Fast Constructions of Quantum Codes Based on Residues Pauli Block Matrices

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 𝐺𝐹(𝑞), Euler’s criterion ℒ_𝑞(𝑥) for the Legendre symbol is defined by

ℒ_𝑞(𝑥)=𝑥^(𝑞−1)/2mod𝑞, (1.1) where 𝑞 is a power of an odd prime number. Namely, ℒ_𝑞(0)=0, ℒ_𝑞(𝑥)=1 if 𝑥=𝑦² for some element 𝑦∈𝐺𝐹(𝑞), and ℒ_𝑞(𝑥)=−1 if 𝑥≠𝑦² for any element in 𝐺𝐹(𝑞). Based on quadratic residues in 𝐺𝐹(𝑞), one defines a matrix

𝑄_𝑞=(𝑎_𝑖𝑗)_𝑞×𝑞, (1.2) with the elements 𝑎_𝑖𝑗=ℒ_𝑞(𝑖−𝑗).

Lemma 1.1. Taking any two rows of 𝑄_𝑞, that is, 𝑎_𝑖=(ℒ_𝑞(𝑥₀),ℒ_𝑞(𝑥₁),…,ℒ_𝑞(𝑥_𝑞−1))and𝑎_𝑖+𝑠=(ℒ_𝑞(𝑥₀+𝑠),ℒ_𝑞(𝑥₁+𝑠),…,ℒ_𝑞(𝑥_𝑞−1+𝑠)) for 𝑠≠0, it follows 𝑎_𝑖⋅𝑎_𝑖+𝑠=^𝑞−1_𝑖=0ℒ_𝑞(𝑥_𝑖)ℒ_𝑞(𝑥_𝑖+𝑠)=𝑞−1mod𝑞. (1.3)

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 𝐺𝐹(𝑞) by employing the matrix 𝑄_𝑛 in (1.2). Namely, we first replace all entries of 𝑄_𝑛 with Pauli matrices and obtain the Pauli block matrix 𝒬_𝑛. 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 𝒬_𝑛 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 𝒫={𝜎_𝑗∶0≤𝑗≤3}, where

𝜎₀=1001,𝜎₁=,𝜎0110₂=0−𝑖𝑖0,𝜎₃=,100−1 (2.1) where 𝑖=√−1. For simplicity, we denote the 2×2 identity matrix 𝜎₀ by a block matrix ℐ throughout this paper.

Pauli matrices in 𝒫 have the following basic properties:

𝜎²₁=𝜎²₂=𝜎²₃𝜎=ℐ,₁𝜎₂=𝑖𝜎₃,𝜎₂𝜎₃=𝑖𝜎₁,𝜎₃𝜎₁=𝑖𝜎₂. (2.2)

2.2. Pauli Block Matrice

Definition 2.1. Denote 𝒬=([𝑎]_𝑖𝑗)_𝑘×𝑡, then 𝒬 is a Pauli block matrix if and only if all entries [𝑎]_𝑖𝑗 belong to 𝒫, that is, [𝑎]_𝑖𝑗∈𝒫.
Based on the matrix 𝑄_𝑞 in (1.2), we propose several approaches for constructions of Pauli block matrices for any two entries 𝜎_𝑖,𝜎_𝑗∈𝒫⧵{𝜎₀}.

Construction. Taking a matrix 𝑄_𝑞 in (1.2), it follows two kinds of Pauli block matrices: (1)𝒬_𝑞⁽¹⁾, which is constructed by replacing “0,1” in 𝑄_𝑞 with 𝜎_𝑖 and “−1” with 𝜎_𝑗,(2)𝒬_𝑞⁽²⁾, which is constructed by replacing “1” in 𝑄_𝑞 with 𝜎_𝑖 and “−1,0” with 𝜎_𝑗.
Specially, one achieves two types of Pauli block matrices.

Construction. If 𝑞=4𝑚+3 for any positive integer 𝑚, then the (𝑞+1)×(𝑞+1) matrix can be constructed as 𝑄_𝑞+1=𝐼+0𝑡^𝑇−𝑡𝑄_𝑞, (2.3) where 𝑡 denotes the all-1 column vector of the length 𝑞 and 𝐼 is the (𝑞+1)×(𝑞+1) identity matrix. As a result, there are two types of Pauli block matrices: (1)𝒬⁽¹⁾_𝑞+1, which is constructed by replacing “0,1” in 𝑄_𝑞+1 with 𝜎_𝑖 and “−1” with 𝜎_𝑗,(2)𝒬⁽²⁾_𝑞+1, which is constructed by replacing “1” in 𝑄_𝑞+1 with 𝜎_𝑖 and “−1,0” with 𝜎_𝑗.

Construction. If 𝑞=4𝑚+1 for any positive integer 𝑚, then one constructs the 2(𝑞+1)×2(𝑞+1) matrix 𝑄_2(𝑞+1) by replacing “0” in the matrix 0𝑡^𝑇𝑡𝑄_𝑞 (2.4) with the block matrix [𝑅₀] and “±1” with the matrix [𝑅_±1], where [𝑅₀]=1−1−1−1,[𝑅_±1]=±111−1. (2.5) Then, there are two Pauli block matrices: (1)𝒬⁽¹⁾_2(𝑞+1), which is constructed by replacing “0,1” in 𝑄_2(𝑞+1) with 𝜎_𝑖 and “−1” with 𝜎_𝑗,(2)𝒬⁽²⁾_2(𝑞+1), which is constructed by replacing “1” in 𝑄_2(𝑞+1) with 𝜎_𝑖 and “−1,0” with 𝜎_𝑗.

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 𝐴=(𝑎_𝑖𝑗)_𝑟×𝑙 and 𝐵=(𝑏_𝑖𝑗)_𝑠×𝑡, that is,

𝑎𝐴⊗𝐵=¹¹𝐵𝑎₁₂𝐵⋯𝑎_1𝑙𝐵𝑎₂₁𝐵𝑎₂₂𝐵⋯𝑎_2𝑙𝐵𝑎⋮⋮⋮⋮_𝑟1𝐵𝑎_𝑟2𝐵⋯𝑎_𝑟𝑙𝐵. (2.6) 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 𝒬_𝑞 and 𝒬_𝑝 are two Pauli block matrices. For any nonnegative integer numbers 𝑠 and 𝑚, a large-order block Jacket matrix 𝒬_{𝑞𝑠𝑝𝑚} may be constructed (or decomposed) in the following way: 𝒬_{𝑞𝑠𝑝𝑚}={ℐ_𝑞𝑠⊗(^𝑚−1_𝑖=0ℐ_{𝑝𝑚−𝑖−1}⊗𝒬_𝑝⊗ℐ_𝑝𝑖)}×{(^𝑠−1_𝑖=0ℐ_{𝑞𝑠−𝑖−1}⊗𝒬_𝑞⊗ℐ_𝑞𝑖)⊗ℐ_𝑝𝑚}={ℐ_𝑞𝑠⊗(^𝑚_𝑖=1ℐ_{𝑝𝑚−𝑖}⊗𝒬_𝑝⊗ℐ_𝑝𝑖−1)}×{(^𝑠_𝑖=1ℐ_{𝑞𝑠−𝑖}⊗𝒬_𝑞⊗ℐ_𝑞𝑖−1)⊗ℐ_𝑝𝑚}. (2.7)

Proof. Based on an arbitrary Pauli block matrix 𝒬_𝑟, the large-order Pauli block matrices 𝒬_𝑟𝑙 for 𝑙≥2 can be obtained by using the recursive relations: 𝒬_𝑟𝑙=𝒬_𝑟𝑙−1⊗𝒬_𝑟=^𝑙−1_𝑖=0ℐ_{𝑟𝑙−𝑖−1}⊗𝒬_𝑟⊗ℐ_𝑟𝑖=^𝑚_𝑖=1ℐ_{𝑟𝑙−𝑖}⊗𝒬_𝑟⊗ℐ_𝑟𝑖−1, (2.8) where 𝑟∈{𝑝,𝑞} and 𝑙∈{𝑠,𝑚}. According to the properties of the Kronecker product, it is easy to calculate 𝒬_{𝑞𝑠𝑝𝑚}=𝒬_𝑞𝑠⊗𝒬_𝑝𝑚=(ℐ_𝑞𝑠⊗𝒬_𝑝𝑚)⋅(𝒬_𝑞𝑠⊗ℐ_𝑝𝑚), (2.9) and then this completes the proof of the theorem.

Employing Pauli Block matrices 𝒬_𝑛 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.

Table 1 

Computation complexity of the fast algorithm based on Pauli block matrices. For simplicity, Add and Mul denote the number of additions and multiplications. The notations 𝐿^𝜎_𝑝 and 𝐿^𝜎_𝑞 express the number of nonidentity matrices 𝜎₀ in Pauli block matrices 𝑄_𝑝 and 𝑄_𝑞.

As an example, the construction of 𝒬₁₅=𝒬₃⊗𝒬₅ is depicted in Figure 1. According to Table 1, the computation of the Pauli block matrix 𝒬₁₅ 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 𝒫^⊗𝑛 denote the set of the 𝑛-fold tensor products (the Kronecker product) of Pauli operators (matrices) in 𝒫 [13]. Then 𝒫^⊗𝑛, together with possible multiplicative factors in {±𝑖,±1}, form a group of 𝑛-qubit operations, denoted by 𝒢_𝑛. An arbitrary operation 𝛼_𝑢∈𝒢_𝑛 can be uniquely expressed by

𝛼_𝑢=𝑖^𝜆[𝜎^𝑥𝑢1₁𝜎^𝑧𝑢1₂]⊗⋯⊗[𝜎^𝑥𝑢𝑛₁𝜎^𝑧𝑢𝑛₂], (3.1) where 𝑥_𝑢𝑡,𝑧_𝑢𝑡∈{0,1} for 1≤𝑡≤𝑛. Omitting factor 𝑖^𝜆, we denote 𝛼_𝑢 by a concatenated 2𝑛-dimensional vector 𝛼_𝑢 [6]:

𝛼_𝑢=(𝑥_𝑢∣𝑧_𝑢)=(𝑥_𝑢1,…,𝑥_𝑢𝑛∣𝑧_𝑢1,…,𝑧_𝑢𝑛). (3.2)

The Hamming weight of 𝛼_𝑢 is the number of (𝑥_𝑢ℎ∣𝑧_𝑢ℎ) (1≤ℎ≤𝑛) such that (𝑥_𝑢ℎ∣𝑧_𝑢ℎ)≠(0∣0). The symplectic inner product of any two vectors 𝛼_𝑢=(𝑥_𝑢∣𝑧_𝑢) and 𝛼_𝑣=(𝑥_𝑣∣𝑧_𝑣) is defined by

𝛼_𝑢⋅𝛼_𝑣=𝑥_𝑢⋅𝑧_𝑣+𝑧_𝑢⋅𝑥_𝑣, (3.3) where 𝑥_𝑢⋅𝑧_𝑣=∑^𝑛_𝑖=1𝑥_𝑢𝑖𝑧_𝑣𝑖 and 𝑧_𝑢⋅𝑥_𝑣=∑^𝑛_𝑖=1𝑧_𝑢𝑖𝑥_𝑣𝑖. According to [6], two operations 𝛼_𝑢 and 𝛼_𝑣 commute if and only if

𝛼_𝑢⋅𝛼_𝑣=0. (3.4)

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

Assume that each row of a Pauli block matrix 𝒬_𝑛 is denoted by (𝜎_𝑖1,…,𝜎_𝑖𝑛) for 1≤𝑖≤𝑛, from which an 𝑛-qubit operation, called as the row operator, can be obtained as

𝛼_𝑖=𝜎_𝑖1⊗⋯⊗𝜎_𝑖𝑛. (3.5)

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

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

Proof. Employing Pauli block matrices 𝒬_𝑛 that are constructed via substituting Pauli matrices for the entries of the Hadamard matrices, all row operators of 𝒬_𝑛 can be expressed by the concatenated vectors in (3.2), from which the 𝑛×2𝑛 matrices (𝐻_𝑥∣𝐻_𝑧) can hence be constructed. According to the properties of the Hadamard matrices, it is easy to calculate 𝐻_𝑥⋅𝐻^𝑇_𝑧+𝐻_𝑧⋅𝐻^𝑇_𝑥=0mod2, (3.6) which implies that all independent 𝑛-qubit row operators of 𝒬_𝑛 are commuting [5].

Corollary 3.2. Considering any two given Pauli block matrices 𝒬_𝑝 and 𝒬_𝑞, all independent 𝑝𝑞-qubit row operators of the Kronecker product 𝒬_𝑝𝑞=𝒬_𝑝⊗𝒬_𝑞 are commuting.

Example 3.3. We consider 𝐺𝐹(3)={0,1,2} with the nonzero squares 1²=1mod3 and 2²=1mod3, and hence ℒ₃(0)=0,ℒ₃(1)=1, and ℒ₃(2)=−1. With the rows and columns of a matrix 𝑄₃ being indexed by {0,1,2}, one obtains 𝑄₃=01−1−1011−10. (3.7) According to Construction 1, one gains the Pauli block matrix: 𝒬₃⁽¹⁾=𝜎_𝑖𝜎_𝑖𝜎_𝑗𝜎_𝑗𝜎_𝑖𝜎_𝑖𝜎_𝑖𝜎_𝑗𝜎_𝑖. (3.8) Taking 𝑖=1 and 𝑗=3, one has the concatenated matrix: 𝐻₃=(𝐻^𝑥₃∣𝐻^𝑧₃)=110001011100101100. (3.9) It is easy to check that 𝐻^𝑥₃⋅𝐻^𝑧₃^𝑇+𝐻^𝑧₃⋅𝐻³_𝑥^𝑇=0mod2, which means that three row operators of 𝒬₃⁽¹⁾𝛼₁=𝜎₁⊗𝜎₁⊗𝜎₃,𝛼₂=𝜎₃⊗𝜎₁⊗𝜎₁,𝛼₃=𝜎₁⊗𝜎₃⊗𝜎₁ (3.10) are commuting.
For another, based on Construction 2, one obtains
𝑄₄=1111−111−1−1−111−11−11, (3.11) from which the Pauli block matrix 𝒬₄⁽¹⁾ can be achieved: 𝒬₄⁽¹⁾=𝜎_𝑖𝜎_𝑖𝜎_𝑖𝜎_𝑖𝜎_𝑗𝜎_𝑖𝜎_𝑖𝜎_𝑗𝜎_𝑗𝜎_𝑗𝜎_𝑖𝜎_𝑖𝜎_𝑗𝜎_𝑖𝜎_𝑗𝜎_𝑖. (3.12) Taking 𝑖=1 and 𝑗=3, one has the concatenated matrix: 𝐻₄=11110000011010010011110001011010, (3.13) from which it is easy to check that all row operators of 𝒬₄⁽¹⁾ are commuting.

Example 3.4. According to 𝐺𝐹(5) with ℒ₅(1)=ℒ₅(4)=1mod5 and ℒ₅(2)=ℒ₅(3)=−1mod5, if the rows and columns of 𝑄₅ are indexed by 𝐺𝐹(5), one gets 𝑄₅=01−1−11101−1−1−1101−1−1−11011−1−110. (3.14) Employing Construction 1, we get the Pauli block matrix: 𝒬₅⁽¹⁾=𝜎_𝑖𝜎_𝑖𝜎_𝑗𝜎_𝑗𝜎_𝑖𝜎_𝑖𝜎_𝑖𝜎_𝑖𝜎_𝑗𝜎_𝑗𝜎_𝑗𝜎_𝑖𝜎_𝑖𝜎_𝑖𝜎_𝑗𝜎_𝑗𝜎_𝑗𝜎_𝑖𝜎_𝑖𝜎_𝑖𝜎_𝑖𝜎_𝑗𝜎_𝑗𝜎_𝑖𝜎_𝑖. (3.15) Taking 𝑖=1 and 𝑗=3, one has 𝐻₅=11001001101110000011011101000100111110001001101100, (3.16) which means that five row operators of 𝒬₅⁽¹⁾ for 𝑖=1 and 𝑗=3 are commuting.
Furthermore, according to Construction 3 with respective 𝑄₅ in (3.14), one obtains a Pauli block matrix 𝒬⁽¹⁾₁₀ for 𝑖=1 and 𝑗=3 with the concatenated matrix 𝐻₁₀ expressed as
𝐻₁₀=10110000110100111100001001011011011010011110110000000100111110001001010111011010001110110011000100110110001001100111011000001110111111000100010110001010100111011100001110001111000110010110000110100111. (3.17) It is obvious that all row operators of 𝒬⁽¹⁾₁₀ are commuting.

4. The Stabilizer Quantum Codes

In this section, we construct quantum codes 𝐶(𝒮) by using Pauli block matrices 𝒬_𝑛 with commutative row operators, from which 𝑛−𝑘 independent row operators can be selected as generators of an Abelian group 𝒮.

Given an Abelian subgroup 𝒮 of 𝒢_𝑛, the stabilizer quantum code 𝐶(𝒮) is a set of 𝑛-qubit quantum states {|𝜓⟩} associated with 𝒮, that is,

𝐶(𝒮)={|𝜓⟩∶𝑀|𝜓⟩=|𝜓⟩,∀𝑀∈𝒮}, (4.1) which is the subspace fixed by 𝒮 (called as the stabilizer). For an [[𝑛,𝑘,𝑑]] stabilizer quantum code, which encodes 𝑘 logical qubits into 𝑛 physics qubits, 𝐶(𝒮) has dimension 2^𝑘 and 𝒮 has 2^𝑛−𝑘 independent operators.

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

Theorem 4.1. Given a Pauli block matrix 𝒬_𝑛 with commutative row operators, the stabilizer quantum code 𝐶(𝒮) can be constructed with parameters [[𝑛,𝑘,𝑑]] where the stabilizer 𝒮 is a set of 𝑛-qubit operators generated from 𝑛−𝑘 independent row operators of Pauli block matrix 𝒬_𝑛.

Proof. Suppose that there are 𝑟 (𝑟≥𝑛−𝑘) independent rows of 𝒬_𝑛. Then 𝑛−𝑘 generators of the stabilizer 𝒮 can be generated by selecting 𝑛−𝑘 rows from these 𝑟 independent rows of 𝒬_𝑛 provided 𝑛−𝑘≤𝑟. Namely, any 𝑛−𝑘 operators 𝛼_𝑖1,𝛼_𝑖2,…,𝛼_{𝑖𝑛−𝑘} can be selected to generate an Abelian group: 𝒮=⟨𝛼_𝑖1,𝛼_𝑖2,…,𝛼_{𝑖𝑛−𝑘}⟩, (4.2) which is a stabilizer in essence. By making use of (4.1), a stabilizer quantum code [[𝑛,𝑘,𝑑]] can be generated from 𝒮. According to the quantum Singleton bound proposed in [6], it follows that 𝑘≤𝑛−2𝑑+2. This completes the proof of the theorem.

Example 4.2. We consider the Pauli block matrix 𝒬₅ in (3.15) with the concatenated matrix 𝐻₅=(𝐻^𝑥₅∣𝐻^𝑥₅) in (3.16). It is known that all rows of 𝐻₅ are independent. Thus, any 5−𝑘 rows of 𝐻₅, denoted by ℋ₅=(ℎ^𝑥₅∣ℎ^𝑧₅), can be selected to generate the stabilizer 𝒮 with 5−𝑘 generators. According to the construction conditions of quantum codes in [5], we get the generator matrix of quantum codes 𝒢₅=(𝑔^𝑥₅∣𝑔^𝑧₅) satisfying ℎ^𝑥₅⋅𝑔^𝑧₅+ℎ^𝑧₅⋅𝑔^𝑥₅=0, (4.3) which can be rewritten as ℋ₅⋅𝑅⋅𝒢₅=0, (4.4) where 𝑅=(^{05×5𝐼5×5}_{𝐼5×505×5}). To construct such a quantum code, we assume that there exists one unitary matrix 𝑈 such that 𝑈(ℎ𝑅)=(𝐼_{(5−𝑘)×(5−𝑘)}∣Λ_{(5−𝑘)×(5+𝑘)}). (4.5) According to (4.4), the generator matrix 𝑔 is calculated: 𝒢₅=(Λ^𝑇_{(5−𝑘)×(5+𝑘)}∣𝐼_{(5+𝑘)×(5+𝑘)}), (4.6) from which quantum codes can be constructed with the parameters [[5,0,4]], [[5,1,2]], [[5,2,1]], [[5,3,1]], and [[5,4,1]].
Taking the quantum code [[5,1,2]] as an example, we select 4 rows to generate the matrix:
ℋ₅=1100100110111000001101110100010011111000. (4.7) From (4.4), we get 𝒢₅=011010000000010100001101001000101100010010000000100011000001. (4.8) Therefore, a quantum code [[5,1,2]] can be constructed from (4.8), where 𝑑=2, the Hamming weight of 𝒢₅, can be calculated from the Hamming weight of the bitwise or of 𝑔^𝑥₅ with 𝑔^𝑧₅.

Example 4.3. Suppose 𝑛=10, and then we consider Pauli block matrix 𝒬⁽¹⁾₁₀ for 𝑖=1 and 𝑗=3 with the concatenated matrix in (3.17). It is obvious that all rows of 𝐻₁₀ are independent and commutative. Selecting any 10−𝑘 (0≤𝑘≤10) row operators 𝛼_𝑖1,𝛼_𝑖2,…,𝛼_{𝑖10−𝑘} from 𝒬⁽¹⁾₁₀, we obtain the stabilizer 𝒮=⟨𝛼_𝑖1,𝛼_𝑖2,…,𝛼_{𝑖10−𝑘}⟩, from which quantum codes can be constructed with the parameters [[10,0,4]], [[10,1,4]], [[10,2,4]], [[10,3,3]], [[10,4,3]], [[10,5,3]], [[10,6,3]], [[10,7,3]], [[10,8,1]], and [[10,9,1]].

5. Conclusion

A family of quantum codes is investigated with fast Pauli block transforms by using quadratic residues in the finite field 𝐺𝐹(𝑞). 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 𝒮 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.

Figure 1 

Signal flow graph for Pauli block transform for 𝒬15.

Acknowledgment

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