Asymmetric Entanglement-Assisted Quantum Error-Correcting Codes and BCH Codes

The concept of asymmetric entanglement-assisted quantum error-correcting code (asymmetric EAQECC) is introduced in this article. Codes of this type take advantage of the asymmetry in quantum errors since phase-shift errors are more probable than qudit-flip errors. Moreover, they use pre-shared entanglement between encoder and decoder to simplify the theory of quantum error correction and increase the communication capacity. Thus, asymmetric EAQECCs can be constructed from any pair of classical linear codes over an arbitrary field. Their parameters are described and a Gilbert-Varshamov bound is presented. Explicit parameters of asymmetric EAQECCs from BCH codes are computed and examples exceeding the introduced Gilbert-Varshamov bound are shown.


Introduction
In the last decades the interest in quantum computation has grown exponentially, mainly because it transforms some intractable problems into tractable ones as showed the polynomial time algorithms given by Shor for discrete logarithms and prime factorization [41].
The usage of subatomic particles to hold memory and the application of quantum mechanics determine the behavior of quantum computers.These computers (the current implementations) are less reliable than the classical ones and produce more errors.Another inconvenient with this computers is decoherence and, even when one cannot clone quantum information [12,45], both challenges can be addressed with quantum error correction [42,43].
The first steps in the construction of quantum error-correcting codes corresponded to the binary case [8,9,22] (see also [2,3,25]).Afterwards and especially because of their interest in fault-tolerant computation the non-binary case was also studied [28] (some more references are [5,6,24,29,36]).Most of the quantum error-correcting codes are stabilizer codes where the error group is determined by eigenspaces with eigenvalue 1.
Unitary operators, usually denoted X and Z, are used to provide quantum (errorcorrecting) codes and the minimum distance d of such codes indicates that one can correct up to ⌊(d − 1)/2⌋ phase-shift and qudit-flip errors.In [27], the authors noticed that phaseshift errors happened more likely than qudit-flip errors, thus it was desirable to construct quantum codes where two minimum distances d x and d z , for detecting qudit-flip and phaseshift errors, respectively, were considered and provide results for addressing their behavior.As a consequence, in the last years asymmetric quantum error-correcting codes have been studied giving rise to codes suitable when dephasing occurs more often than relaxation [14,15,16,31,32,40].Most of the asymmetric quantum codes come from the CSS construction of quantum stabilizer codes and, for them, there is also a Gilbert-Varshamov bound [35].In addition, the existence of an asymmetric quantum error-correcting code coming from the CSS construction can also be applied to linear ramp secret sharing and communication over wiretap channels of type II [19].
To provide an asymmetric (or symmetric) quantum code requires some type of selforthogonality of the classical constituent code (or an inclusion of a constituent code into the dual of other constituent one) and, then, many good classical codes cannot be considered for that purpose.For overcoming this restriction and boosting the rate of transmission, it was proposed in [7] (for the symmetric case) to share entanglement between encoder and decoder.Some constructions of this type for binary codes (and also for codes over finite fields F p , p prime) can be found in the literature [26,34,44].The case when the codes are supported in an arbitrary finite field has been described in [21].
It seems clear that it remains to consider entanglement-assisted quantum error-correcting codes (EAQECCs) for the asymmetric case.To the best of our knowledge this task had not been performed yet.Section 2 of this paper is devoted to explain how to construct and which are the parameters of an asymmetric EAQECC obtained from any two linear classical codes.Theorem 3 and Theorem 4 (for nested constituent codes) are the main results in this section.Section 3 gives a Gilbert-Varshamov bound for asymmetric EAQECCs; we state and prove this bound for both the finite and the asymptotic case.In Section 4 we present the explicit computation of the parameters of asymmetric EAQECCs coming from BCH codes, see Theorem 9 and Corollary 10.Finally, our Section 6 provides examples of asymmetric EAQECCs which exceed the Gilbert-Varshamov bound before stated.Notice that asymmetric EAQECCs give rise to (symmetric) EAQECCs and in this section we show also examples of EAQECCs obtained with our procedure exceeding the Gilbert-Varshamov bound for EAQECCs.

Asymmetric EAQECCs
Let q = p r a positive power of a prime number p and set F q the finite field of order q.A q-ary stabilizer quantum code is the linear space of (C q ) n given by the intersection of the eigenspaces with eigenvalue 1 corresponding to some subgroup S of the error group G n generated by the matrices corresponding to a basis of Hom ((C q ) ⊗n , (C q ) ⊗n ), that is G n is determined by the product [28,Lemma 11] that an error in G n is detectable by the stabilizer code if and only if it belongs to the group generated by the subgroup S and the center of G n or the error is not in the centralizer of S in G n .
The above facts can be regarded in terms of additive codes in F 2n q .In order to do this, we introduce the trace-symplectic form for two vectors (a|b) , (a ′ |b ′ ) ∈ F 2n q as follows: where tr q|p is the trace map and • the inner product in F n q .Then (in the linear case) an [[n, k, d]] q stabilizer quantum code exists if and only if there is a linear code C ⊆ F 2n q of dimension n − k such that C ⊆ C ⊥ts , where C ⊥ts stands for the dual code with respect to the • ts product.Here the minimum distance d is determined by the minimum symplectic weight swt(C ⊥ts \ C).It is convenient to recall that for (a|b) as above, swt (a|b) = # {j ∈ {1, 2, . . ., n}|(a j , b j ) = (0, 0)} , # meaning cardinality.
A particular case in the above construction follows from the so-called CSS (Calderbank-Shor-Steane) procedure [43,9].Here we need two linear codes C 1 and C 2 in F n q such that C 2 ⊆ C ⊥ 1 , ⊥ means Euclidean duality, and then the code q provides a stabilizer quantum code whose parameters depend on those of C 1 and C 2 .Some classical references are [5,6,8,9,10].
The fact that dephasing usually happens much more often that relaxation [27] motivated the study and searching of asymmetric quantum error-correcting codes [14,15,16,30,31,32,33].For this purpose, the most used procedure is the CSS construction because it easily allows us to get parameters d z and d x such that our previous stabilizer code detects phase-shift (respectively, qudit-flip) errors up to weight q be linear codes.The CSS construction gives rise to an asymmetric quantum code with parameters ).The previously mentioned stabilizer and asymmetric quantum codes require self-orthogonality conditions with respect to trace-symplectic duality and not every classical linear code can be used for providing those quantum codes.The self-orthogonality condition can be bypassed if encoder and decoder share some quantity of entanglement [7] giving rise to the so called entanglement-assisted quantum error-correcting codes (EAQECCs).In the binary case the construction of these codes is described in [26] (third paragraph of Section II).This construction also holds for codes over finite fields of the type F p , p being a prime number (see [44, Remark 1] and [34] for a proof).There it is proved that one can obtain an EAQECC from a classical code C ⊆ F 2n p such that C ⊆ C ⊥ts and the set of detectable quantum errors is given by On F 2n q , q = p r , one can also define a symplectic product: Using a suitable basis of F q over F p , an isomorphism of F p -linear spaces φ : F 2r p → F 2 q can be given, providing an isomorphism of F p -linear spaces With the help of φ E , in [21], the results of EAQECCs over F p can be extended to F q and the product • s instead of • ts .Indeed, the following result holds: q be a linear code over F q of dimension (n − k).Denote by H = (H X |H Z ) a generator matrix for C. Let C ′ ⊆ F 2(n+c) q be a linear code over F q whose projection to the coordinates 1, 2, . . ., n, n + c + 1, n + c + 2, . . ., 2n + c equals C and such that C ′ ⊆ (C ′ ) ⊥s , c being the minimum required number of maximally entangled quantum states in C q ⊗ C q .Then, The encoding quantum circuit is constructed from C ′ , and it encodes k + c logical qudits in n physical qudits using c maximally entangled pairs.The minimum distance is In this paper we are interested in the asymmetric case and we desire to construct asymmetric EAQECCs from two linear codes C 1 and C 2 over an arbitrary finite field F q .Assume that H 1 (respectively, H 2 ) is a generator matrix of C 1 (respectively, C 2 ).
The above described construction of stabilizer codes over F p following the CSS procedure determines asymmetric EAQECCs coming from any two linear codes where ⊥ denotes the Euclidean dual.Notice that in this case • ts = • s .The set of detectable errors is .

Defining
(1) , where wt means minimum Hamming weight, it is clear we are able to construct an asymmetric EAQECC which can detect up to d x −1 qudit-flip errors and up to d z −1 phase-shift errors.
These results can be extended to any finite field F q using again the above described isomorphism φ E and [21, Proposition 1] which relates • st and • s .The general result being: Theorem 3. Consider linear codes C i ⊆ F n q of dimension k i and generator matrix H i , i = 1, 2. Set d x and d z as in (1).
Then C 1 × C 2 ⊆ F 2n q gives rise to an asymmetric EAQECC which encodes n − k 1 − k 2 + c logical qudits into n physical qudits which can correct up to ⌊(d x − 1)/2⌋ qudit-flip errors and up to ⌊(d z − 1)/2⌋ phase-shift errors.The minimum required of maximally entangled pairs is . As a consequence, we obtain an We end this section with a result that assumes that our constituent linear codes are nested.We will see that the asymmetric EAQECC comes from puncturing a code in F 2n q .Theorem 4. Let C 1 and C 2 be F q -linear codes such that Then, there exists an asymmetric EAQECC with parameters where d z (respectively, d x ) is the minimum Hamming weight of the elements in the set

A Gilbert-Varshamov bound for asymmetric EAQECCs
We devote this section to provide a finite and an asymptotic Gilbert-Varshamov-type (GV) bound for asymmetric EAQECCs.We start with the finite case.
Proof.For simplicity sake, in this proof C ′ 2 will be used instead of C ⊥ 2 .Consider integer numbers n, k 1 , k 2 and c as in the statement.Define where we recall that # means cardinality.Let us see a proof.Denote by GL(n, q) the set of invertible matrices on F n q and for a fixed with M ′ 1 span(v 1 ) ⊥ = span(v 2 ) ⊥ , where M ′ 1 span(v 1 ) ⊥ stands for the linear space given by the products M ′ 1 w such that w ∈ span(v 1 ) ⊥ .Then we have We also claim that #B x (v 1 ) = #B x (v 2 ).Indeed, Proposition 4] and the fact that the rank of a matrix and its transpose coincide, we deduce that

Next we will count the quantity of triples
Then, we observe that . Thus the total number of such triples is On the other hand, we can count the total number of triples as for any fixed nonzero v.This implies a (the least one) of each minimal cyclotomic coset which we denote I a .Then {I a } ∈A is the set of minimal cyclotomic cosets with respect to q, A being the set of representatives above mentioned.In addition set i a := #(I a ).For convenience, we will write We will use the following two results which can be found in [20,18].
Proof.Consider the linear codes , where ∆ ′ = H \ s j=0 I a j [18].Hence, the minimum required of maximally entangled pairs is The minimum distance of the dual codes satisfies d(C ⊥ 1 ) ≥ a t+1 + 1 (by Proposition 8) and d(C ⊥ 2 ) ≥ a s+1 + 1 because C 2 contains s + 1 consecutive cyclotomic cosets and it is equivalent to a code as in Proposition 8.
Finally, applying Theorem 3, we get an asymmetric EAQECC with parameters as in the statement.
From the previous result, we can deduce the following one.
Corollary 10.Keeping the above notation where q = p r , assume that Let s ∈ {0, 1, . . ., z} such that s < t.Then we can construct an asymmetric EAQECC with parameters Proof.It follows from the proof of [1,Theorem 10] where it is showed that if Inequalities (3) and ( 4) hold, then the number t of non-zero cyclotomic cosets considered is and all of them have cardinality ℓ/r.
Remark 11.Notice that the parameters of the asymmetric EAQECC given in Corollary 10 can be written as follows:

Entanglement and Minimum Distances
Assume that C is a (standard) asymmetric quantum code over a field F q coming from the CSS construction with parameters [[n, k, d z /d x ]] q .Considering entanglement and a suitable extension of constituent codes, it is possible to increase the value of d z (or d x ) keeping the length and information rate.Therefore, one may increase the asymmetry ratio (the ratio between d z and d x ).Indeed, for both the standard case and the entanglement-assisted case, one considers two linear codes C 1 and C 2 and it holds that ).However, for the standard case it must be imposed that ⊥ , the information rate is kept, one of the minimum distances is the same and the other one increases.
Let us illustrate the above technique with a small example.Keep the notation as in Section 4 and assume that ℓ = r, that is we do not consider subfield-subcodes in this example.Let C i be the code we deduce that the value d x for the standard case (C 1 , C 2 ) and the entanglement-assisted case (C ′ 1 , C ′ 2 ) is the same.However, in the standard case, and then, when one considers entanglement, by the BCH bound.As a consequence, to share entanglement allows us to increase the value d z and therefore the asymmetry ratio.

Examples of Asymmetric EAQECC
Table 1 shows the different values involved in the construction of asymmetric EAQECCs, over several finite fields, constructed as in Theorem 9.The last two columns display the representatives of the cyclotomic cosets used to define the codes C 1 and C 2 .Notice that the parameters of our asymmetric EAQECCs follow immediately from Theorem 3 and are [[n, n − k 1 , d z /d x ; c]] q .All these codes exceed the asymmetric Gilbert-Varshamov bound proved in Theorem 5.In addition, for fixed values (q, n, k 1 , k 2 , c), we consider the set P of pairs (d 1 , d 2 ) of Z-minimum and X-minimum distances of asymmetric EAQECCs such that (d 1 , d 2 ) does not exceed the bound in Theorem 5 but either (d 1 + 1, d 2 ) or (d 1 , d 2 + 1) beat it; we have noticed that frequently the cardinality of P is one.Let (d z , d x ) be the maximum of P with respect to the lexicographical order where (1, 0) > (0, 1).We would like to add that our construction from BCH codes (regarded as in Section 4), using Theorem 3, may produce good symmetric EAQECCs as well.We use the fact that an asymmetric EAQECC provides a symmetric EAQECC with the same parameters but its minimum distance, which is the minimum of the two minimum distances d x and d z .In all our examples, both minimum distances are equal.Thus, Table 2 displays values of codes coming from our construction, giving rise to symmetric EAQECCs whose parameters are [[n, k = n − k 1 − k 2 + c, d = d z = d x ; c]] q .We have used the Hartmann-Tzeng bound [38] in our computations.All codes in this table exceed the Gilbert-Varshamov bound for symmetric EAQECCs [21].Table 2 also contains, for each code C, the minimum distance d of a symmetric EAQECC with the same parameters (q, n, k, c) as C and such that d does not beat the above mentioned symmetric Gilbert-Varshamov bound but d + 1 does.In several cases, our codes exceed the value d by more than one unit.

2
and 2c ≤ wt(C \ 0) − 1, and the result follows from[21, Theorem 7]  and(1).Notice that the above asymmetric EAQECC comes from the punctured code defined asP (C) = (pr(a), pr(b) | (a, b) ∈ C , pr being the projection to the first n − c coordinates.In fact, according to the proof of [21, Theorem 9] dim P (C) − dim P (C) ∩ P (C) ⊥s = 2c, which by Theorem 2 shows that c is the number of maximally entangled pairs.

3. 1 .Theorem 5 .
The finite GV bound.Let us start with our result.Consider positive integer numbers n, k 1 , k 2 , d z , d x and c such that k 1 ≤ n, k 2 ≤ n and k 1 + k 2 − n ≤ c ≤ min{k 1 , k 2 }, which satisfy the following inequality has dimension t j=t ′ i a j .Proposition 8.The minimum distance of the (Euclidean) dual of the subfield-subcode E ∆ | Fq , where ∆ = ∪ t j=0 I a j is larger than or equal to a t+1 + 1 (BCH bound).Next we state the main result in this section.Theorem 9.With the above notation consider two different indices s, t ∈ {0, 1, . . ., z} and assume that s < t.Then we can construct an asymmetric EAQECC with parameters we may consider a new pair of linear codes, C ′ 1 and C ′ 2 , by enlarging C 1 or C 2 , in such a way that either C ′ 2 ⊆ (C ′ 1 ) ⊥ or C ′ 1 ⊆ C ⊥ 2 do not hold any more, but this new pair gives an asymmetric EAQECC with better parameters.Hence, taking into account that

Table 1 .
Table 1 displays the threshold (d z , d x ) as well.Note that many times our codes exceed both values d z and d x by one or two units.Asymmetric EAQECC coming from Theorem 9