I Introduction
In classical information theory, there are mainly two types of error models: the independent noise model proposed by Shannon and the adversarial noise model considered by Hamming. Errors in these two models are usually called random errors and burst errors. Correspondingly, there are random errorcorrecting codes (RECC) and burst errorcorrecting codes (BECC) to deal with these two different types of errors [1]. In reality, channels tend to introduce errors which are localized in a short interval, i.e., the burst errors. These errors could be commonly found in communication systems and storage mediums, as a result of a stroke of lightning in wireless channels or scratch on a storage disc.
In the quantum regime, quantum errors can be independent or correlated in space and time. Hence there are counterparts of quantum random errorcorrecting codes (see, e.g., [2, 3, 4, 5, 6]) and quantum burst errorcorrecting codes [7, 8, 9, 10]. Analogous to the classical case, quantum channels commonly have memory [11] or introduce errors which are localized [12], i.e., quantum burst errors. Vatan et al. [8]
first considered spatially correlated qubit errors and constructed families of QBECCs using CSS construction
[3, 4]. However, CSS construction yields QBECCs with inferior code rate. In [9], a quantum interleaver for QBECCs was proposed so that long QBECCs could be produced from short ones. However, this method highly relies on short efficient QBECCs, which are lacking at this moment. In
[10], QBECCs of length up to were found using computer search.The construction and investigation of QBECCs have received far less attention, compared to the development of standard QECCs or entanglementassisted QECCs [13, 5, 14, 6, 15, 16, 17]. Many important questions remain open. Currently, there is no general upper bound for correctable quantum burst errors, analygous to the classical Reiger bound: , of an classical BECC, where is the code length, is the message size, and is the correctable length of burst errors. In addition, there is an interesting class of quantum codes, called degenerate codes, that have no classical correspondences. They can potentially store more quantum information or correct more quantum errors than nondegenerate codes. However, degenerate QBECCs have never been explored.
In this paper we generalize the theory of standard QECCs to QBECCs. We develop the stabilizer formalism for QBECCs and prove the corresponding quantum Reiger bound: , for an QBECC that corrects a quantum burst error of length or less. The quantum Reiger bound further generalizes the quantum Singleton bound in QECCs. We obtain many new QBECCs in the stabilizer formalism via computer heuristic search, and these codes are better than existing QBECCs with the same code lengths. We show that the burst errorcorrecting abilities of most of these codes can achieve the quantum Reiger bound and thus are perfect codes. In particular, several of our constructed QBECCs, that attain the quantum Reiger bound, are degenerate codes. Additionally, we propose a new concatenation construction of long QBECCs from two short component codes based on the quantum tensor product code structure [7] and the interleaving technique. Since only one of the component codes of QTPCs needs to satisfy the dual containing constraint, this construction method can largely facilitate the systematical construction of QBECCs. Finally, we perform numerical experiments on two of our constructed QBECCs by measuring the entanglement fidelity over Markovian correlated depolarizing quantum memory channels [12]. It is known that the correlation errors in memory channels can lower the performance of the entanglement fidelity of standard QECCs (see [18, 12]). But if we consider the extra burst error correction abilities of them, they can indeed outperform the best QECCs of the same lengths for random errors.
Ii Theory of Quantum Burst Error Correction Codes
In this section, we introduce background of QECCs and develop the stabilizer formalism for QBECCs.
Iia Quantum Burst Error Correction Codes
In a twodimensional complex Hilbert space , a qubit can be written as where and are complex numbers satisfying . Two states and , that are different up to a global phase , are considered to be the same in this paper. The Pauli matrices
form a basis of the linear operators on , where and . An qubit is then a quantum state in the th tensor product of , i.e., .
Since it is possible to discretize quantum errors [19, 20], we only need to consider a discrete set of quantum errors of qubits, described by the following error group
(1) 
Furthermore, it is sufficient to consider the quotient group of since the global phase in is not important. Let and . We define the burst length of to be , denoted by , if the nonidentity matrices in are confined to at most consecutive ’s.
The idea of a QECC is to encode quantum information into a subspace of some larger Hilbert space. An QECC is defined to contain the subspace of dimension in . If , then is also written as . A QECC can correct arbitrary errors from an error class [21, 22] if
(2) 
for all and for all , where and , and is a constant which depends only on and . If for all and for all , then is called a nondegenerate quantum code.
The above errorcorrecting condition (2) can be generalized to the burst error case.
Proposition 1
The code can correct any quantum burst errors of length or less if and only if
(3) 
for all and for all , where and , and , and is a constant which depends only on and .
If for all and for all , where , then is a nondegenerate QBECC.
According to the group theoretic framework for QECCs in [4, 23], we can also get the stabilizer formalism for QBECCs. Let and
be two vectors in
, the symplectic inner product of them is given by(4) 
For a subspace of , the symplectic dual space of is given by
(5) 
We define the symplectic burst length of a nonzero vector to be the largest integer such that and for some . We denote by .
According to [23], each element can be written uniquely as where , , , and for . It is easy to verify that . Then we have the group framework for quantum burst error correction codes.
Theorem 1
Suppose that there exists an dimensional linear subspace of which is contained in its symplectic dual , i.e., . Let be the largest integer such that for arbitrary two vectors whose symplectic burst length there is . Then there exists a quantum burst error correction code which can correct arbitrary quantum burst errors of length or less. If all the , then is a nondegenerate quantum burst error correction code.
Proof:
As shown in [23], binary quantum codes can be constructed by using additive codes over . Define the trace inner product of two vectors by
(6) 
Let be an additive code over , then the trace dual of with respect to the trace inner product is defined by
(7) 
Then Theorem 1 can also be reformulated by using additive codes over and by replacing the symplectic inner product with trace inner product.
Theorem 2
Suppose that is an additive code over which is contained in its trace dual , i.e., . Let be the largest integer such that for arbitrary two vectors whose burst length there is . Then there exists a binary quantum burst error correction code which can correct arbitrary quantum burst errors of length or less. If all the , then is a nondegenerate quantum burst error correction code.
The CSS code construction [3, 2] provides a direct way to construct QECCs from classical linear codes. The CSS construction for QBECCs can be obtained from Theorem 1.
Corollary 1 (CSS Construction)
Let and be two binary linear codes which have and burst error correction abilities, respectively, and such that . Let be the largest integer such that for arbitrary two vectors whose burst length there is . Then there exists a binary quantum burst error correction code which can correct arbitrary quantum burst errors of length or less and if , then is a nondegenerate code.
IiB Quantum Reiger Bound
For a classical code which can correct random errors or can correct any burst errors of length , there exists two important upper bounds called the Singleton bound and the Reiger bound that constrain the random error correction and burst error correction abilities of , respectively (see [1]). In quantum codes, let be a QECC which can correct quantum random errors, there exists the quantum Singleton bound which is an upper bound for the quantum random error correction ability of code (see [19, 23]).
In the following, we derive the quantum Reiger bound (QRB) which is an upper bound for the quantum burst error correction ability of code .
Theorem 3 (Quantum Reiger Bound)
If an QBECC can correct quantum burst errors of length , then it satisfies
(8) 
Proof:
The proof follows closely by that of the quantum Singleton bound given by Preskill (see [24, p.32] and [19, p.568]).
First of all, Lemma 1 in the Appendix says that if can correct burst errors, then it must satisfy , a consequence following from the quantum nocloning principle.
Then we introduce a qubit ancilla system , and construct a pure state that is maximally entangled between the system and the codewords of the QBECC :
(9) 
where denotes an orthonormal basis for the dimensional Hilbert space of the ancilla, and denotes an orthonormal basis for the dimensional code subspace. It is obvious that
(10) 
where is the von Neumann entropy of a density operator .
Next we divide the qubit QBECC into three disjoint parts so that and consist of qubits each and consists of the remaining qubits. If we trace out and , the reduced density matrix that we obtained must contain no correlations between and the ancilla , a consequence following from Lemma 2 in the Appendix. This means that the entropy of system is additive:
(11) 
Similarly,
(12) 
Furthermore, in general, the von Neumann entropy is subadditive, so that
(13)  
(14) 
Combining these inequalities with the equalities above, we find
(15)  
(16) 
Both inequalities can be simultaneously satisfied only if
(17) 
Finally, we have
(18) 
since is bounded above by its dimension . We then conclude the quantum Reiger bound.
Iii Construction of Quantum Burst Error Correction Codes
In this section we provide two methods for constructing QBECCs: one by using computer search based on the stabilizer formalism, and the other by concatenating and interleaving quantum tensor product codes.
Iiia Stabilizer QBECCs Constructed by Using Computer Search
We create a program using Magma software (version V2.1216) to search all possible cyclic codes to a reasonable length () according to Proposition 1
and make the codes as close as possible to the quantum Reiger bound. For simplicity, we only consider the construction of QBECCs from cyclic codes with odd length. We list new codes that are near or saturate the quantum Reiger bound found by this program in Table
I. The bold numbers “” stand for the coefficients and the superscript numbers stand for the exponents in the generator polynomials of the corresponding classical cyclic codes [1]. Notice that the burst errorcorrecting abilities of most QBECCs in Table I can saturate the quantum Reiger bound. In particular, we get several degenerate QBECCs which are the first class of QBECCs until now and they can saturate the quantum Reiger bound. Moreover, some of the constructed QBECCs are better than the QBECCs in Ref. [10], e.g., the codes , and in Table I have larger dimensions than the codes , and in Ref. [10], respectively, but have the same burst errorcorrecting ability.We remark that some QBECCs of length up to 51 have been found by using computer search in Ref. [10]. However, only CSS type QBECCs were considered and no degenerate codes were obtained in Ref. [10].
Generator Polynomials  QRB  Degenerate?  

3  3  False  
3  3  False  
4  4  False  
4  4  True  
2 

3  False  
5  5  False  
6  6  True  
5  5  False  
7 

7  True  
2  2  False  
3  4  False  
4  4  False  
5  5  False  
6 

7  False  
10 

10  True 
IiiB Concatenation Construction of QBECCs Based on Quantum Tensor Product Codes
In this section we give a concatenation construction of long QBECCs from two short component codes based on the quantum tensor product codes structure [7, 25].
Firstly we present a brief review of classical and quantum tensor product codes. Details could be found in, e.g., [26, 7].
Let be a linear code with a parity check matrix and let be the number of check symbols. Let be a linear code over the extension filed with a parity check matrix . Then the tensor product code (TPC) of and is denoted by , and the parity check matrix is given by
(19) 
By selecting different types of component codes, TPCs can be designed to provide different error control abilities.
In [7], a framework for the construction of quantum tensor product codes (QTPC), which can provide a wide variety of quantum errorcorrecting, errordetecting or errorlocating properties, was proposed. In particular, if one of the component codes is selected as a BECC, then QTPCs can have multiple quantum burst errorcorrecting abilities, but provided these bursts fall in distinct subblocks.
Theorem 4 ([7])
Let be an burst error correction code, and let be an burst error correction code over the extension field , and the numbers of check symbols are and , respectively. If and or if and , where is the Hermitian dual code of , then there exists a QTPC which can correct or fewer bursts of burst errors each is a burst of length or less, provided these bursts fall in distinct subblocks.
Although we can use QTPCs to correct a single burst of errors since QTPCs have multiple burst error correction abilities, they are not efficient enough any more. To overcome this problem, we can interleave the encoded qubits before sending into the quantum channels, and deinterleave after receiving the qubits. The whole interleaving/deinterleaving procedure is summarized as follows:

After the quantum encoding, we arrange the encoded qubits into an code array.

Instead of transmitting the encoded qubits sequentially one by one, we do an interleaved transmission. Denote by the burst error correction ability of . If and the component code can correct endaround (see [1]) burst errors, then we divide the code array into subblocks by rows. We do the transmission subblock by subblock, and in each subblock, we transmit the qubits column by column sequentially (each column contains qubits).

After receiving all the qubits, we deinterleave the qubits into an code array so that the quantum decoding can be processed next, and the deinterleaving is just the inverse of the interleaving. The deinterleaving/interleaving procedure can be accomplished by using quantum SWAP gates (see [19]).
Suppose that a single burst errors of length at most happens among the interleaved qubits. After the quantum transmition and deinterleaving, the qubits are recovered to their original positions, but the single burst errors of length at most has been dispersed into or fewer consecutive subblocks (end around) and each subblock contains a burst errors of length at most . Thus the resultant QTPC can correct a single burst error of length at most according to Theorem 4 and Ref. [7]. Then we have the following result.
Theorem 5
Let and be two component codes of a QTPC with parameters , where is an burst error correction code and is an burst error correction code (endaround), and the numbers of check symbols are and , respectively. If , then there exists an burst error correction quantum code .
Example 1
We choose as a burst error correction cyclic code with the generator polynomial and it is Hermitian dual containing by Table I. Let be an MDS code over the extension field with and . Then there exists a burst error correction QTPC with parameters .
Iv Performance of QBECCs over Markovian Correlated Quantum Memory Channels
In this section we evaluate the performance of two specific QBECCs in the presence of correlated errors.
The channel model that we choose is a Markovian correlated depolarizing quantum channel [12, 18]:
(20) 
where are the Pauli operators, and the conditional probabilities satisfy the normalization condition
(21) 
where for and , and are the error probabilities in the depolarizing channel, is the correlation degree.
Specifically, we show the performance of the two specific codes by measuring the entanglement fidelity as a function of the error probability and the correlation degree [19].
The two specific codes considered here are two QBECCs and in Table I which can correct burst errors of length and of length , respectively. Through the computation, we know that the minimum distances of the two codes are and , then they can also correct and random errors, respectively, and they have achieved the upper bounds in Ref. [27]. We plot the performance by means of entanglement fidelity of the two specific codes with respect to random errors or burst errors, versus the correlation degree or the error probability in Fig. 1 or Fig. 2, respectively. For details about the computation of the entanglement fidelity, see [12, 18, 19], and the computation results are put in the Cloud.
It is shown that the correlation errors do degrade the performance of the entanglement fidelity of the two codes in Fig. 1. If the the correlation degree which means that errors are independent with each other, then the extra burst error correction abilities of the two codes do little help to improve the performance of entanglement fidelity of them, respectively, see Fig. 1 and Fig. 3. However, if we consider the correlated errors when , the performance of the entanglement fidelity can be improved largely, see Fig. 1 and Fig. 2. In particular, in Fig. 2, the code have better performance when considering its extra burst error correction ability compared to the code when only considering the random error correction ability.
Acknowledgment
J. Fan was supported by NJIT (Grant No. YKJ201719) and by NSFC (Grant No. 61403188). M.H. Hsieh was supported in part by an ARC Future Fellowship under Grant FT140100574 and in part by U.S. the Army Research Office for Basic Scientific Research under Grant W911NF1710401.
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