Non-Additive Quantum Codes from Goethals and Preparata Codes
NNon-Additive Quantum Codes fromGoethals and 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 extend the stabilizer formalism to a class of non-additive quantum codes which are constructed from non-linearclassical codes. As an example, we present infinite families of non-additive codes which are derived from Goethals and Preparatacodes.
I. I
NTRODUCTION
Recently, several new non-additive quantum error-correctingcodes (QECCs) have been constructed that have higher dimen-sion than additive QECCs with the same length and minimumdistance [5], [16], [17]. The first example of such a code is thecode ((5 , , of [11] which has been found via numericaloptimization. Afterwards, the code has been identified as thespan of a particular state and its image under five unitarytransformations (see also [7]). The recently discovered codes ((9 , , and ((10 , , (see [16], [17]) start with a so-called graph state which corresponds to a stabilizer state, i. e.,a stabilizer code with parameters [[ n, , d ]] (see [8], [13]). Abasis of the quantum code is obtained by this initial statetogether with its image under some tensor products of Paulimatrices and identity (Pauli operators). The distance betweenany pair of these states can be defined as the minimal weightof a Pauli operator transforming one state into the other (seebelow). The problem of finding a code of high dimensioncan be stated as finding a maximal clique in a search graph whose vertices are all images of the initial state. There isan edge between two states if their distance is at least theprescribed minimum distance. Using the formalism of graphstates, Cross et al. show in [5] that it is sufficient to consider Z -only operators in order to define the search graph, but thishas the disadvantage that the distance between two of thesestates is not necessarily equal to the number of Z operators inthe tensor product. Moreover, constructing a code (( n, K, d )) requires to find a clique of size K in a search graph with n vertices. Fixing one basis state, the graph can be slightlysimplified. However, for the code ((10 , , the simplifiedgraph still has 678 vertices and 149.178 edges.A different approach for constructing non-additive QECCsbased on Boolean functions and projection operators hasbeen presented in [1]. Evaluating the Boolean function at theprojection operators, the code is given as the sum of the imageof products of the projections. Finally, we mention the non-additive QECCs obtained by the method given in [12]. Those codes have the same parameters as the additive CSS codesthat can be obtained using the same underlying binary codes.In this paper, we extend the approach of combining unitaryimages of stabilizer states to arbitrary stabilizer codes asstarting point. For a stabilizer code [[ n, k, d ]] , the search graphhas only n − k vertices, and a clique of size K yields aquantum code of dimension K × k . What is more, we presentsome infinite families of non-additive quantum codes whichare constructed using non-linear binary codes, avoiding theNP-hard problem of finding a maximal clique in the searchgraph. Finally, we show how to obtain encoding circuits forthe resulting non-additive codes using encoding circuits forthe underlying stabilizer codes.II. A BRIEF REVIEW OF THE STABILIZER FORMALISM
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., [4], [6]). 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 a r X i v : . [ qu a n t - ph ] J a n F (4) that are self-orthogonal with respect to a symplecticinner product [3], [4]. 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 ∗ .The stabilizer together with the logical operators Z i generate aself-dual code. In terms of the correspondence to vectors over GF (4) , the stabilizer and normalizer correspond to an additivecode over GF (4) and its dual with respect to an symplecticinner product, respectively, which we will also denote by C and C ∗ . The minimum distance d of the quantum code isgiven as the minimum weight in the set C ∗ \ C which islower bounded by the minimum distance d ∗ of C ∗ . If d = d ∗ ,the code is said to be pure , and for d ≥ d ∗ , the code is saidto be pure up to d ∗ .III. T HE UNION OF STABILIZER CODES
Note that we have defined a stabilizer code C as the jointeigenspace of the commuting operators S i generating thestabilizer S . The term stabilizer suggests that the code isthe joint +1 eigenspace of the operators. However, for eachof the generators S i we may choose either the eigenspacewith eigenvalue +1 or the eigenspace with eigenvalue − .This gives rise to a decomposition of the space ( C ) ⊗ n into n − k mutually orthogonal spaces which can be labeled by theeigenvalues of the n − k generators S i , or equivalently, by thecharacters χ of the stabilizer group S . Moreover, the n -qubitPauli group P n acts transitively on these spaces.From now on we fix the code C as the the joint +1 eigenspace corresponding to the trivial character. Let t ∈ P n be an arbitrary n -qubit Pauli operator. Then we can define acharacter of S on the generators S i as follows χ t ( S i ) := (cid:40) +1 if t and S i commute, − if t and S i anti-commute. As the elements of the normalizer N commute with allelements of the stabilizer S , two elements t and t definethe same character if t − t ∈ N . Hence the set of characterscorresponds to the cosets of N in P n . If T is a set of cosetrepresentatives, we can write the decomposition of the fullspace as ( C ) ⊗ n = (cid:77) t ∈T t C . (2)Note that measuring the eigenvalues of the generators S i projects onto one of these space t C , corresponding to thecharacter χ t given by the sequence of eigenvalues. In termsof the classical codes, the eigenvalues correspond to an error-syndrome which is obtained by computing the symplecticinner product of the received vector with the n − k vectorscorresponding to the generators of the stabilizer, i. e., a basis ofthe code C . For all vectors of the dual code C ∗ correspondingto the normalizer N , the inner product is zero. So the differentspaces t C correspond to cosets C ∗ + t of the code C ∗ .As for a fixed code C two spaces t C and t C are eitheridentical or orthogonal, we can define the distance of them asfollows: dist( t C , t C ) := min { wgt( p ) : p ∈ P n | pt C = t C} . (3)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 distance (3) can alsobe expressed in terms of the associated vectors over GF (4) . Lemma 1:
The distance of the spaces t C and t C equalsthe minimum weight in the coset C ∗ + t − t , where we use t i to denote both an n -qubit Pauli operator and the correspondingvector over GF (4) . 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 } . With this preparation, we are ready to present the generalconstruction of the union of stabilizer codes (see also [7]).The quantum code will be defined as the span of some of thesummands in (2).
Definition 2 (union stabilizer code):
Let C = [[ n, k, d ]] be a stabilizer code and let T = { t , . . . , t K } be a subsetof the coset representatives of the normalizer N of the code C in P n . Then the union stabilizer code is defined as C = (cid:77) t ∈T t C . Without loss of generality we assume that T contains identity.With the union stabilizer code C we associate the (in generalnon-additive) union normalizer code given by C ∗ = (cid:91) t ∈T C ∗ + t = { c + t i : c ∈ C ∗ , i = 1 , . . . , K } , where C ∗ denotes the additive code associated with thenormalizer N of the stabilizer code C . We will refer to S X S Z ... ... S Xn − k S Zn − k Z X Z Z ... ... Z Xk Z Zk X X X Z ... ... X Xk X Zk t X t Z ... ... t XK t ZK 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 binaryvectors as shown in Fig. 1. The first n − k rows correspond tothe binary vectors (cf. (1)) associated with the generators S i of the stabilizer S of the code C . They generate the classicalcode C . The next k rows correspond to the logical operators Z j , followed by the k logical operators X i . The last K rowscorrespond to the K translations t i defining the cosets of theclassical code C ∗ and the unitary images of the stabilizercode C , respectively. We use curly brackets to stress the factthat the set of operators T need not be closed under groupoperation. In general, the quantum code is not invariant underthese generalized logical X -operators . On the other hand, if T is closed under group operation, the resulting code will bea stabilizer code where a basis of T defines an additional setof logical X -operators. Theorem 3:
Let C be a union stabilizer code as in Definition2. The dimension of C is |T | k = K k , and the minimumdistance is lower bounded by the minimum distance d of theunion normalizer code C ∗ . Proof: As T is a subset of the coset representativesof the normalizer N , the spaces t i C , each of which hasdimension k , are mutually orthogonal. Hence the dimensionof the union code is K k . Fixing an orthonormal basis {| c j (cid:105) : j = 1 , . . . , k } of the stabilizer code C , the set { t i | c j (cid:105) : i =1 , . . . , K, j = 1 , . . . , k } is an orthonormal basis of the unionstabilizer code. Let E ∈ P n be an n -qubit Pauli error of weight < wgt( E ) < d . For basis states | c i,j (cid:105) = t i | c j (cid:105) ∈ t i C and | c i (cid:48) ,j (cid:48) (cid:105) = t i (cid:48) | c j (cid:48) (cid:105) ∈ t i (cid:48) C we consider the inner product (cid:104) c i,j | E | c i (cid:48) ,j (cid:48) (cid:105) = (cid:104) c j | t † i Et i (cid:48) | c j (cid:48) (cid:105) . (4)For i (cid:54) = i (cid:48) , we have dist( t i C , t i (cid:48) C ) = min { wgt( c + t i − t i (cid:48) ) : c ∈ C ∗ } ≥ d , and hence (4) vanishes for wgt( E ) < d . For i = i (cid:48) , we get (cid:104) c j | t † i Et i | c j (cid:48) (cid:105) = ±(cid:104) c j | E | c j (cid:48) (cid:105) . (5)As the code C is pure up to the minimum distance of C ∗ ⊂ C ∗ , equation (5) vanishes as well. Remark 4:
We note that similar to stabilizer codes, the trueminimum distance of a union stabilizer code might be higher.The true minimum distance is given by min { dist( c + t i , c (cid:48) + t i (cid:48) ) : t i , t i (cid:48) ∈ T ,c, c (cid:48) ∈ C ∗ | c + t i − ( c (cid:48) + t i (cid:48) ) / ∈ (cid:101) C } = min { wgt( v ) : v ∈ ( C ∗ − C ∗ ) \ (cid:101) C } ,where ( C ∗ − C ∗ ) = { a − b : a, b ∈ C ∗ } denotes the set of alldifferences of vectors in C ∗ , and (cid:101) C is the symplectic dual ofthe additive closure of C ∗ .In order to construct union quantum codes, we may start witha stabilizer code C and use a search graph whose verticesare the mutually orthogonal translates { t C : t ∈ T } of thestabilizer code. Two vertices are connected by an edge if andonly if the distance between them is at least d , where d isthe desired minimum distance. For simplicity, we also requirethat the code C is pure up to d ≥ d . The distance betweentwo translates can be computed using Lemma 1. This allowsto use stabilizer codes C of arbitrary dimension, and henceallows to go beyond the case of stabilizer states (or graphstates) as, e. g., in [5], [17]. We note that the construction ofnon-additive quantum codes of [2] is also based on taking theunion of orthogonal images of a stabilizer code.IV. U NION S TABILIZER C ODES FROM B INARY C ODES
A. CSS-like codes
Given two linear binary codes C = [ n, k , d ] and C =[ n, k , d ] with C ⊥ ⊂ C , the so-called CSS construction(see, e. g., [14]) yields a quantum error-correcting code C =[[ n, k + k − n, d ]] with d ≥ min( d , d ) . Starting with thisCSS code, we consider unions of cosets of the binary codes C i , i. e., (cid:101) C i = (cid:91) t ( i ) ∈T i C i + t ( i ) such that the minimum distance of the codes (cid:101) C i is at least (cid:101) d ≤ d . Using the translations { ( t (1) | t (2) ) : t (1) ∈ T , t (2) ∈T } we obtain a CSS-like union stabilizer code of dimension |T | · |T | · k + k − n whose minimum distance is at least (cid:101) d .If G = (cid:0) H G (cid:1) and G = (cid:0) H G (cid:1) are generator matrices ofthe 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. Enlargement construction
Steane has presented a construction that allows to increasethe dimension of a CSS code [14]. For this, he starts with theCSS construction applied to a binary code C = [ n, k, d ] whichcontains its dual, yielding a CSS code C = [[ n, k − n, d ]] .Using a code C (cid:48) = [ n, k (cid:48) > k + 1 , d (cid:48) ] which contains C , heobtains a quantum code [[ n, k + k (cid:48) − n, min( d, (cid:100) d (cid:48) / (cid:101) )]] . The H H G G t (1)1 t (2)1 ... ... t (1)1 t (2) K ... ... t (1) K t (2)1 ... ... t (1) K t (2) K Fig. 2. Arrangements of the vectors associated with a CSS-like unionstabilizer code. resulting code can also be considered as a union stabilizercode. If D is a generator matrix of the complement of C in C (cid:48) and A is a fixed point free, invertible linear map, thetranslations can be defined as T = { ( vD | vAD ) : v ∈ F k (cid:48) − k } . The key observation [14] for proving the lower bound onthe minimum distance is that the weight of an operator g =( g X | g Z ) can be expressed in terms of the Hamming weightof the binary vectors and their sum: wgt(( g X | g Z )) = 12 (wgt( g X ) + wgt( g Z ) + wgt( g X + g Z )) . As T is closed under addition and the properties of A ensurethat (cid:54) = t X (cid:54) = t Z (cid:54) = 0 for any non-zero element ( t X | t Z ) ∈ T ,the weight of all three binary vectors is lower bounded by d (cid:48) .V. Q UANTUM C ODES FROM R EED -M ULLER , G
OETHALS , AND P REPARATA C ODES
Using the CSS-like construction of the previous section, wenow construct some families of non-additive quantum codes.For this, we use the Goethals codes G ( m ) and the Preparatacodes which are nonlinear binary codes of length n = 2 m for m ≥ , m even. Some of the properties of these codes aresummarized 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 − , m ) . Furthermore, they are nestedsubcodes of RM( m − , m ) , i. e., RM( m − , m ) ⊂ G ( m ) ⊂ P ( m ) ⊂ RM( m − , m ) . • The parameters of the codes are
RM( m − , m ) = R ( m ) = [2 m , m − (cid:0) m (cid:1) − m − , G ( m ) = (2 m , m − m +1 , P ( m ) = (2 m , m − m , m − , m ) = [2 m , m − m − , . X ZZX (cid:110) T (cid:111) Q −→ I
000 0 I I (cid:110) T Q (cid:111) Q c −→ I
000 0 I I c ... ... c K Fig. 3. Transformation of the union stabilizer code given by the inverseencoding circuits Q and Q c . Steane has constructed a family of additive quantum codesfrom Reed-Muller codes [15]. The codes are obtained applyingthe enlargement construction of [14] to the chain of codes
RM( r, m ) ⊂ RM( r, m ) ⊥ = RM( m − r − , m ) ⊂ RM( m − r, m ) . In particular, for r = 2 and m ≥ this yields additive QECCs C = [[2 m , m − (cid:0) m (cid:1) − m − , , while using only the CSSconstruction, one obtains C = [[2 m , m − (cid:0) m (cid:1) − m − , .As the Goethals code G ( m ) is the union of K G = 2( m ) − m +2 cosets of R ( m ) , we can construct a CSS-like union stabilizercode based on C . The minimum distance of the resultingnon-additive code is and its dimension is K G dim( C ) =2 m − m +2 .Replacing the Goethals code by the Preparata code P ( m ) ,we have K P = 2( m ) − m +1 cosets of R ( m ) . This results in aCSS-like union stabilizer code with minimum distance anddimension K P dim( C ) = 2 m − m .Both the union stabilizer codes based on Goethals codesand those based on Preparata codes are superior to the additivecodes derived from Reed-Muller codes. The parameters of thefirst codes in these families are as follows:enlarged RM Goethals Preparata [[64 , , , , , , , , , , , , , , , , , , However, applying the enlargement construction to extendedprimitive BCH codes results in stabilizer codes with parame-ters [[2 m , m − m − , and [[2 m , m − m − , (see[14]). VI. E NCODING C IRCUITS
In [9] we have shown how to compute a quantum circuitconsisting of Clifford gates only that transforms any stabilizer S given by the binary ( n − k ) × n matrix ( X | Z ) into thestabilizer of a trivial code given by (0 | I , where I is anidentity matrix of size n − k . The corresponding trivial codecorresponds to the mapping | φ (cid:105) (cid:55)→ | . . . (cid:105)| φ (cid:105) . We denotethe resulting quantum circuit that corresponds to the inverseencoding circuit of ( X | Z ) by Q . Note that we can apply Q to all the operators defining the code as illustrated in Fig. 3.Further note, that for the trivial stabilizer code, the “encoded” ψ i (cid:105) X • Z • Z • P Y • Z • Z • P Y • X • HHHHH (cid:103) •• (cid:103) • (cid:103) • (cid:103) •• (cid:103) • (cid:103) • (cid:103) • | (cid:105)| (cid:105) | i (cid:105) , i = 0 , . . . , Fig. 4. Inverse encoding circuit for the non-additive code ((5 , , . The first set of gates including the Hadamard transformations implements the inverseencoding circuit Q for the stabilizer code [[5 , , , followed by 5 CNOT and 2 Toffoli gates implementing the classical circuit Q c . X - and Z -operators are weight-one Pauli operators σ x and σ z ,respectively, acting on the last k qubits. As the transformedtranslations T Q define cosets of the normalizer, we maychoose them such that they are tensor products of operators σ x and identity acting on the first n − k qubits only. Then we havethe trivial union code spanned by a set of K k basis statesof the form | c i (cid:105)| j (cid:105) , where {| j (cid:105) : j = 0 , . . . , k − } is thecomputational basis of k qubits and { c i : i = 0 , . . . , K − } is a set of bit strings of length n − k . In order to obtain astandard basis for our input space of dimension K k , we needa quantum circuit Q c mapping | c i (cid:105) (cid:55)→ | i (cid:105) for i = 0 , . . . , K − .Note that this is a purely classical circuit which can berealized, e. g., using σ x , CNOT gates, and Toffoli gates.We illustrate this for the non-additive code ((5 , , whichis a union stabilizer code derived from the stabilizer state C = [[5 , , . For a stabilizer code with k = 0 , there areno encoded X - and Z -operators. So the code ((5 , , isspecified by five generators of the stabilizer and six transla-tions. Using an inverse encoding circuit Q , these operatorsare transformed as follows: X X X X XX X Z I ZX Z I Z XY I Y Z ZY Z Z Y I
I I I I II I Z Z XI I I X XI I I Z YI I Z Y YI I Z X Z Q −→ (6)It remains to find a (classical) quantum circuit Q c that mapsthe six binary strings on the right hand side of (6) to saythe binary representations of i = 0 , . . . , . Using a breath-firstsearch among all circuits composed of σ x , CNOT , and Toffoligates, we found the minimal realization shown together withthe circuit Q in Fig. 4.VII. C ONCLUSIONS
The approach presented in this paper generalizes naturallyto the construction of non-additive quantum codes for higherdimensional systems. In order to obtain other families of non-additive quantum codes, it is interesting to study classical non-linear codes which can be decomposed into cosets of linearcodes, similar to the Preparata and Goethals 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. Thiswork was partially supported by the FWF (project P17838).R
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