3.1. Definitions and Terms
Based on the general definition of a square matrix, in this section, we presented a construction method of quantum codes over binary matrices. As the following, for obtaining the stabilizer generators of quantum codes over the complex number field with a bigger size, we consider a class of binary cocyclic Jacket matrices based on a circular permutation matrix to construct its concatenated generator matrix, i.e.,
here,
and
are generator matrices of the encoding bit and phase, respectively.
Since a cocyclic approach was introduced in [
24], some interesting codes come from useful cocyclic matrices, such as the Hadamard matrix that is the matrix representation of a cocycle [
25].
Definition 1. If set G is a finite group of order v and is an Abelian group of order w, respectively, a mapping: φ: G × G is called a cocycle while it satisfies the following particular cocycle equation From the definition, it is easy to show that , so , as we call it, is a normalized cycle.
According to the cocycle mapping, if the rows and columns of a square matrix can be indexed by the elements of G in terms of some distribution, the matrix is called a cocyclic matrix, which is described as
where the map
denotes a position
of the entry in matrix
. If the cocycle
is symmetric,
will be a symmetric matrix. Furthermore, on the basis of the cocylic matrices, some cocyclic codes are primitively derived from Hadamard matrices or from related block designs [
33]. In this case, a
Hadamard matrix
H with entries
such that
is used to construct codes with length
for integer
. If 1s and −1 s are replaced by 1 s and 0 s, respectively, the Hadamard matrix becomes a binary one. A main superiority of cocyclic codes is that the long-length codes may be easily gained with a circulant manner; hence, it may be applicable to the generalized classical case and further quantum codes.
Definition 2. A square matrix , if its inverse is obtained simply by element-wise inverse, i.e., it satisfieswhere a is the normalized constant, we call matrix a block Jacket matrix. Obviously, the Hadamard matrix belongs to the defined Jacket matrix. Furthermore, if there is a permutation matrix , and its inverse matrix can be described as , matrix is a quasi-Jacket block matrix [34]. This class of matrices may be generalized by a simple binary transformation. Matrix decomposition with a Kronecker product of identity matrices can successively design a large length of codes that resorts to lower-order coefficient matrices. Meanwhile, this factorization decomposition generates a family of block circular sparse Jacket matrices which have good performance of LDPC codes. In the applications, the derived circulant permutation matrices with a Jacket pattern could lead to a very simple encoding algorithm. 3.2. Cocyclic Matrix Derived from Jacket Block Matrices
On the basis of the above definitions, we shall apply the index mapping to construct generator matrices and based on cocyclic Jacket matrices with order . Firstly, we form an Abelian group by factoring a quasi-Jacket block matrix, of which elements will be used to construct a stabilizer.
Theorem 1. Given and are two cocyclic Jacket matrices, their Kronecker product will be also a cocyclic Jacket matrix with size [35]. Proof. Because
and
are Jacket matrices, in terms of the definition of Jacket matrix, it has
where
are two constants. As a result, their Kronecker product is
for
. Correspondingly, we have
i.e.,
is a Jacket matrix.
On the other hand, suppose that the row and column index orders of the two matrices as
where
, and
(or
) and
(or
) refer to the first row and column indices of matrix
(or
), respectively. Hence, the row and column index orders of matrix
are
Correspondingly, define
Because both
and
are cocyclic matrices,
and
satisfy the condition shown in Equation (
12). As a result, it is verified that
also meets it with the index orders shown in Equation (
18). It means that
also is a cocyclic matrix. □
More generally, by making use of the mathematical methods of induction, it is easily verified that a large-length matrix
with order
is still a Jacket matrix and cocylic matrix, while Jacket matrix
is a cocylic matrix for
. Specially, if
, the following matrix
with a Kronecker product also achieves the definitions of Jacket matrix and cocylic matrix. In the following, we further illustrate that the generator matrix may be derived from a recursive cocyclic matrix.
Theorem 2. A recursive p-order cocyclic matrix for a integer m can be composed of a family of cocyclic matrices with conventional multiplication as followswherefor that, is a cocyclic matrix and is an identity matrix. Proof. We use the mathematical induction method to prove the result. In fact, while
, the conclusion is obviously valid, i.e.,
. Since the hypothesis is true for integer
, we consider that it holds for
. Namely, the recursive equation
holds. According to the property of the Kronecker product, it obeys the following formula
for any square matrices
and
D. Hence,
It means that Equation (
24) is equal to Equation (
26), i.e., the case of
also holds. □
In terms of the above conclusion, based on its decomposition factors, we can form a factor group . Resorting to the factor group , we consider constructing the stabilizer with two isomorphic classes of groups.
Lemma 1. Take factor group in Equation (23) and its isomorphic factor group with matrix . In terms of the property of the Kronecker product, if matrices and satisfy the commuting condition, i.e., their symplectic inner product is equal to zero, it is easy to check that any pair of factors and () in the two groups also satisfies the commuting condition (7). It means that any pair of and can be taken as the generator matrices and of quantum codes, respectively, when two matrices and commute.
Example 1. If the prime integer is given , let . It has vectors and in group .
3.2.1. Quasi-Jacket Block Matrices with Size by
Resorting to the binary matrixes, we consider building the generator matrix of quantum stabilizer codes. On the one hand, we define a class of fundamental matrices
and
derived from binary Hadamard matrix
H as follows, respectively:
It is easy to check that the introduced matrices have the following properties
Furthermore, define a permutation matrix with
order as
Based on the above fact, with simple computation, we can obtain a 4-order matrix and its inverse matrix
which is a quasi-Jacket block matrix in terms of the definition.
Similarly, we may also obtain the 16-order quasi-Jacket block matrix as
and its inverse matrix
According to its definition, the matrix obviously is a quasi-Jacket block matrix.
Generally, by resorting to the recursive relationships based on
defined by (
30), we can obtain the quasi-Jacket block matrices with larger size
by using the Kronecker product, i.e.,
which may be used to generate the generator matrices
(or
).
On the other hand, to obtain the matrix
(or
), we consider another class of matrix called the permutation matrix
,
for an even number
. According to the definition, it is obviously a finite cyclic group
formed by a family of matrices
is an Abelian group based on the matrix-multiplication operation, and
is its unit element. Here, given that the number
p is an even number and
h is an odd number, respectively, a sequence of matrices whose exponents are odd can form a subset
of Abelian group
.
Based on the generated two classes of matrices, i.e., the cocyclic Jacket matrix and permutation matrix, we consider obtaining the stabilizer for quantum codes. Let
with a size of
be obtained by the cocyclic Jacket matrix described before. In the following, we consider how to construct another matrix
. Firstly, we take two 4-order basic permutation matrices
in Equation (
34) with parameters
respectively, i.e.,
, such that
Then, it provides that
, and it takes
. As a result, the concatenated generator matrix
G is built.
In the following, we use the mathematical induction to deduce that two matrices
and
in the obtained concatenated generator matrix
G satisfy self-orthogonal and commuting conditions, i.e., the obtained generator matrix should meet the conditions of self-orthogonal and commuting, so that it may be used to construct a stabilizer of quantum codes. In fact, it is obvious that matrix
satisfies the self-orthogonal, so we firstly consider that the case of
for
meets the self-orthogonal constrain. Namely, in the following matrix
any two rows in
and
are self-orthogonal and commuting. Assuming it meets the case of
, we consider the result for
m. In fact, if any two rows in concatenated block submatrix
meet the conditions, it can also do so in
. Without a loss of generality, here, we only consider the rows in submatrix
. By making use of the Kronecker product of any two row vectors in
, it is easy to check that they meet the orthogonal condition.
And then, the symplectic inner product in Equation (
6) is proved as following. Define complement operators
and
, according to number theory; then, it is possible to easily check that matrices
with parameters
commute. Namely, any two row vectors in concatenated matrices
satisfy the commuting condition in Equation (
7). In fact, it can be checked that the elements in the Abelian group
commute when their cycle indexes both are odd or even numbers. The matrices satisfy isomorphic
when its corresponding orthogonal matrix also satisfies isomorphic
. The complement of
is an element in the circulant permutation group
; hence, two pairs of vectors
,
and
,
all belong to this group. The product of any two vectors in
and
is equal to 0 or 1 simultaneously; i.e., it means that two pairs of vectors of the inner product with module 2 is equal to 0.
Similarly, we consider the generator matrix based on the block matrix
and 16-order permutation matrix stemmed from
.
Generally, the
N-order generator matrix for
can be generated by
On the other hand, from the factorization of a large number,
, in Equation (
22), we can obtain a group
with Formula (
23) whose element
. Any element
in the group can be taken as the generator
. Similarly, we can form another group
based on
, and take the element
(same structure to
) as the generator
. As a result, a large-length generator matrix with size
by
is gained.
3.2.2. Quasi-Jacket Block Matrices with Size by
Furthermore, to obtain the more general length of code, we define a matrix
It is easily checked that
Furthermore, in terms of the basic properties, we consider matrix
and its inverse matrix
It is obvious that the constructed matrix
is also a quasi-Jacket matrix which can be seen as a
binary cocyclic matrix. It means that a quantum code with a length of
may be obtained. Similarly, the recursive relationships may be used to obtain the large-length block matrices with a larger size
by using the Kronecker product, i.e.,
such that the more general case of the generator matrix with a long size is achieved. In a similar manner, the obtained large-size matrix also satisfies the conditions of self-orthogonal and commuting like the process before proving the case of size
. We take two matrices based on parameters
in
or in
as
and
, respectively. It is evident that the obtained matrices commute each other.
3.3. Quantum LDPC Codes Based on Cocyclic Block Matrix
To encode
k information qubits into a QC quantum code
with parameter
, one should firstly gain two
cocyclic circulant permutation matrices
G based on the previously constructed matrices
and
of satisfying the orthogonal condition. According to the recursive relationship, we take any
rows of the two generated matrices as the generators
and
. It means that the concatenated matrix
in Equation (
3) meets the construction condition of quantum code.
In classic coding theory, a class of very important linear codes called LDPC code is widely explored and used in practice. Its theoretical importance mainly embodies that its good performance of linear-time decoding can achieve Shannon capacity [
14,
16]. Its quantum version, known as quantum LDPC (QLDPC) codes, plays a very important role in quantum information. However, compared to its classical counterparts, the achieved fruits of quantum LDPC codes are still far less. The main problem is the lack of efficient methods for obtaining iterative coding. Generally, the proportion of 1s in a binary matrix is called the matrix’s destiny. Given that
and
are the numbers of 0s and 1s in a matrix, respectively, the destiny is
According to the construction method described in Equation (
41), it obvious that the total number of 0s and 1s in a
matrix is
. Furthermore, each of the 12 matrices
and
in
contain
p 1s; hence, there are
1s in this matrix: i.e, the 1s density in
is
From the ratio, it means that the constructed cocylic matrix has a good performance of low-density property. On the other hand, there are
additions and
multiplications for the computation complexity in processing the obtained matrix with size
. The check matrix
H of LDPC codes is generally represented by a simpler and more intuitive Tanner graph with a one-to-one map. If there are equal degrees of all variable nodes in the Tanner graph, its corresponding LDPC code is called a regular code. Otherwise, it is called an irregular code. Denote by
and
the variable and check the nodes’ degrees corresponding respectively to the minimum weights of the row and column of the check matrix. Assume that the girth
g in the graph refers to the length of the shortest cycle [
36,
37]. Then, the following bounds should be satisfied, i.e.,
for girth
, and
for girth
, where
x is an integer, it can be seen from the above bounds that the code length will increase exponentially, while the cycle length and weights also are rising.
To obtain the good performance, having no 4-cycles in the Tanner graph of the quantum LDPC codes is required. A cycle is composed of a group of interconnected vertices in a Tanner graph, with one of these vertices serving as both the starting and ending points, and passing through each vertex only once. If the length of a cycle is called the number of lines it contains, the girth of a graph is defined as the minimum cycle length in the graph. The traditional cycle detection methods are generally based on block matrix sequences [
15]. When using this algebraic description to detect cycles in a given check matrix, there is redundancy due to the possibility that different block matrix sequences may form the same girth. To avoid redundancy caused by repeated traversal, we propose the following method for detection.
Theorem 3. Based on the cocyclic block matrices in Equations (38) and (43), the constructed quantum codes have no cycles in the Tanner graph. Proof. According to the proposed construction method, matrix derived from matrices and obviously cannot form a cycle. If it considers a quasi-Jacket block matrix sequence with a cycle of girth , as the loop is cyclic. Here, elements represent the block matrix of row and column for . The formation of a cycle means that any block matrix sequence is different from the original sequence.
Assume new coordinates to describe any block matrix in a block matrix sequence. For a cycle with a length of , the block matrices in the sequence can be located at most in the i-th row. Without losing generality, the sequence of block matrices that may generate a girth is classified according to the number of possible rows. Assume that i block matrices are located in the row, and u rows are marked in order from top to bottom . The new row coordinate ranges for each block matrix, as adjacent block matrices cannot be in the same row. Therefore, we can obtain , where . When , the i-th block matrix is adjacent not only to the i-th block matrix but also to the first block matrix, so can be obtained. For column coordinates, making , “1” does not mean that the block matrix is in the first column, but rather that the column in which the block matrix is located is the first occurrence in the sequence. The subsequent block matrix is marked with new column coordinates in the order. We implement this by setting the parameter , where represents the number of columns distributed in the first v block matrices. Then, for the v-th block matrix, will indicate that the column can be in a different column from the first block matrix. For the th block matrix located in row , the first block matrix will be , i.e., the new coordinate is (1,1) and . For the v-th block matrix, we may obtain that ; hence, the new coordinates of the current block matrix sequence have possibilities. After traversing each possibility and determining all the branches of the i-th block matrix, we can obtain that the cycle of the constructed matrix is bigger than 4 when the parameter in construction method meets .
According to the above description, after determining all the branches of the i-th block matrix, a tree diagram of the th layer will be obtained, thereby avoiding redundancy caused by repeated traversal. □
In the classical coding field, we firstly compare the performance of the classical codes obtained from the cocyclic block matrices with the classical quasi-cyclic (QC) codes in
Figure 1. In light of the character of the quasi-Jacket block matrix, the gained generator matrix of classical code has sparse property; hence, the parity check matrix of low weight can be obtained. In this figure, we take the different cocyclic matrices presented in this paper to analyze their bit error ratio (BER) and consider the coding performance of a single one among the generated pair classical codes. Here, given the different prime numbers
p are taken
with the construction method of the quasi-Jacket block matrix in this paper, their corresponding quantum code lengths are
for
, respectively. Assuming the binary phase shift keying (BPSK) modulation is over an additive white Gaussion noise (AWGN) channel, the bit error performance of the selected quantum codes is shown in this figure. As a result, the gained codes have the good properties of a sparse matrix. In decoding the performance of LDPC, short cycles will degrade the result under an iterative algorithm. The fewer short cycles of code makes its performance better; hence, the prime numbers
more than 4 are taken to avoid
cycles in the Tanner graph.
It can be seen from the result that the constructed quantum LDPC codes have no apparent performance in the low signal-to-noise ratio (SNR) region, but the cocyclic codes are better than quasi-cyclic codes at high SNR. Furthermore, we also can see that the difference becomes more pronounced when the code length is larger. Therefore, the designed cocyclic codes may be better applied to construct large-length quantum codes.
Generally, three types of independent Pauli errors occur in a realistic depolarizing communication channel, i.e., bit-flip errors X, phase–flip errors Z and Y, and errors that are the combination of X and Z. Assuming that the total probability denotes as f, the probabilities of the three classes of errors are equally owing to their independence. Namely, the distributed marginal flip probability can be depicted as . As a result, the errors of types X and Z in the obtained generator matrices and are separately corrected by reason of their isomorphism.
Furthermore, in the construction of quantum LDPC codes, the binary generator matrix of a quantum code is constructed by matrices and which can be seen as the parity-check matrices of two two classical codes and , respectively. The spaces of degenerate X-codewords and Z-codewords are derived from the rows of matrices and . The minimum distance of the quantum code is defined as the smaller of two-types distances and . According to the structure of the generator matrices, at least the minimum distances are bounded above by as the code length .
In the following, we compare the proposed quantum cocyclic codes to the quantum Reed-Solomon coding method with the previous work in ref. [
38] under the 0.5 coding rate of constructed codes in
Figure 2. The designed quantum codes with lengths
, respectively, and similar length of CSS-type quantum Reed-Solomon (RS) codes are taken for comparison. Here, we take the generator in terms of the similar method from Equation (
31) with size
for parameter
respectively. Furthermore, similar lengths of quantum Reed–Solomon codes are constructed according to the circulant permutation of two classical codes in its construction method. Here, we take the prime number
, so that the codes of size
, respectively, are gained for comparison.
In the SNR range, it is shown that the taken two shorter lengths of quantum codes have better performance from 0.2 to 0.4 dB than the longer one. However, owing to the close cycles and their bit-flip error correction capability, the higher length of code shows its superiority with increasing length.