identity . The operators S i generate an Abelian group S with 2 n − k elements, called the stabilizer of the code. It is a subgroup of the n -qubit P auli group P n which itself is generated by the tensor product of n Pauli matrices and identity . W e further require that S does not contain any non- trivial multiple of identity . The normalizer of S in P n , denoted by N , acts on the code C = [[ n, k , d ]] . It is possible to identify 2 k logical operators X 1 , . . . , X k and Z 1 , . . . , Z k such that these 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 so do the operators Z j . The operator X i anti-commutes with the operator Z j if i = j and otherwise commutes with it. It has been sho wn that the n -qubit P auli group corresponds to a symplectic geometry and that one can reduce the problem of constructing stabilizer codes to finding additive codes ov er GF (4) that are self-orthogonal with respect to a symplectic inner product [3], [4]. Up to a scalar multiple, the elements of P 1 can be expressed as σ a x σ b z where ( a, b ) ∈ F 2 2 is a binary vector . Choosing the basis { 1 , ω } of GF (4) , where ω is a primitiv e element of GF (4) with ω 2 + ω + 1 = 0 , we get the following correspondence between the Pauli matrices, elements of GF (4) , and binary vectors of length two: operator GF (4) F 2 2 I 0 (00) σ x 1 (10) σ y ω 2 (11) σ z ω (01) This mapping extends naturally to tensor products of n Pauli matrices being mapped to vectors of length n over GF (4) or binary vectors of length 2 n . W e rearrange the latter in such a way that the first n coordinates correspond to the exponents of the operators σ x and write the vector as ( a | b ) , i. e., g = σ a 1 x σ b 1 z ⊗ . . . ⊗ σ a n x σ b n z ˆ = ( a | b ) = ( g X | g Z ) . (1) T w o 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, and the normalizer corresponds to the symplectic dual code C ∗ . The stabilizer together with the logical operators Z i generate a self-dual code. In terms of the correspondence to vectors over GF (4) , the stabilizer and normalizer correspond to an additive code over GF (4) and its dual with respect to an symplectic inner product, respectiv ely , which we will also denote by C and C ∗ . The minimum distance d of the quantum code is giv en as the minimum weight in the set C ∗ \ C which is lower bounded by the minimum distance d ∗ of C ∗ . If d = d ∗ , the code is said to be pur e , and for d ≥ d ∗ , the code is said to be pur e up to d ∗ . I I I . T H E U N I O N O F S TA B I L I Z E R C O D E S Note that we have defined a stabilizer code C as the joint eigenspace of the commuting operators S i generating the stabilizer S . The term stabilizer suggests that the code is the joint +1 eigenspace of the operators. Howe ver , for each of the generators S i we may choose either the eigenspace with eigen value +1 or the eigenspace with eigenv alue − 1 . This giv es rise to a decomposition of the space ( C 2 ) ⊗ n into 2 n − k mutually orthogonal spaces which can be labeled by the eigen values of the n − k generators S i , or equiv alently , by the characters χ of the stabilizer group S . Moreov er , the n -qubit Pauli group P n acts transitiv ely on these spaces. From now on we fix the code C as the the joint +1 eigenspace corresponding to the tri vial character . Let t ∈ P n be an arbitrary n -qubit Pauli operator . Then we can define a character of S on the generators S i as follows χ t ( S i ) := ( +1 if t and S i commute, − 1 if t and S i anti-commute. As the elements of the normalizer N commute with all elements of the stabilizer S , two elements t 1 and t 2 define the same character if t − 1 2 t 1 ∈ N . Hence the set of characters corresponds to the cosets of N in P n . If T is a set of coset representativ es, we can write the decomposition of the full space as ( C 2 ) ⊗ n = M t ∈T t C . (2) Note that measuring the eigen v alues of the generators S i projects onto one of these space t C , corresponding to the character χ t giv en by the sequence of eigen v alues. In terms of the classical codes, the eigen values correspond to an error- syndrome which is obtained by computing the symplectic inner product of the recei ved vector with the n − k vectors corresponding to the generators of the stabilizer , i. e., a basis of the code C . For all vectors of the dual code C ∗ corresponding to the normalizer N , the inner product is zero. So the different spaces t C correspond to cosets C ∗ + t of the code C ∗ . As for a fixed code C two spaces t 1 C and t 2 C are either identical or orthogonal, we can define the distance of them as follows: dist( t 1 C , t 2 C ) := min { wgt( p ) : p ∈ P n | pt 1 C = t 2 C } . (3) Here wgt( p ) is the number of tensor f actors in the n -qubit Pauli operator p that are different from identity . Clearly , dist( t 1 C , t 2 C ) = dist( t − 1 2 t 1 C , C ) . The distance (3) can also be expressed in terms of the associated vectors over GF (4) . Lemma 1: The distance of the spaces t 1 C and t 2 C equals the minimum weight in the coset C ∗ + t 1 − t 2 , where we use t i to denote both an n -qubit Pauli operator and the corresponding vector over GF (4) . Pr oof: Direct computation shows dist( t 1 C , t 2 C ) = dist( C ∗ + t 1 , C ∗ + t 2 ) = dist( C ∗ + ( t 1 − t 2 ) , C ∗ ) = min { wgt( c + t 1 − t 2 ) : c ∈ C ∗ } = min { wgt( v ) : v ∈ C ∗ + t 1 − t 2 } . W ith this preparation, we are ready to present the general construction of the union of stabilizer codes (see also [7]). The quantum code will be defined as the span of some of the summands in (2). Definition 2 (union stabilizer code): Let C 0 = [[ n, k, d 0 ]] be a stabilizer code and let T 0 = { t 1 , . . . , t K } be a subset of the coset representativ es of the normalizer N 0 of the code C 0 in P n . Then the union stabilizer code is defined as C = M t ∈T 0 t C 0 . W ithout loss of generality we assume that T 0 contains identity . W ith the union stabilizer code C we associate the (in general non-additiv e) union normalizer code given by C ∗ = [ t ∈T 0 C ∗ 0 + t = { c + t i : c ∈ C ∗ 0 , i = 1 , . . . , K } , where C ∗ 0 denotes the additiv e code associated with the normalizer N 0 of the stabilizer code C 0 . W e will refer to                         S X 1 S Z 1 . . . . . . S X n − k S Z n − k Z X 1 Z Z 1 . . . . . . Z X k Z Z k X X 1 X Z 1 . . . . . . X X k X Z k                                t X 1 t Z 1 . . . . . . t X K t Z K        Fig. 1. Arrangements of the vectors associated with a union stabilizer code. both, the vectors t i and the corresponding unitary operators, as translations . A union stabilizer code can be defined in terms of binary vectors as sho wn in Fig. 1. The first n − k rows correspond to the binary vectors (cf. (1)) associated with the generators S i of the stabilizer S of the code C 0 . They generate the classical code C 0 . The next k rows correspond to the logical operators Z j , followed by the k logical operators X i . The last K ro ws correspond to the K translations t i defining the cosets of the classical code C ∗ 0 and the unitary images of the stabilizer code C 0 , respectively . W e use curly brackets to stress the fact that the set of operators T 0 need not be closed under group operation. In general, the quantum code is not in variant under these gener alized logical X -operators . On the other hand, if T 0 is closed under group operation, the resulting code will be a stabilizer code where a basis of T 0 defines an additional set of logical X -operators. Theor em 3: Let C be a union stabilizer code as in Definition 2. The dimension of C is |T 0 | 2 k = K 2 k , and the minimum distance is lower bounded by the minimum distance d of the union normalizer code C ∗ . Pr oof: As T 0 is a subset of the coset representati ves of the normalizer N 0 , the spaces t i C 0 , each of which has dimension 2 k , are mutually orthogonal. Hence the dimension of the union code is K 2 k . Fixing an orthonormal basis {| c j i : j = 1 , . . . , 2 k } of the stabilizer code C 0 , the set { t i | c j i : i = 1 , . . . , K, j = 1 , . . . , 2 k } is an orthonormal basis of the union stabilizer code. Let E ∈ P n be an n -qubit Pauli error of weight 0 < wgt( E ) < d . For basis states | c i,j i = t i | c j i ∈ t i C 0 and | c i 0 ,j 0 i = t i 0 | c j 0 i ∈ t i 0 C 0 we consider the inner product h c i,j | E | c i 0 ,j 0 i = h c j | t † i E t i 0 | c j 0 i . (4) For i 6 = i 0 , we hav e dist( t i C 0 , t i 0 C 0 ) = min { wgt( c + t i − t i 0 ) : c ∈ C ∗ 0 } ≥ d , and hence (4) v anishes for wgt( E ) < d . For i = i 0 , we get h c j | t † i E t i | c j 0 i = ±h c j | E | c j 0 i . (5) As the code C 0 is pure up to the minimum distance of C ∗ 0 ⊂ C ∗ , equation (5) vanishes as well. Remark 4: W e note that similar to stabilizer codes, the true minimum distance of a union stabilizer code might be higher . The true minimum distance is giv en by min { dist( c + t i , c 0 + t i 0 ) : t i , t i 0 ∈ T 0 , c, c 0 ∈ C ∗ 0 | c + t i − ( c 0 + t i 0 ) / ∈ e C 0 } = min { wgt( v ) : v ∈ ( C ∗ − C ∗ ) \ e C 0 } , where ( C ∗ − C ∗ ) = { a − b : a, b ∈ C ∗ } denotes the set of all differences of vectors in C ∗ 0 , and e C 0 is the symplectic dual of the additiv e closure of C ∗ . In order to construct union quantum codes, we may start with a stabilizer code C 0 and use a sear ch graph whose vertices are the mutually orthogonal translates { t C 0 : t ∈ T 0 } of the stabilizer code. T wo vertices are connected by an edge if and only if the distance between them is at least d , where d is the desired minimum distance. For simplicity , we also require that the code C 0 is pure up to d 0 ≥ d . The distance between two translates can be computed using Lemma 1. This allows to use stabilizer codes C 0 of arbitrary dimension, and hence allows to go beyond the case of stabilizer states (or graph states) as, e. g., in [5], [17]. W e note that the construction of non-additiv e quantum codes of [2] is also based on taking the union of orthogonal images of a stabilizer code. I V . U N I O N S T A B I L I Z E R C O D E S F R O M B I N A R Y C O D E S A. CSS-like codes Giv en two linear binary codes C 1 = [ n, k 1 , d 1 ] and C 2 = [ n, k 2 , d 2 ] with C ⊥ 2 ⊂ C 1 , the so-called CSS construction (see, e. g., [14]) yields a quantum error-correcting code C = [[ n, k 1 + k 2 − n, d ]] with d ≥ min( d 1 , d 2 ) . Starting with this CSS code, we consider unions of cosets of the binary codes C i , i. e., e C i = [ t ( i ) ∈T i C i + t ( i ) such that the minimum distance of the codes e C i is at least e d ≤ d . Using the translations { ( t (1) | t (2) ) : t (1) ∈ T 1 , t (2) ∈ T 2 } we obtain a CSS-like union stabilizer code of dimension |T 1 | · |T 2 | · 2 k 1 + k 2 − n whose minimum distance is at least e d . If G 1 =  H 2 G 12  and G 2 =  H 1 G 21  are generator matrices of the codes C i , where H i is a generator matrix of the dual code C ⊥ i , the corresponding vectors are as shown in Fig. 2. B. Enlar gement construction Steane has presented a construction that allo ws to increase the dimension of a CSS code [14]. For this, he starts with the CSS construction applied to a binary code C = [ n, k , d ] which contains its dual, yielding a CSS code C 0 = [[ n, 2 k − n, d ]] . Using a code C 0 = [ n, k 0 > k + 1 , d 0 ] which contains C , he obtains a quantum code [[ n, k + k 0 − n, min( d, d 3 d 0 / 2 e )]] . The          H 2 0 0 H 1 G 12 0 0 G 21                                   t (1) 1 t (2) 1 . . . . . . t (1) 1 t (2) K 2 . . . . . . t (1) K 1 t (2) 1 . . . . . . t (1) K 1 t (2) K 2                          Fig. 2. Arrangements of the vectors associated with a CSS-like union stabilizer code. resulting code can also be considered as a union stabilizer code. If D is a generator matrix of the complement of C in C 0 and A is a fixed point free, in vertible linear map, the translations can be defined as T 0 = { ( v D | v AD ) : v ∈ F k 0 − k 2 } . The key observation [14] for proving the lower bound on the minimum distance is that the weight of an operator g = ( g X | g Z ) can be expressed in terms of the Hamming weight of the binary vectors and their sum: wgt(( g X | g Z )) = 1 2 (wgt( g X ) + wgt( g Z ) + wgt( g X + g Z )) . As T 0 is closed under addition and the properties of A ensure that 0 6 = t X 6 = t Z 6 = 0 for any non-zero element ( t X | t Z ) ∈ T 0 , the weight of all three binary vectors is lower bounded by d 0 . V . Q UA N T U M C O D E S F R O M R E E D - M U L L E R , G O E T H A L S , A N D P R E PA R A TA C O D E S Using the CSS-lik e construction of the previous section, we now construct some families of non-additi ve quantum codes. For this, we use the Goethals codes G ( m ) and the Preparata codes which are nonlinear binary codes of length n = 2 m for m ≥ 4 , m ev en. Some of the properties of these codes are summarized as follows (see, e. g., [10]): • Both the Goethals code G ( m ) and the Preparata code P ( m ) are unions of cosets of the Reed-Muller code R ( m ) := RM( m − 3 , m ) . Furthermore, they are nested subcodes of RM( m − 2 , m ) , i. e., RM( m − 3 , m ) ⊂ G ( m ) ⊂ P ( m ) ⊂ RM( m − 2 , m ) . • The parameters of the codes are RM( m − 3 , m ) = R ( m ) = [2 m , 2 m −  m 2  − m − 1 , 8] G ( m ) = (2 m , 2 2 m − 3 m +1 , 8) P ( m ) = (2 m , 2 2 m − 2 m , 6) RM( m − 2 , m ) = [2 m , 2 m − m − 1 , 4] .     X Z Z X     n T 0 o Q 1 − →     00 I 0 00 0 I 0 I 00     n T Q 1 0 o Q c − →     00 I 0 00 0 I 0 I 00        c 1 0 00 . . . . . . c K 0 00    Fig. 3. Transformation of the union stabilizer code given by the in verse encoding circuits Q 1 and Q c . Steane has constructed a family of additive quantum codes from Reed-Muller codes [15]. The codes are obtained applying the enlargement construction of [14] to the chain of codes RM( r , m ) ⊂ RM( r, m ) ⊥ = RM( m − r − 1 , m ) ⊂ RM( m − r , m ) . In particular , for r = 2 and m ≥ 5 this yields additiv e QECCs C 1 = [[2 m , 2 m −  m 2  − 2 m − 2 , 6]] , while using only the CSS construction, one obtains C 0 = [[2 m , 2 m − 2  m 2  − 2 m − 2 , 8]] . As the Goethals code G ( m ) is the union of K G = 2 ( m 2 ) − 2 m +2 cosets of R ( m ) , we can construct a CSS-like union stabilizer code based on C 0 . The minimum distance of the resulting non-additiv e code is 8 and its dimension is K 2 G dim( C 0 ) = 2 2 m − 6 m +2 . Replacing the Goethals code by the Preparata code P ( m ) , we have K P = 2 ( m 2 ) − m +1 cosets of R ( m ) . This results in a CSS-like union stabilizer code with minimum distance 6 and dimension K 2 P dim( C 0 ) = 2 2 m − 4 m . Both the union stabilizer codes based on Goethals codes and those based on Preparata codes are superior to the additiv e codes derived from Reed-Muller codes. The parameters of the first codes in these families are as follows: enlarged RM Goethals Preparata [[64 , 35 , 6]] ((64 , 2 30 , 8)) ((64 , 2 40 , 6)) [[256 , 210 , 6]] ((256 , 2 210 , 8)) ((256 , 2 224 , 6)) [[1024 , 957 , 6]] ((1024 , 2 966 , 8)) ((1024 , 2 984 , 6)) Howe ver , applying the enlargement construction to extended primitiv e BCH codes results in stabilizer codes with parame- ters [[2 m , 2 m − 5 m − 2 , 8]] and [[2 m , 2 m − 3 m − 2 , 6]] (see [14]). V I . E N C O D I N G C I R C U I T S In [9] we hav e shown ho w to compute a quantum circuit consisting of Clifford gates only that transforms any stabilizer S gi ven by the binary ( n − k ) × 2 n matrix ( X | Z ) into the stabilizer of a trivial code gi ven by (0 | I 0) , where I is an identity matrix of size n − k . The corresponding trivial code corresponds to the mapping | φ i 7→ | 0 . . . 0 i| φ i . W e denote the resulting quantum circuit that corresponds to the in verse encoding circuit of ( X | Z ) by Q 1 . Note that we can apply Q 1 to all the operators defining the code as illustrated in Fig. 3. Further note, that for the tri vial stabilizer code, the “encoded” | ψ i i                X • Z • Z • P Y • Z • Z • P Y • X • H H H H H g • • g • g • g • • g • g • g • | 0 i | 0 i    | i i , i = 0 , . . . , 5 Fig. 4. In verse encoding circuit for the non-additive code ((5 , 6 , 2)) . The first set of gates including the Hadamard transformations implements the inv erse encoding circuit Q 1 for the stabilizer code [[5 , 0 , 3]] , followed by 5 CNOT and 2 T offoli gates implementing the classical circuit Q c . X - and Z -operators are weight-one Pauli operators σ x and σ z , respectiv ely , acting on the last k qubits. As the transformed translations T Q 1 0 define cosets of the normalizer, we may choose them such that they are tensor products of operators σ x and identity acting on the first n − k qubits only . Then we hav e the trivial union code spanned by a set of K 2 k basis states of the form | c i i| j i , where {| j i : j = 0 , . . . , 2 k − 1 } is the computational basis of k qubits and { c i : i = 0 , . . . , K − 1 } is a set of bit strings of length n − k . In order to obtain a standard basis for our input space of dimension K 2 k , we need a quantum circuit Q c mapping | c i i 7→ | i i for i = 0 , . . . , K − 1 . Note that this is a purely classical circuit which can be realized, e. g., using σ x , CNOT gates, and T offoli gates. W e illustrate this for the non-additive code ((5 , 6 , 2)) which is a union stabilizer code deriv ed from the stabilizer state C 0 = [[5 , 0 , 3]] . F or a stabilizer code with k = 0 , there are no encoded X - and Z -operators. So the code ((5 , 6 , 2)) is specified by five generators of the stabilizer and six transla- tions. Using an in verse encoding circuit Q 1 , these operators are transformed as follows:       X X X X X X X Z I Z X Z I Z X Y I Y Z Z Y Z Z Y I                      I I I I I I I Z Z X I I I X X I I I Z Y I I Z Y Y I I Z X Z                Q 1 − →       00000 10000 00000 01000 00000 00100 00000 00010 00000 00001                      00000 00000 01010 00000 11011 00000 01111 00000 11100 00000 10010 00000                (6) It remains to find a (classical) quantum circuit Q c that maps the six binary strings on the right hand side of (6) to say the binary representations of i = 0 , . . . , 5 . Using a breath-first search among all circuits composed of σ x , CNOT , and T offoli gates, we found the minimal realization sho wn together with the circuit Q 1 in Fig. 4. V I I . C O N C L U S I O N S The approach presented in this paper generalizes naturally to the construction of non-additive quantum codes for higher dimensional systems. In order to obtain other families of non- additiv e quantum codes, it is interesting to study classical non- linear codes which can be decomposed into cosets of linear codes, similar to the Preparata and Goethals codes. A C K N O W L E D G M E N T S W e ackno wledge fruitful discussions with V aneet Aggarwal and Robert Calderbank. Markus Grassl would like to thank NEC Labs., Princeton for the hospitality during his visit. This work was partially supported by the FWF (project P17838). R E F E R E N C E S [1] V . Aggarwal and A. R. Calderbank, “Boolean Functions, Projection Operators and Quantum Error Correcting Codes, ” in Pr oceedings of the 2007 IEEE International Symposium on Information Theory , Nice, France, June 24 – June 29 2007, pp. 2091–2095, preprint cs/0610159. [2] V . Arvind, P . P . Kurur , and K. R. 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