Fast Constructions of Quantum Codes Based on Residues Pauli Block Matrices

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.


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 /2 mod q, 1.
where q is a power of an odd prime number.Namely, L q 0 0, L q x 1 if x y 2 for some element y ∈ GF q , and L q x −1 if x / y 2 for any element in GF q .Based on quadratic residues in GF q , one defines a matrix with the elements a ij L q i − j .

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Lemma 1.1.Taking any two rows of Q q , that is, − → a i L q x 0 , L q x 1 , . . ., L q x q−1 and − → a i s L q x 0 s , L q x 1 s , . . ., L q x q−1 s for s / 0, it follows − → a i • − → a i s q−1 i 0 L q x i L q x i s q − 1 mod q. 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.

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.

Pauli Matrices
Pauli matrices are defined by P {σ j : 0 ≤ j ≤ 3}, where where i √ −1.For simplicity, we denote the 2 × 2 identity matrix σ 0 by a block matrix I throughout this paper.
Pauli matrices in P have the following basic properties: 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 \ {σ 0 }.

Construction 2.1
Taking a matrix Q q in 1.2 , it follows two kinds of Pauli block matrices: , which is constructed by replacing "0, 1" in Q q with σ i and "−1" with σ j , 2 Q 2 q , 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 2.2
If q 4m 3 for any positive integer m, then the q 1 × q 1 matrix can be constructed as 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: 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 2.3
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 with the block matrix R 0 and "±1" with the matrix R ±1 , where Then, there are two Pauli block matrices: 1 which is constructed by replacing "1" in Q 2 q 1 with σ i and "−1, 0" with σ j .

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, 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 q s p m may be constructed (or decomposed) in the following way:

2.7
Proof.Based on an arbitrary Pauli block matrix Q r , the large-order Pauli block matrices Q r l for l ≥ 2 can be obtained by using the recursive relations: x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 where r ∈ {p, q} and l ∈ {s, m}.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 Q n in Constructions 2.1, 2.2, and 2.3 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 Q 15 Q 3 ⊗ Q 5 is depicted in Figure 1.According to Table 1, the computation of the Pauli block matrix Q 15 requires 26 additions and 34 multiplications.Compared with the directed computation, the proposed algorithm is obviously faster.

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 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 : The Hamming weight of − → α u is the number of According to 6 , two operations α u and α v 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 S ⊆ G n .
Assume that each row of a Pauli block matrix Q n is denoted by σ i 1 , . . ., σ i n for 1 ≤ i ≤ n, from which an n-qubit operation, called as the row operator, can be obtained as 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 2.1 (also for Constructions 2.2 and 2.3), 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 which implies that all independent n-qubit row operators of Q n are commuting 5 .

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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.

3.8
Taking i 1 and j 3, one has the concatenated matrix:

3.9
It is easy to check that x T 0 mod 2, which means that three row operators of Q For another, based on Construction 2.2, one obtains Advances in Mathematical Physics from which the Pauli block matrix Q 1 4 can be achieved:

3.12
Taking i 1 and j 3, one has the concatenated matrix:

3.14
Employing Construction 2.1, we get the Pauli block matrix:

3.15
Taking i 1 and j 3, one has 3.17 It is obvious that all row operators of Q 1 10 are commuting.

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, 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 .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 provided n − k ≤ r.Namely, any n − k operators α i 1 , α i 2 , . . ., α i n−k can be selected to generate an Abelian group: .To construct such a quantum code, we assume that there exists one unitary matrix U such that According to 4.4 , the generator matrix g is calculated: 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 .

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 largeorder Pauli block matrices are structurally constructed via the fast Pauli block constructing transforms based on the recursive relationship of identity matrices and successively lowerorder 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.

1 Figure 1 :
Figure 1: Signal flow graph for Pauli block transform for Q 15 .

1 4 are commuting. Example 3 . 4 .
is easy to check that all row operators of Q According to GF 5 with L 5 the rows and columns of Q 5 are indexed by GF 5 , one gets

means that five row operators of Q 1 5
for i 1 and j 3 are commuting.Furthermore, according to Construction 2.3 with respective Q 5 in 3.14 , one obtains a Pauli block matrix Q 1 10 for i 1 and j 3 with the concatenated matrix H 10 expressed as

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 .
quantum code 5, 1, 2 can be constructed from 4.8 , where d 2, the Hamming weight of G 5 , can be calculated from the Hamming weight of the bitwise or of g x 5 with g z 5 .

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 σ 0 in Pauli block matrices Q p and Q q .
. . ., α i n−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.We consider the Pauli block matrix Q 5 in 3.15 with the concatenated matrix H 5 H x 5 | H x 5 in 3.16 .It is known that all rows of H 5 are independent.Thus, any 5 − k rows of H 5 , denoted by H 5 h x 5 | h z 5 , 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 G 5