A new approach to codeword stabilized quantum codes using the algebraic structure of modules
Douglas Frederico Guimarães Santiago, Geraldo Samuel Sena Otoni
aa r X i v : . [ qu a n t - ph ] M a y A NEW APPROACH TO CODEWORDSTABILIZED QUANTUM CODES USING THEALGEBRAIC STRUCTURE OF MODULES
DOUGLAS FREDERICO GUIMAR ˜AES SANTIAGOand GERALDO SAMUEL SENA OTONI
Instituto de Ciˆencia e Tecnologia - ICTDiamantina, BrasilEmail: [email protected]
Abstract —In this work, we study the Codeword StabilizedQuantum Codes (CWS codes) a generalization of the stabilizersquantum codes using a new approach, the algebraic structure ofmodules, a generalization of linear spaces. We show then a newresult that relates CWS codes with stabilizer codes generalizingresults in the literature.
I. I
NTRODUCTION
With the development of quantum computing, as well as inclassical computing, the emergence of mechanisms to detectand correct errors should be implemented, then follows theneed of the theory of quantum error correction codes ([1], [2],[3], [4], [5],[6], [7], [8]). Protection against quantum errorinvolves different challenges than protecting against classicmistakes, but despite this, much of the classical theory of error-correcting codes can be harnessed for quantum codes.A quantum code is a subspace of a Hilbert space andis usually represented by the parameters (( n, K, e )) d . Theparameter d is the amount of quantum levels being considered,e.g, the number of linearly independent states a single quditcan present. The parameter n is the dimension of the largerHilbert space, K is the dimension of the code. The parameter e is the number of qudits that the code can detect.A class of quantum codes much explored in the literature isthe class of stabilizer codes ([9], [10] ). In these, the subspacewhich defines the code is the intersection of the subspacesassociated with the eigenvalue of a set of operators thatform a subgroup of the Pauli group. This group is called thestabilizer group S .In a CWS code (Codeword Stabilized Quantum Codes)with parameters (( n, K, e )) d , the stabilizer group stabilizes asingle quantum state (up global phase) and the basis elementsare constructed by applying distinct Pauli Operators in thestabilizer state ([11], [12], [13], [14], [15], [16], [17]). TheCWS codes are a generalization of the stabilizers codes, sinceit has been proved that every stabilizer code can be seen as aCWS code. Conversely, it was also proved that a CWS codesatisfying certain conditions is actually a stabilizer code. Thereare several results in the literature about the CWS codes andone of these allow us to construct quantum CWS codes withthe higher possible parameter K with parameters n and e fixed. The problem of constructing good CWS (with parameter K large) becomes then the problem of constructing good classicalcodes to correct a particular set of errors. The theory of CWScodes then presents a method to create new quantum codes(stabilizer or not) based on classical codes.This work present a new approach in the study the CWScodes by the algebraic structure of modules and the general-ization of the concept of parity check matrix. We also presentTheorem 4, that generalizes results found in the literature andhelps to determine when a CWS code is a stabilizer code.In the second section, we explain in more details the struc-ture of CWS codes, based mainly in [11]. In the third sectionwe introduce and generalize the notion of parity matrix. In thefourth section, we present some necessary results on the theoryof modules. In the fifth section we prove some known resultsabout stabilizer spaces using the concept of parity matrix asdone in [18], but using also the structure of modules. In thesixty section we present our main result (Theorem 4). TheCorollaries 4 and 5 concerning this theorem represent wellknown results in the literature, although we also have notfound a prove on qudits for these results.II. S TRUCTURE OF
CWS
CODES
For a qudit, the Pauli group G d is generated by X , Z , wherethe commute relation is given by ZX = q d XZ and q d = e i πd . Note that setting this way, for a qubit ( d = 2 )the Pauli group G , which in the binary case we also representby G , is given by G = { I, − I, Z, − Z, X, − X, ZX, − ZX } . (1)There is a representation of G d and a basis {| k i} d − k =0 such that Z | k i = q kd | k i , X | k i = | k + 1 i , para todo k ∈ Z d . (2)It follows that Z j X k = q jkd X k Z j and general relation ([19])is given by ( q i d Z j X k )( q i d Z j X k ) (3) = q j k − k j d ( q i d Z j X k )( q i d Z j X k ) onsidering these commute relations, an element of thePauli group G nd = G d |{z} ⊗ . . . , ⊗ G d |{z} n may be written as αZ V X U where α = q kd where V e U represent vectors in Z nd indicatingthe power of Z and X on each qudit respectively. Extendingthe commute relation 3 we have ( Z U X U )( Z V X V ) (4) = q h U , V i−h U , V i d ( Z V X V )( Z U X U ) where h ., . i denotes the canonical inner product restricted to Z nd , which is not necessarily a linear space. If d is prime Z nd is a linear space, and in this case, Z d is a field, otherwise Z nd has the structure of a module.Disregarding global phase, We represent one Pauli operator E = αZ U X U as expanded vector in Z nd . This is done byapplying the function R defined as follows: Definition 1:
Let G nd be the qudit Pauli Group with n entriesand the Z d -module, Z nd . The function R is defined as R : G nd → Z nd αZ U X U ( U | U ) . Clearly the function R is well defined, is surjective butnot injective, since the information contained in the phase α is lost. the function R is also a group homomorphism, e.g, R ( g g ) = R ( g ) + R ( g ) and R ( g † ) = − R ( g ) . Using therepresentation of the elements of G nd given by the function R ,we can determine the phase that appears in the general com-mute relation (4) through the operator of dimension n × n times defined by Λ = (cid:20) I − I (cid:21) , (5)where and I refers to zero and identity submatrices, respec-tively of dimension n × n . Using this operator, we note that anytwo Pauli operators in P and P obey the commute relation P P = q R ( P )Λ R T ( P ) d P P . (6)The operation R ( P )Λ R T ( P ) is known as symplecticproduct ([19]).We can, according to [11], construct a CWS code by twosets:1) An Abelian group S = h s , . . . , s r i of order | S | = d n not containing multiples of the identity except the iden-tity itself ( this group stabilizes, disregarding globalphase, a single state | ψ i ∈ H nd );2) A set W = { w i } Ki =1 where { w i } are Pauli operatorssuch that β = { w i | ψ i} represents the code base. R ( P ) is a line vector R T ( P ) is the transposed line vector. Moreover, we can verify that these conditions guarantee thatall operators S are simultaneously diagonalizable, e.g, thereexists a common basis of eigenvectors to all operators of S .Regardless of S stabilize a single state or not, let’s call thisgroup Stabilizer Group .III. P
ARITY C HECK M ATRIX
We can represent a collection of Pauli operators through amatrix as the way the theory of classical error correction codesdoes ( [20], [21]). We will call this matrix by
Parity CheckMatrix , or simply
Parity Matrix . This matrix is already usedin the formalism of stabilizer codes ([1]) with the generatorsof the stabilizer group, but here we define in general, for anyset of Pauli operators.
Definition 1:
Given a collection of Pauli operators C = { p , . . . , p r } in G nd , we call parity check matrix of C , R ( C ) ,the matrix of size r × n where each row of the matrix R isthe vector ( p i ) .Given a stabilizer group S with generators S = { s , . . . , s r } , R ( S ) will be the parity check matrix of size r × n about the collection of generators of S . The Z d -modulegenerated by the rows of the parity check matrix over S willbe denoted by h R ( S ) i . It is easy to verify that | S | = h R ( S ) i ,where the symbol denotes the cardinality of the set. It is useful at this point to enunciate the most important the-orem for CWS codes using the parity check matrix definition.This theorem allow us to create quantum CWS codes lookingfor classical codes [12], [13]
Theorem 1:
Let Q be a CWS code with stabilizer gener-ators S = { s , . . . , s r } , codewords W = { w i } Ki =1 , w = I , ǫ = { E } a set of Pauli errors and let Cl S be the function Cl S ( P ) = R ( S )Λ R T ( P ) . (7)Then the code Q detects errors in ǫ if and only if Cl S ( W ) detects errors in Cl S ( ǫ ) and moreover, if Cl S ( E ) = 0 then Ew i = w i E (8)for all i . IV. M ODULES
The algebraic structure of modules can be seen in [22].In this work, we use repeatedly that, given a homomorphismfrom Z d -modules represented by a matrix T , the cardinalityof the module generated by the rows of T , which we denoteby hT i is equal to the cardinality of the module generated bythe columns of T , which can be represented by the module Im ( T ) . We will refer to these modules as row-modules andcolumn-modules, respectively.The isomorphisms theorems for modules [22] will be usedfrequently in the proofs of this work Theorem 2:
Let A be a ring. The concept of parity check matrix will also be used on the collection of codewords W = { w i } Ki =1 , generating a parity check matrix R ( W ) of size K × n . ) If φ : M → M is a A -module homomorphism, thenthere is a isomorphism: Im ( φ ) ≃ M Ker ( φ ) .
2) If N , N are submodules of a A -module M , there is aisomorphism: N + N N ≃ N N ∩ N .
3) If
N, P are submodules of a A -module M and P ⊂ N ⊂ M so P is an submodule of N and there is aisomorphism: M/N ≃ M/PN/P .
The definitions of elementary operations used in this workare Definition 2:
The elementary operations are given by:1) Exchange two columns/rows ( C i ↔ C j or R i ↔ R j ).2) Add a column/row with the multiple in Z d of othercolumn/row ( C i → C i + βC j or R i → R i + βR j ).Given a matrix T with entries in Z d , we will first prove thatelementary operations in their rows or column do not changethe cardinality of the row and column modules. For this weassume a matrix T = [ C , C , ..., C n ] , where C i is a columnvector in Z kd .Then the column module is: Im ( T ) = { X C + ... + X n C n /X i ∈ Z d } , Clearly exchange between two columns do not change thecardinality of Im ( T ) . Neither the operation C i → C i + βC j ,as we will see below. Without loss of generality, assume theoperation using the first and second column, then Im ( T ′ ) = { X ′ ( C + βC ) + ... + X ′ n C n /X ′ i ∈ Z d } , But Im ( T ′ ) ⊂ Im ( T ) . To see this, make X ′ = X , X ′ =( X − βX ) , X ′ = X , ..., X ′ n = X n .We also have Im ( T ) ⊂ Im ( T ′ ) . To see this, make X = X ′ , X ′ = ( X + βX ) , X = X ′ , ..., X n = X ′ n .Elementary operations with the lines also do not affect thecolumn module. To see this, consider the matrix T as T = R R ... R k where each R i is a row vector in Z nd and R ij yourcomponents. The Kernel of T is: Ker ( T ) = { [ X , X , ..., X k ] (9) /R i X + R i X + ... + R in X n = 0 ∀ i ∈ (1 . . . k ) } Clearly exchanging the lines do not change the cardinalityof
Ker ( T ) . With a analogous proof for the column module we It’s important to remark that all elementary operations are made in Z d see that R i → R i + βR j also do not change it. How Ker ( T ) do not change with elementary operations with lines, by thefirst isomorphism theorem 2, the cardinality of Im ( T ) alsodo not change which means that the number of elements ofthe column module also do not change.To show that the elementary operations do not changethe line module we can make a procedure analogous to theprevious one using the transposed matrix T T .The next step is to make elementary operations until T geta form in wich we can see that both cardinalities are equal.We will need the lemma: Lemma 1:
Let a and b be integers, ≤ a, b ≤ d − and a = 0 , so there exist q, r ∈ Z d satisfying ≤ r < a and r = a q + b Proof:
Using the Euclid’s division algorithm,there exist q ′ and ≤ r < d − satisfying b = a q ′ + r then b = a q ′ + r inthis case, how ≤ b < d , we have also ≤ q ′ < d . Take q = d − q ′ and we have r = b + a q The next proposition allow us to obtain an equivalent togaussian elimination through elementary operations.
Proposition 1:
Through elementary operations (with thecolumn elements) we can transform V = ( v , . . . , v n ) ∈ Z nd with ≤ v i < d and at least one not null entry, in V ′ = ( a, , . . . , wih only the first entry a assuming a notnull entry Proof:
Repeat the process:1) Let v j be one entry with the least not null absolute value.Exchange the j entry with the first. Then rename theentries to V = ( a , . . . , a n ) .2) For each j = 1 use Lemma 1 to obtain in the j entry, r j = a j + q j a onde ≤ r j < a .
3) Repeat procedures 1 and 2 until get the result.In the next proposition we get the statement about theequality of the cardinalities of the row and columns modules.
Proposition 2:
Let T ( m × n ) an matrix with entries in Z d rep-resenting an Z d -module homomorphism, then the cardinalitiesof the row and columns modules are equal, Im ( T ) = hT i . Proof:
How elementary operations do not change thecardinalities of the row and columns modules, just follow theprocedure:1) Consider together all the first row an first column valuesof T . Take the least of them and through exchangeelementary operations, put it on the (1 , position.) Still considering together all the first row an first col-umn values of T , make how Proposition 1. After thisprocedure, we make null all the first row an first columnvalues of T but the (1 , position.repeating this procedure to the others rows and columns, weobtain an matrix T ′ in wich only the ( i, i ) positions with i ∈ , . . . least ( n, m ) may assume not null values. Clearlythis matrix satisfies Im ( T ′ ) = hT ′ i How elementaryoperations do not change the cardinalities of the row andcolumns modules, we get the statement.V. S
TABILIZER S PACES
A first question that arises is if the fact that the stabilizergroup S be abelian, not containing multiple of the identity butthe identity itself and | S | = d n are necessary and sufficientconditions to S stabilize a single phase state | ψ i . The answerto this question is positive. For the binary case the result isdemonstrated in [1] and makes use of the parity check matrix R ( S ) . We can extend this statement for the case d prime. Wealso can prove that the result holds for any d using ideascontained in [23] and [18]. Here, we chose to make a newapproach, similar to that made for qubits, using the paritycheck matrix R ( S ) and the interpretation of the matrix R ( S )Λ as a homomorphism between Z d -modules. Lemma 2:
Let S = h s , . . . , s r i be an abelian subgroupof the Pauli group G nd not containing multiple of the identityother than the identity itself. If | S | < d n , then we can addan element P ∈ G nd \ S such that S = h s , . . . , s r , P i is stillan abelian group not containing multiple of the identity otherthan the identity itself. Proof:
The Λ operator does not change the cardinality ofthe row-module, so we have | S | = h R ( S ) i = h R ( S )Λ i
Let S = h s , . . . , s r i be an abelian subgroupof the Pauli group G nd not containing multiple of the identityother than the identity itself. Then | S | ≤ d n . Proof:
The demonstration follows similar to the proof ofLemma 2. Suppose that | S | > d n . By the first isomorphismtheorem for modules we have Im ( R ( S )Λ) ≃ Z nd Ker ( R ( S )Λ) , the phrase stabilize a single phase state should be considered alwaysdisregarding a global phase where Ker ( R ( S )Λ) < d n , which is a contradiction becauseall elements of h R ( S ) i belong to Ker ( R ( S )Λ) .The next theorem relates the order of the stabilizer groupwith the dimension of the stabilized quantum code Q . To un-derstand it, we will start now an argument that will culminatewith the theorem.All Pauli operator P is an isomorphism between linearspaces, so if Q is a quantum code, P Q is a quantum codewith the same dimension of Q . If Q is stabilized by S = h s , . . . , s r i then according to the formalism of stabilizers, P Q is stabilized by S ′ = P SP † . The generators of S ′ are S ′ = h q d − α d s , . . . , q d − α r d s r i where the vector ( d − α . . . , d − α r ) is obtained using theequation 6 according to the following operation R ( S )Λ R T ( P † ) . If Q is stabilized by S , then Q is the eigenspace associatedto the eigenvalue 1 of each operator S = { S i } , then P Q isthe eigenspace associated with the eigenvalues { q α i d } of eachoperator on S . Considering then the homomorphism betweenmodules represented by the matrix R ( S )Λ we have that every element x on the image of this homomor-phism, x ∈ Im ( R ( S )Λ) , represents a distinct subspace of H nd .We know that they are distinct because subspaces associatedto distinct eigenvalues has only trivial intersection.By lemmas 2 and 3 we know that we can completethe stabilizer group S = h s , . . . , s r i such that S ′ = h s , . . . , s r , P , . . . , P M i is a stabilizer group and has order | S ′ | = d n . Let S ′ = { s , . . . , s r , P , . . . , P M } . Since | S ′ | = h R ( S ′ ) i = h R ( S ′ )Λ i and the cardinality of the column-module is equal to the row-module, so Im ( R ( S ′ )Λ) = d n . As each element x = ( α , . . . , α r , β , . . . , β m ) of Im ( R ( S ′ )Λ) represents a distinct subspace of same dimen-sion, and the dimension of the whole space is dim ( H nd ) = d n , it follows that every subspace V x stabilized by S ′ = h q d − α d s , . . . , q d − α r d s r , q d − β d P , . . . , q d − β m d P M i has dimen-sion 1 and the union of these ones covers the whole H nd .Since each V x is a subspace of the space stabilized by S = h q d − α d s , . . . , q d − α r d s r i , they also cover H nd , and thesame has trivial intersection, so the subspace Q stabilized by S has dimension dim ( Q ) = d n | S | . Thus, we demonstrate thefollowing theorem. Theorem 3:
Let S = h s , . . . , s r i an abelian subgroup ofthe Pauli group G nd where { S i } ri =1 are independent generators,which does not contain multiples of the identity than the iden-tity itself. Then the subspace stabilized by S has dimension d n | S | .We will get now three important corollaries of the precedingtheorem. The first two are used in the proof of Theorem 1.The third result establishes the number of generators of S if d is prime. orollary 1: Let S = h s , . . . , s r i be an abelian subgroupof the Pauli group G nd not containing multiple oh the identitythan the identity itself. If | S | = d n then S is a maximal setof Pauli operators that stabilizes a single state | ψ i .This corollary says that every Pauli operator P ∈ G nd stabilizing | ψ i is in S . The proof is given below. Proof:
By Theorem 3 we have S stabilizes a single state | ψ i . Suppose there is P ∈ G nd and P / ∈ S that stabilizes | ψ i .Clearly P t also stabilizes | ψ i for any t ∈ N , so there is no t ∈ N such that P t = αI with α =
1. In addition, P commuteswith all elements of S since otherwise, there would be s ∈ S and β = | ψ i = P | ψ i = P s | ψ i = βsP | ψ i = β | ψ i which may not occur. Therefore, S = h s , . . . , s r , P i is anabelian group not containing multiple of the identity than theidentity with | S | > d n and stabilizes | ψ i , a contradiction toTheorem 3. Corollary 2:
Under the same assumptions, unless phase, S is a maximal set of Abelian operators. Proof:
Suppose that P ∈ G nd is a Pauli operator thatcommutes with all elements of S . Then it follows that S stabilizes | ψ i and P | ψ i , but these vectors can not be linearlyindependent by Theorem 3. Logo P | ψ i = α | ψ i , e.g, theoperator α † stabilizes P | ψ i . By the previous corollary, itfollows that α † P ∈ S . Corollary 3:
Let S = h s , . . . , s r i an abelian subgroup ofthe Pauli group G nd not containing multiples of the identitythan the identity itself and d prime. Then S stabilizes a singlestate | ψ i if and only if r = n . Proof: If d is prime, then the order of each generator is o ( s i ) = d ∀ i , and it follows that | S | = d r . Then S stabilizesa single state if and only if r = n .For d not prime, we may have a generator S i of S withorder less than d , so that the required amount r of generatorsis greater than n . The maximum number of generators is n ,as cited in [24], n ≤ r ≤ n .VI. CWS CODES AND S TABILIZERS CODES
This section establishes relationships between CWS codesand stabilizer codes. There are several examples of codes thatare not built with the CWS formalism, as we see in [25], [26],[27] and [28]. There are also several CWS codes that are notstabilizers, how can we check in [11], [13], [15], [16], [17].Every stabilizer code is in fact a CWS code and all CWScode with codewords W forming a group, is a stabilizer code.These results are shown in the binary case [13] and for graphstates for any d [29], but we did not found in the literaturea general statement, valid for any d , and being not based on graph states , so we did a demonstration based on the structureof the parity check matrix (Definition 1). Given a set of Paulioperators C , the parity check matrix with coefficients in Z d , R ( C ) . If the number of operators in C is l , R ( C ) represents ahomomorphism between Z d -modules, Z nd → Z ld , so it makessense to speak of kernel and image modules, respectively Ker ( R ( C )) and Im ( R ( C )) . Lemma 4:
Let Q be a CWS code with stabilizer S gener-ated by S = { s , . . . , s r } and codewords W = { w i } Ki =1 . Thenthe cardinality of the centralizer of W in S , C S ( W ) and thecardinality Z d -module h R ( S ) i T Ker ( R ( W )Λ) are the same,e.g C S ( W ) = h R ( S ) i \ Ker ( R ( W )Λ) Proof:
It suffices to show that the function f : C S ( W ) → R ( S ) i T Ker ( R ( W )Λ) g R ( g ) is well defined, and is bijective.1) f is well defined because if g = g ∈ C S ( W ) , then R ( g † g ) = and so R ( g ) = R ( g ) .2) f is injective, because if R ( g ) = R ( g ) then R ( g † g ) = , then g † g = αI and α = 1 because there is nomultiple of the identity other than the identity itself in S .3) f is surjective. Let v ∈ h R ( S ) i T Ker ( R ( W )Λ) . Thereis g ∈ G nd such that R ( g ) = v . As v ∈ h R ( S ) i and upto phase phase h R ( S ) i is a maximal abelian set, there is g = αg with g ∈ S and R ( g ) = v .As v ∈ Ker ( R ( W )Λ) , R ( W )Λ R T ( g ) = it fol-lows that g commutes with all elements of W , then g ∈ C S ( W ) . Theorem 4:
Let Q be a CWS code with stabilizer S gen-erated by S = { s , . . . , s r } . and codeword operators W = { w i } Ki =1 with w = I . Then Q is a stabilizer code if andonly if it satisfies h R ( W ) i h R ( W ) i∩h R ( S ) i ) = K . Proof:
Let | ψ i be the state stabilized by S and β = { w i | ψ i} Ki =1 a basis for Q . Q is a stabilizer code if and onlyif there exists a abelian subgroup H ≤ G nd , not containingmultiples of the identity than the identity itself that stabilizes Q . In particular H need to stabilize | ψ i . How S is a maximalsubgroup that stabilizes | ψ i (Corollary 1), then H ≤ S .Moreover, every element h ∈ H must satisfy hw i = w i h for all i , then the subgroup H is the centralizer of W in S , e.g H = C S ( W ) . It remains to show that d n | C S ( W ) | = h R ( W ) i h R ( W ) i∩h R ( S ) i , so according to Theorem 3, C S ( W ) stabilizes Q if and only if h R ( W ) i h R ( W ) i∩h R ( S ) i = K .According to Lemma 4, we have C S ( W ) = h R ( S ) i \ Ker ( R ( W )Λ) Since S is up to phase a maximal abelian set in G nd (Corollary 2), we have h R ( S ) i = Ker ( R ( S )Λ) , from whichit follows that h R ( S ) i \ Ker ( R ( W )Λ) = Ker ( R ( S )Λ) \ Ker ( R ( W )Λ)= Ker ( M ) This condition is not restrictive since all CWS code is equivalent to a w = I . Just do w ′ i = w † w i . here M = (cid:20) R ( S )Λ R ( W )Λ (cid:21) = (cid:20) R ( S ) R ( W ) (cid:21) Λ . Then weestimate Ker ( M ) .We have h M i = h R ( S )Λ i + h R ( W )Λ i . By the secondisomorphism theorem for modules, we have h R ( S )Λ i + h R ( W )Λ i R ( S )Λ ≃ h R ( W )Λ ih R ( W )Λ i ∩ h R ( S )Λ i , and how the operator Λ does not change the cardinality of therow-module, we have h M i = h R ( S ) i h R ( W ) i h R ( W ) i∩h R ( S ) i and thereforeas h R ( W ) i h R ( W ) i∩h R ( S ) i = K and h R ( S ) i = | S | = d n , wehave h M i = Kd n . As seen, the cardinality of the row-module is equal to the cardinality of the column-module, so Im ( M ) = Kd n . Finally, by the first isomorphism theorem,we have Ker ( M ) = d n Kd n = d n K . Example
Take the ((3 , , code with stabilizer S = h s , s , s i where s = XZI , s = ZXZ and s = IZX and codewords W = { I, ( XZ ) ⊗ Z ⊗ Z , ( XZ ) ⊗ Z ⊗ Z } . We haverespectively: R ( S ) = e R ( W ) = the row-module, h R ( W ) i is represented by the followingvectors: where the left are those belonging to h R ( S ) i ∩ h R ( W ) i .Then we see that h R ( W ) i h R ( W ) i∩h R ( S ) i = K , then the code isstabilizer by Theorem 4. Actually, we can see that the codeis equivalent to the code [[3 , , in [14] with stabilizer S ′ = h ZXZ, XZ X i .From Theorem 4, follows two Corollaries representingresults usually found in the literature. Corollary 4:
Let Q be a CWS code with stabilizer S = h s , . . . , s r i and codewords W = { w i } Ki =1 forming a group.Then Q is a stabilizer code. Proof: If W is a group, then the rows of R ( W ) also forman additive group, then h R ( W ) i = W = K . Moreover,we have by the construction of CWS codes that, h R ( W ) i ∩h R ( S ) i = { } , so h R ( W ) i h R ( W ) i∩h R ( S ) i = K Corollary 5:
Let Q be a CWS code with stabilizer S = h s , . . . , s r i , codewords W = { w i } Ki =1 with w = I andstabilized state | ψ i . If the classic words Cl S ( W ) form a group,then the code is a stabilizer one. Proof:
To show that h R ( W ) i h R ( W ) i∩h R ( S ) i = K is enoughto show that every element r w ∈ h R ( W ) i is of the form r w = R ( w j ) + r s with r s ∈ h R ( S ) i . The transformation Cl S has domain in G nd . each element of G nd has a representationon Z nd . As already seen (equation 7), we can describe thetransformation Cl S over Z nd as a homomorphism of modulesrepresented by the matrix T = R ( S )Λ .Take then r w ∈ h R ( W ) i , so r w = α R ( w ) + . . . + α k R ( w k ) and T ( r w ) = α T ( R ( w i )) + . . . + α k T ( R ( w k ))= α c + . . . + α k c k . As Cl S ( W ) form a group, the last summation is T ( R ( w j )) = c j ∈ Cl S ( W ) , e.g T ( r w ) = T ( R ( w j )) so r w = R ( w j ) + r s where r s ∈ Ker ( T ) = h R ( S ) i .It also follows that any stabilizer code can be seen as aCWS code, as shown in the following theorem Theorem 5:
All stabilizer code Q is a CWS code. Proof:
Let S ′ = h s , . . . , s m i be the stabilizer groupof the code Q and let dim ( Q ) = K . As already dis-cussed, S ′ can then be extended to a maximal group S = h s , . . . , s m , g , . . . , g L i with cardinality | S | = d n . This groupstabilizes a single state | ψ i ∈ Q . Consider now the paritycheck matrix R ( S ) . We have h R ( S ) i = d n . As the cardinal-ity of the row-module is the same of the column-module, wehave Im ( R ( S )) = d n and in turn also Im ( R ( S )Λ) = d n .This equality implies that for every x ∈ Im ( R ( S )Λ) , thereis a Pauli operator P x such that H nd = L P x | ψ i and eachstate P x | ψ i is the intersection of the eigenspaces associatedwith eigenvalues q x i d for each generator of S . Since Q is astabilizer code and dim ( Q ) = K we know that there are K of these Pauli operators forming a set W = { P x i } Ki =1 thatform a basis for Q . Then just take the set W as codewords VII. C
ONCLUSION
In Section V, was demonstrated for qudits, that a stabilizergroup S of order | S | stabilizes a subspace of H nd of dimension d n | S | . Although there is already a demonstration of this result,we created a proof which generalizes the one for qubitscontained in [1] and makes use of the parity check matrixof Definition 1 and the interpretation of the matrix R ( S )Λ asa homomorphism of Z d -modules.In Section VI we use the parity check matrix R ( S ) andthe interpretation of the matrix R ( S )Λ as a homomorphismfrom Z d -modules to prove Theorem 4 which generalizes theresults contained in Corollaries (4 and 5). These corollaries areaccepted results in the literature, but hard to find for qudits.A CKNOWLEDGMENT
The authors would like to thank FAPEMIG.
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