Inhomogeneous Quantum Codes(III):The Asymmetric Case?

Weiyang WANG Keqin FENG

1 Introduction

After the works of Shor[11]and Steane[12–13]in 1995–1996,the theory of quantum errorcorrecting codes developed rapidly.In 1998,Calderbank et al.[2]presented systematic methods to construct binary quantum codes,called stabilizer codes or additive codes,from classical errorcorrecting codes.At the same time,the stabilizer method was generalized to non-binary quantum codes and new methods were found to construct non-additive quantum codes.Recently,a number of new types of quantum codes,such as convolutional quantum codes,subsystem quantum codes and asymmetric quantum codes,were studied and the stabilizer method was extended to these variations of quantum codes.In particular,there were intensive activities in the area of asymmetric quantum codes(see[1,3,8–9,16]).

The current paper concentrates on the asymmetric quantum codes which deal with the case,in which dephasing errors(Z-errors)happen more frequently than qubit-f l ipping errors(X-errors)(see[12–13]).Such codes are used in fault tolerant operations of a quantum computer carrying controlled and measured quantum information over asymmetric channels(see[14]).The characterization of non-additive symmetric quantum codes was given in[5–6]to the asymmetric case,and several examples of good asymmetric quantum codes were shown in[15].

In this paper,we deal with the asymmetric case of the more general quantum codes,called inhomogeneous quantum codes.An inhomogeneous quantum code is a subspace ofwhereq1,···,qnmay take different positive integers.Inhomogeneous quantum codes were researched as early as in 1999(see[7,9]),and it seems that such kind of quantum codes has not been well-developed since then.The general formulation of symmetric inhomogeneous quantum codes was defined,and the stabilizer(additive)construction and nonadditive construction were extended to the symmetric inhomogeneous case in[4,15].The mainaim of this paper is to establish the stabilizer and non-additive constructions for asymmetric inhomogeneous quantum codes and present several series of good asymmetric inhomogeneous quantum codes by using these constructions.

This paper is organized as follows.In Section 2,we recall the basic facts on mixed classical codes and inhomogeneous(symmetric)quantum codes introduced in[4].Then we define asymmetric inhomogeneous quantum codes as a slight variation of the symmetric ones.In Section 3,we establish the stabilizer construction of asymmetric inhomogeneous quantum codes and show some examples of good(additive)asymmetric inhomogeneous quantum codes by using the stabilizer method.In Section 4,we present a new characterization of asymmetric inhomogeneous quantum codes and show some non-additive asymmetric inhomogeneous quantum codes as an application of the new characterization.

2 Preliminaries

2.1 Mixed classical codes

We recall some basic facts on mixed classical codes introduced in[4].These facts will be used to construct asymmetric inhomogeneous quantum codes in the next two sections.LetA=⊕⊕···⊕be a finite abelian group and|=(1≤i≤n).We assume that

Fora=(a1,···,an)andb=1≤i≤n,we define the Hamming weight ofaby

and the Hamming distance betweenaandbby

Definition 2.1A(classical)mixed code C on A is a subset of A with the size K=|C|≥2.The minimal distance of C is defined by

We call(A,K,d)the parameters ofC.A mixed codeConAis called additive ifCis a subgroup ofA.For the additive codeC,we haved(C)=min{wH(C):0A mixed codeCwith the minimal distancedcan detect≤d?1 digits of errors or correctdigits of errors.One of the basic problems in the classical coding theory is to construct mixed codeshaving larger efficiency and largerd=d(C).We have the following Singleton bound(see[4,Lemma 3.2]):

One of the best classes of mixed codes is called MDS code which meets the Singleton bound:K=Some examples of the mixed MDS(algebraic geometry)codes were presented in[4,Theorem 4.3].

Let=Hom(A,C?)be the group of characters ofA.It is known that we have an isomorphism of groupsA→,aχaand such isomorphism can be chosen to satisfy

For an additive codeConA,the dual codeofCis defined by

whereis additive and|C|·||=|A|.In the next section,we use the following “symplectic”mapping(,)sto analyse and construct asymmetric inhomogeneous quantum codes

where forv=(a|b),=∈A),

It can be seen that this mapping is a non-degenerate pairing which means that

(1)

(2)(v,=1 for allif and only ifv=0.

For a subgroupGof,the set

is also a subgroup of,called the symplectic dual ofG.It can be seen that|G|and=G.

2.2 Asymmetric inhomogeneous quantum codes

Now we recall the notations on(symmetric)inhomogeneous quantum codes given in[4].Let

For eachi,let{|:c∈}be a fixed orthonormal basis ofNamely,forc,∈,

wheredenotes the Hermitian inner product on the complex vector spaceThenVhas the following orthonormal basis:

and|ciis called thei-th quantum digit qubit of|c.AnA-ary(inhomogeneous)quantum state is a non-zero vector inVwhich is uniquely expressed by

In quantum physics,two quantum states|v=are called indistinguishable if|v=α|ufor some nonzero complex numberα,namely,?(c)=αψ(c)for allc∈A.Such two quantum states are assumed the same in the quantum world.On the other hand,|vand|uare called totally distinguishable if==0,where(c)is the complex conjugate of?(c).

Now we introduce the quantum error group acting onV.Each quantum error is a unitary(linear)operator on the complex vector spaceV.At each qudit,there are two types of errorsX(ai)andacting ondefined by their action on the basis{|:c∈Ai}:

On the quantum state spaceV=we have quantum error operatorsX(a)andZ(b),a=(a1,···,an),b=(b1,···,bn)∈Adefined by their action on the basis(2.4)as,for

Letmbe the exponent of the groupA.Namely,mis the smallest positive integer,such thatma=0 for alla∈A.Then the values of all charactersχ∈are the power ofω=eThe(non-abelian)quantum error group ofVis

The mapping

is an epimorphism from the groupEnto the additive groupA2=A⊕A.The following result is a starting point on the stabilizer construction of inhomogeneous quantum codes.

Lemma 2.1A subgroup G of Enis abelian if and only ifis a symplective self-orthogonal subgroup of A2(namely,

Definition 2.2For a quantum error e=X(a)Z(b)∈and=(a|b)∈(a=(a1,···,an),b=(b1,···bn),ai,bi∈A),we define their quantum weight wQ,X-weight wXand Z-weight wZby

respectively.Namely,wQis the number of qubits,where the quantum error XI occurs.wXand wZare the numbers of qubits,where the X-error and the Z-error occur respectively.

Now we define the inhomogeneous quantum codes overA.

Definition 2.3An inhomogeneous quantum code Q over A is a complex vector subspace of V with dimension K=dimCQ≥1.Each non-zero vector in Q is called a codeword.

ForK≥2,we call that a quantum errore∈Encan be detected byQ,if for anyand|vinQsuch thatare totally distinguishable),we have=0(i.e.,are distinguishable).More generally,for a subsetSofEn,we call thatScan be detected byQ,if eache∈Scan be detected byQ.

A(symmetric)inhomogeneous quantum codeQoverAis called to have parameters(A,K,d)(d≥1)ifK=dimQand the set of quantum errors

can be detected byQ.

The Asymmetric quantum code is a slight variation of the symmetric one.Let,≥1.

Definition 2.4An inhomogeneous quantum code Q over A is called asymmetric with parametersA,Kif the set of quantum errors

can be detected by Q.Such a code Q is called pure forin Q(not necessarily=0)and any e∈?1,(e)≥1,we have=0.When K=1,an asymmetric inhomogeneous quantum code with parametersis always purefor

One of the basic problems in the quantum code theory is to construct asymmetric inhomogeneous quantum codes with large efficiency(N=|A|=···)and larger(good ability to detect and correct quantum errors).As the cases of usual classical and quantum codes,we have bounds of the parameters to judge the goodness of asymmetric inhomogeneous quantum codes.LetQbe a pure quantum code with parametersBy the definition of pure quantum codes and the inequalitiesandwZ(e)≤wZ(e)+wZ()fore,∈En,we have that theN(,)subspaces ofV

are orthogonal to each other,where

with

Therefore,we get the following Hamming bound for pure quantum codes:

The quantum code with parametersis called perfect if|A|=K·N(dx)N(dz).On the other hand,we will show that for some asymmetric quantum codes constructed in Sections 3 and 4,we have the following Singleton bound:

It seems that this Singleton bound may be true for all asymmetric inhomogeneous quantum codes.As in the classical case,an asymmetric inhomogeneous quantum code is called an MDS code,if the equality in(2.7)holds.

In the next section,we establish the stabilizer construction of asymmetric inhomogeneous quantum codes and present examples of perfect or MDS codes by using this construction.

At the end of this section,we remark that the asymmetric quantum code with parametersA,has more powerful ability than the symmetric quantum code with parameters(A,K,d),since the detected setEn?1is usually larger than the setEn(d?1).

3 Stabilizer(Additive)Asymmetric Inhomogeneous Quantum Codes

In this section,we give the stabilizer construction of asymmetric inhomogeneous quantum codes as a generalization of such construction in the symmetric case(see[2,4])and the asymmetric usual case(see[16]for(1≤i≤n)).As in Section 2,letA=A1⊕···⊕be an abelian group andWe fix an isomorphismA→such that formula(2.3)holds.

Theorem 3.1If there exists a classical additive mixed code C in A⊕A andthen there exists an asymmetric inhomogeneous quantum code Q in V withdimQ=and S=?1or?1}can be detected by Q.Particularly,Q has parametersA,

where for a subset S of A⊕A,

Moreover,Q is pure for={0}).

ProofThe following proof is a minor modification of the proof in[2;4,Theorem 4.1].We omit some details.Firstly by Lemma 2.2 and the assumptionC?we can liftCto an abelian subgroupGofEn,such thatand|G|=|C|.Then the complex vector spaceV=V1?V2?···?has the following orthogonal decomposition:

where for eachχ∈,

We show that eachQ=V(χ)is a quantum code with the required parameters

In order to determine the dimension dimQ=dimV(χ),it can be proved that each quantum error operatore∈Enis a permutation of the setand the groupEnacts on Σ transitively.Therefore,allhave the same dimension,so that dimQ=dimV(χ)==K.

Next we determine the parametersdxanddz.We need to show that if∈Q=V(χ)and=0,then for any=A⊕Awith?1 and?1,we have=0.∈C=so that=0.which means that there exists ane∈C,such that∈Q=V(χ),

whereTherefore,is a mapping fromQ=V(χ)toV(χ.Since|v∈V(χ),andV(χ)⊥V(χ),we have=0.

At last we show thatQ=V(χ)is pure fordx=and{0}).By the definition,for any∈Q,any 0≤?1 and?1,we need to show that=0.From the assumption,we know thatThen,by the above argument,so that=0.This completes the proof of Theorem 3.1.

As a usual case,the asymmetric inhomogeneous quantum codes constructed by Theorem 3.1 are called additive codes since they come from the classical additive codes overA⊕A.We needCto be symplectic self-orthogonal andto be determined by the minimum quantum weightswXandwZofC.The next result shows that we can get asymmetric inhomogeneous quantum codes from a pair of classical additive codesC1andC2overAwithwhereis the usual dual ofC2andcan be determined by the Hamming weights ofC1andC2.

Theorem 3.2If there exist(mixed)additive codes and C2with parameters(A,K1,d1)and(A,K2,d2)respectively andthen there exists an asymmetric inhomogeneous quantum code Q with parametersA,where dx=and dz=Moreover,Q is pure for

ProofConsider the additive codeC=overA⊕A.The assumptionimplies=C.By Theorem 3.1,we have the asymmetric quantum codeQwith parametersA,where

andQcan detect the set

Thus we can takedx=andMoreover,for any∈Qwe have=0 for anyin the set

Therefore,Qis pure forand

Remark 3.1In Theorem 3.2,if=>and{0})>thenQis pure quantum code with parametersFor the classical mixed codesandwe have the Singleton bound

Thus the asymmetric quantum codeQsatisfies the Singleton bound(2.7),

In fact,Qis an MDS code if and only if bothC1andC2are MDS codes.On the other hand,we can see similarly that the quantum codeQis perfect(|A|=K·Nif and only if both the classical codesC1andC2are perfect(|A|=and|A|=

(2)TakingC=in the proof of Theorem 3.2,we get the asymmetric quantum codeQwith parameters wheredx=dz=wH,and the codeis pure fordx=d1anddz=d2.

Example 3.1(Perfect Quantum Codes)Suppose that there exists a perfect(MDS)additive codeCinAwith parameters(A,K,d),K=TakeC1=CandC2=Ain Theorem 3.2.Then={0}C1andC2is a trivial perfect(MDS)code with parameters(A,|A|,1).By Remark 3.1,we get a perfect(MDS)quantum code with parameterswhere{,}={d,1}.Such a quantum code can detect only theX-error(Z-error)fordx=danddz=1(for=1 and=d).

It is known that for the usualq-ary case(A=A1⊕ ···⊕An=1≤i≤n),all the nontrivial parameters of perfect additive classical codes are

For the more general mixed case,Herzog and Schonheim[17]presented a group-partition method to construct classical mixed codes withd=3.We introduce this construction briefly now.

LetGbe a finite(additive)abelian group andG1,···,Gnbe subgroups ofG.{G1,···,Gn}is called a partition ofG,ifGi{0}(1≤i≤n)is a partition ofG{0}.Namely,

which implies that

For a partition{G1,···,Gn}ofG,consider the mapping

Then?is an epimorphism of groups,so thatC=ker?is an additive code inAandK=|C|=By a simple computation and(?),we know thatCis a perfect code with parameters(A,K,3).

It is proved that ifGhas a partition(n≥2),thenGshould be an elementaryp-group.Namely,Gis an additive groupfor some prime numberpandm≥2.Several partitions of(,+)were constructed in[17–18,20–21].From these constructions,we get several perfect quantum codes with parametersA,Kfor some groupA=A1⊕···⊕(1≤i≤n)and={1,3}.

Example 3.2(MDS Quantum Codes)By using the Riemann-Roch theorem for a function fieldMwith a constant field Fq,the following classical mixed(algebraic-geometric)codes were constructed in[4].

Lemma 3.1(see[4,Theorem 3.2]or[19])Let A=(1≤i≤n)and m1≤m2≤···≤mn,M be a function field with a constant fieldFq,g=g(M)be the genus of M,(1≤i≤n)be distinct prime divisors of M,degPi=mi(1≤i≤n),D=P1+P2+···+Pn,and m=degD=m1+m2+···+mn.Let G be a divisor of M and(G)=0(1≤i≤n).Then

(1)IfdegG≤m?1,then

is anFq-linear code with parameters(A,K,d),where K=qk,k=l(G)≥degG+1?g and d≥t,where t is determined by

Moreover,l(G)=degG+1?g ifdegG≥2g?1;and d=t ifdegG>m1+m2+···+mn?t?1+g.

(2)IfdegG≥2g?1,then

is anFq-linear code with parameters(A,where K==l(W+D?G)=degD?degG+g?1+l(G?D)≥degD?degG+g?1and≥whereis determined by

Moreover,=degD?degG+g?1ifdegG≤m?1;anddegG<+···++g?1.

(3)If2g?1≤degG≤m?1,then C(D,G)⊥=C(D,G).

Wang and Feng[4]constructed a class of(symmetric)inhomogeneous quantum codes by the classical mixed algebraic-geometric codes.

Theorem 3.3Let q=ps,where p is a prime number and s≥1.Let t be a positive integer.Let d1,d2,···,dlbe all the positive divisors of t,such that

and m1,m2,···,mnbe the following integers:

where

Nq(e)is the number of monic irreducible polynomials of degree e inFq[x]and the number of fi nite prime divisors of degree e in the rational function fieldFq(x).Let A=A1⊕···⊕An,where(1≤i≤n).Then for each integer k,there exists a mixedadditive quantum code Q with parameters(A,K,d),where K=d is determined by

and Q is pure for d.Moreover,if k=m1+m2+···+then Q is an MDS code.

Actually,the inhomogeneous quantum code constructed in Theorem 3.3 is also an asymmetric inhomogeneous quantum code with parameters(A,In the proof of this theorem in[4],an additive mixed classical codeCkwas constructed.Ckhas parameters(A,,d),=d.Letting==,we can get an asymmetric inhomogeneous quantum code with parameters(A,

4 Non-additive Asymmetric Inhomogeneous Quantum Codes

In this section,we present a new characterization of asymmetric inhomogeneous quantum codes and show some methods to construct such non-additive codes.The new characterization is a generalization of symmetric cases given in[16].

EachA-ary quantum state|v=can be identified with a nonzero mapping?:A→C defined by?(c)=αcfor allc∈A.For a subsetSof{1,2,···,n}andc=(,,···,)∈A(∈is the sub vector ofcwhose coordinate positions belong toS.Namely,cS=can be viewed as a subgroup ofA.For?,ψ:A→C,we define their Hermitian inner product by

wherestands for the conjugate of the complex number?(c).

Let={:a∈A}be the character group ofA.For a functionf:A→C,the Fourier transform offisF:A→C,where

and we have the following inverse transform:

In the proof of the following Theorem 4.2,we need the following two simple facts on Fourier transform.

Lemma 4.1Let F:A→Cbe the Fourier transform of f:A→C.Then

(1)F≡0if and only if f≡0.

(2)F(a)=0for all0aA if and only if f is a constant.

Theorem 4.1(i)There exists an asymmetric inhomogeneous quantum code with parametersA,K(K≥2,dx,dz≥1)if and only if there exist K nonzero mappings

satisfying the following conditions:for each d,1≤d≤min{dx,dz}and each partition of{1,2,···,n},

andwe have the equality

where the complex number fis independent of i.

(ii)There exists a pure asymmetric inhomogeneous quantum code with parameters(K,dx,dz≥1)if and only if there exist K non-zero mappings ?i(1≤i≤K)as shown in(4.1)such that

(a)?i(1≤i≤K)are linear independent,namely,the rank of the K×|A|matrixis K,

(b)for each d,1≤d≤mina partition(4.2)and

ProofWe follow the argument in the proof of[5,Theorem 2.2]or[16,Theorem 2.1].We omit some computational details.

(i)LetQbe aK-dimensional subspace ofV=with the orthogonal basis

Then

For two vectors inQ,

we have

For eache=X(a)Z(b)(a,b∈A)with?1,we can find a partition(4.2),such thatecan be expressed by

The action ofeoncan be computed by(2.5)

By Definition 2.4,Qhas parametersif and only if

SincebSandbZare arbitrary elements inASandAZ,respectively,by Lemma 4.1(1),we know that the above equality is equivalent to

for anyandaZ.Consider the matrix

Our statement now becomes that for anyα,=0 implies=0.It is easy to see that under the assumptionK≥2,M=fI,whereIis the identity matrix andf=∈C.This is the condition(4.3).

(b)can be proved by the same argument as in the proof of[16,Theorem 2.1(ii)].

Now we give an interesting application of Theorem 4.1,where the parametersandare symmetric.

Theorem 4.2Let ,≥1.Then there exists a(pure)quantum code Q with parameters=d2if and only if there exists a(pure)quantum codewith parameters(A,

Proof(i)IfQis a quantum code with parameterswhereK≥2,=and=.By Theorem 4.1,we haveKnonzero mappings:A→C(1≤i≤K)satisfying the condition(4.3)in Theorem 4.1.Let Φi:A→C be the Fourier transform of,

We show that(1≤i≤K)give the required quantum codes.Namely,for each partition(4.2),we need to show that

The left-hand side of(4.6)is

By(4.3),the right-hand side of(4.7)is zero forij,and fori=jit is independent ofi.Therefore,the equality(4.6)is true.IfQis pure,then(1≤i≤K)satisfies condition(4.4).?i(1≤i≤K)are linear independent,so are1≤i≤K).Then we need to show that,for each partition(4.2),

whereis independent ofandcX.

We also have(4.7).Since(1≤i≤K)satisfies(4.4),the right-hand side of(4.7)is

where=This completes the proof of Theorem 4.2.

By this result,from now on we denote the parameterNow we give another application of Theorem 4.1.

Theorem 4.3Let C be a mixed classical additive code in A,be the minimal distance of the dual codeof C,and V=:1≤i≤K}be a set of K distinct vectors in A,such that

Then there exists a pure asymmetric inhomogeneous code with parameters(A,K,

ProofThe proof is similar to that of[16,Theorem 3.2](for the asymmetricq-ary case)or[15,Theorem 3.4](for the inhomogeneous symmetric case).We omit the details.

Example 4.1Letd,n≥2 andA=A1⊕···⊕(1≤i≤n).LetAd(n,2,l)be the maximum size of thed-ary constant weight codes with lengthn,distance 2 and weightl.

LetC(l)be ad-ary constant weight code with lengthn,distance 2,weightland sizeAd(n,2,l).Taking

in Theorem 4.4,we havedV≥2.Then we get a pure quantum code with parameters(A,K,{2,dV}),whereK=

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