#### Abstract

We demonstrate how to fast construct quantum error-correction codes based on quadratic residues Pauli block transforms. The present quantum codes have an advantage of being fast designed from Abelian groups on the basis of Pauli block matrices that can be yielded from quadratic residues with much efficiency.

#### 1. Introduction

The applications of the Legendre symbol have been already suggested in signal processing, communication, image compression, cryptography, and so forth [1, 2].

Provided a finite field , Euler’s criterion for the Legendre symbol is defined by

where is a power of an odd prime number. Namely, , if for some element , and if for any element in . Based on quadratic residues in , one defines a matrix

with the elements .

Lemma 1.1. *Taking any two rows of , that is, for , it follows
*

Currently, the striking development in quantum error-correction codes (QECCs) is the employment of the stabilizer formalism, whereby code words are subspaces in Hilbert space specified by Abelian groups. The problem of constructing QECCs was reduced to that of searching for the classical dual-containing (or self-orthogonal) codes [3–8]. The virtue of this approach is that QECCs can be directly constructed from classical codes with a certain property, rather than developing a completely new coding theory of QECCs from scratch. Unfortunately, the need for dual-containing codes presents a substantial obstacle to the quantum coding theory in its entirety, especially in the context of modern codes, such as low-density parity-check quantum codes [7]. To resolve this problem, we consider the construction of QECCs over the finite field by employing the matrix in (1.2). Namely, we first replace all entries of with Pauli matrices and obtain the Pauli block matrix . After that, we extend this kind of block matrices for the large-size Pauli block matrices by using the recursive techniques with the fast matrix block transforms [9–12]. Since all row operations that are obtained from rows of are independent and commutative, an Abelian group can be generated elegantly. Therefore, a type of quantum code is generated structurally via the stabilizer formalism. This approach provides the great flexibility in designing quantum codes with large code length and hence allows for an advantage of being simply constructed with the low complexity.

This paper is organized as follows. In Section 2, three kinds of Pauli block matrices are constructed. In Section 3, according to the properties of Pauli block matrices, Abelian groups can be generated with efficiency. In Section 4, we investigate constructions of quantum codes based on the stabilizer formalism. Finally, conclusions are drawn in Section 5.

#### 2. Pauli Block Matrices

Pauli matrices have been widely applied in signal processing [11], quantum information and quantum computing [3, 13], and so forth. In this section, we investigate constructions of several types of Pauli block matrices according to the structure of Hadamard transforms based on these Pauli matrices.

##### 2.1. Pauli Matrices

Pauli matrices are defined by , where

where . For simplicity, we denote the identity matrix by a block matrix throughout this paper.

Pauli matrices in have the following basic properties:

##### 2.2. Pauli Block Matrice

*Definition 2.1. *Denote , then is a Pauli block matrix if and only if all entries belong to , that is, .

Based on the matrix in (1.2), we propose several approaches for constructions of Pauli block matrices for any two entries .

*Construction. *Taking a matrix in (1.2), it follows two kinds of Pauli block matrices: (1)*, which is constructed by replacing “” in with and “” with ,*(2)*, which is constructed by replacing “” in with and “” with .*

Specially, one achieves two types of Pauli block matrices.

*Construction. *If for any positive integer , then the matrix can be constructed as
where denotes the all- column vector of the length and is the identity matrix. As a result, there are two types of Pauli block matrices: (1)*, which is constructed by replacing “” in with and “” with ,*(2)*, which is constructed by replacing “” in with and “” with .*

*Construction. *If for any positive integer , then one constructs the matrix by replacing “” in the matrix
with the block matrix and “” with the matrix , where
Then, there are two Pauli block matrices: (1)*, which is constructed by replacing “” in with and “” with ,*(2)*, which is constructed by replacing “” in with and “” with .*

##### 2.3. Fast Constructions of Pauli Block Matrices

To construct the large-order Pauli block matrices, we first introduce the Kronecker product of two matrices and , that is,

With a little abuse, we denote the Kronecker product by using the notation “” throughout this paper.

Making use of the Kronecker product of Pauli block matrices [9–12], a family of Pauli block matrices may be extended.

Theorem 2.2. *Suppose that and are two Pauli block matrices. For any nonnegative integer numbers and , a large-order block Jacket matrix may be constructed (or decomposed) in the following way:
*

*Proof. *Based on an arbitrary Pauli block matrix , the large-order Pauli block matrices for can be obtained by using the recursive relations:
where and . According to the properties of the Kronecker product, it is easy to calculate
and then this completes the proof of the theorem.

Employing Pauli Block matrices in Constructions 1, 2, and 3 with respect to the Kronecker product in (2.7), any large-order Pauli block matrices can be constructed with the fast algorithm. The computational complexity of the proposed algorithm is shown in Table 1.

As an example, the construction of is depicted in igure 1. According to Table 1, the computation of the Pauli block matrix requires 26 additions and 34 multiplications. Compared with the directed computation, the proposed algorithm is obviously faster.

#### 3. Abelian Group Based on Pauli Block Matrices

Let denote the set of the -fold tensor products (the Kronecker product) of Pauli operators (matrices) in [13]. Then , together with possible multiplicative factors in , form a group of -qubit operations, denoted by . An arbitrary operation can be uniquely expressed by

where for . Omitting factor , we denote by a concatenated -dimensional vector [6]:

The Hamming weight of is the number of () such that . The symplectic inner product of any two vectors and is defined by

where and . According to [6], two operations and commute if and only if

The symplectic inner product of two vectors is important since it can be used conveniently to search for generators of an Abelian subgroup .

Assume that each row of a Pauli block matrix is denoted by for , from which an -qubit operation, called as the row operator, can be obtained as

Based on properties of the Kronecker product [11, 14, 15], we achieve the commutativity of row operators for Pauli block matrices .

Theorem 3.1. *For Pauli block matrices proposed in Construction 1 (also for Constructions 2 and 3), all independent row operators of are commuting and hence generate an Abelian group.*

*Proof. *Employing Pauli block matrices that are constructed via substituting Pauli matrices for the entries of the Hadamard matrices, all row operators of can be expressed by the concatenated vectors in (3.2), from which the matrices can hence be constructed. According to the properties of the Hadamard matrices, it is easy to calculate
which implies that all independent -qubit row operators of are commuting [5].

Corollary 3.2. *Considering any two given Pauli block matrices and , all independent -qubit row operators of the Kronecker product are commuting.*

*Example 3.3. *We consider with the nonzero squares and , and hence , and . With the rows and columns of a matrix being indexed by , one obtains
According to Construction 1, one gains the Pauli block matrix:
Taking and , one has the concatenated matrix:
It is easy to check that , which means that three row operators of
are commuting.

For another, based on Construction 2, one obtains

from which the Pauli block matrix can be achieved:
Taking and , one has the concatenated matrix:
from which it is easy to check that all row operators of are commuting.

*Example 3.4. *According to with and , if the rows and columns of are indexed by , one gets
Employing Construction 1, we get the Pauli block matrix:
Taking and , one has
which means that five row operators of for and are commuting.

Furthermore, according to Construction 3 with respective in (3.14), one obtains a Pauli block matrix for and with the concatenated matrix expressed as

It is obvious that all row operators of are commuting.

#### 4. The Stabilizer Quantum Codes

In this section, we construct quantum codes by using Pauli block matrices with commutative row operators, from which independent row operators can be selected as generators of an Abelian group .

Given an Abelian subgroup of , the stabilizer quantum code is a set of -qubit quantum states associated with , that is,

which is the subspace fixed by (called as the stabilizer). For an stabilizer quantum code, which encodes logical qubits into physics qubits, has dimension and has independent operators.

To construct such a quantum code, the sticking point is to search for an Abelian group, the stabilizer , from which the code can be structurally generated through (4.1).

Theorem 4.1. *Given a Pauli block matrix with commutative row operators, the stabilizer quantum code can be constructed with parameters where the stabilizer is a set of -qubit operators generated from independent row operators of Pauli block matrix .*

*Proof. *Suppose that there are () independent rows of . Then generators of the stabilizer can be generated by selecting rows from these independent rows of provided . Namely, any operators can be selected to generate an Abelian group:
which is a stabilizer in essence. By making use of (4.1), a stabilizer quantum code can be generated from . According to the quantum Singleton bound proposed in [6], it follows that . This completes the proof of the theorem.

*Example 4.2. *We consider the Pauli block matrix in (3.15) with the concatenated matrix in (3.16). It is known that all rows of are independent. Thus, any rows of , denoted by , can be selected to generate the stabilizer with generators. According to the construction conditions of quantum codes in [5], we get the generator matrix of quantum codes satisfying
which can be rewritten as
where . To construct such a quantum code, we assume that there exists one unitary matrix such that
According to (4.4), the generator matrix is calculated:
from which quantum codes can be constructed with the parameters , , , , and .

Taking the quantum code as an example, we select rows to generate the matrix:

From (4.4), we get
Therefore, a quantum code can be constructed from (4.8), where , the Hamming weight of , can be calculated from the Hamming weight of the bitwise or of with .

*Example 4.3. *Suppose , and then we consider Pauli block matrix for and with the concatenated matrix in (3.17). It is obvious that all rows of are independent and commutative. Selecting any () row operators from , we obtain the stabilizer , from which quantum codes can be constructed with the parameters , , , , , , , , , and .

#### 5. Conclusion

A family of quantum codes is investigated with fast Pauli block transforms by using quadratic residues in the finite field . We first investigate the construction approaches based on three kinds of Pauli block matrices with commutative row operators. Then the large-order Pauli block matrices are structurally constructed via the fast Pauli block constructing transforms based on the recursive relationship of identity matrices and successively lower-order Pauli block matrices. These Pauli block matrices have such a characteristic that all row operators are independent and commutative, which can generate an Abelian operator group. Finally, an instructive approach for constructions of quantum codes is suggested via the stabilizer formalism according to the Abelian group yielded from Pauli block matrices. This code may provide the great flexibility in designing quantum codes with large block length through implementing the proposed fast construction algorithms.

#### Acknowledgment

This research was supported by the WCU R-32-2008-000-20014-0 NRF, South Korea.