New Constructions of Quantum Stabilizer Codes Based on Difference Sets

: In this paper, new conditions on parameters in difference sets are derived to satisfy symplectic inner product, and new constructions of quantum stabilizer codes are proposed from the conditions. The conversion of the difference sets into parity-check matrices is ﬁrst explained. Then, the proposed code construction is composed of three steps, which are to choose the generators of quantum stabilizer code, to determine the quantum stabilizer groups, and to determine subspace codewords with large minimum distance. The quantum stabilizer codes with various length are also presented to explain the practicality of the code construction. The proposed design can be applied to quantum stabilizer code construction based on combinatorial design.


Introduction
Quantum theory gives the probability of the possible outcomes for a measurement on a physical system [1]. Quantum computers which are based on quantum theory give us the possibility deal on the various tasks such as factoring the large integer number that shows the substantial speed-up in polynomial time over the best classical algorithm [2,3]. However, the effects of noisy and imperfect environments of the quantum channel would reduce the performance advantages. Therefore, quantum error correcting codes (QECCs) have been proposed to protect quantum information from noisy environments. Since the first QECCs were proposed in the 1990s by Shor [4] and Steane [5], the general theory of QECCs has been introduced [6].
After establishment in 1997 [7], quantum stabilizer codes have played a prominent role in QECCs. A stabilizer code appends ancilla qubits to qubits to be protected. The most consequential advantage of stabilizer code is that the errors can be detected and removed from the stabilizer operators [4]. Hence, stabilizer codes had been constructed to enhance the application of stabilizer formalism in quantum mechanics. In addition, the stabilizer theory allows the transfer of classical binary and quaternary codes to corresponding quantum stabilizer codes. Consequently, the various constructions of quantum stabilizer codes based on classical codes have been proposed [3,5,8,9]. The key idea of development of quantum stabilizer code is that the quantum stabilizer code can be established as a parity-check matrix whose binary or quaternary elements satisfy the constraint of symplectic inner product (SIP). Therefore, in [8,9], the authors considered the circulant matrix to construct the parity-check matrix. The modified circulant matrix has been proposed to construct the parity-check matrix and the results for entanglement-assisted quantum error correction codes are explained [8]. In [9], two sub-matrices are proposed to satisfy the constraint of parity-check matrix for quantum stabilizer codes with length seven.
Low-density parity-check (LDPC) codes were first introduced by Gallager [10]. Then, an excellent performance close to Shannon channel capacity was obtained according to large block size of binary |0 = 1 0 , |1 = 0 1 . The general quantum state of a qubit can be represented by a linear superposition of its two orthogonal basis states as |ψ = α|0 + β|1 = α β . The state can be found at both of basis states |0 and |1 at the same time, where the probability of outcome |0 is |α| 2 and the probability of outcome |1 is |β| 2 . According to the norm condition for qubits, the condition |α| 2 + |β| 2 = 1 must be satisfied. In general, n qubits are represented by 2 n dimensional Hilbert space H ⊗n as where i = ∑ n k=1 2 n−k i k . In classical computation, Boolean functions f : {0, 1} → {0, 1} are performed over a single bit. In the case of quantum computation, reversible operation represented by unitary matrices are performed over a qubit. Representative quantum operations are Pauli operators. Four Pauli operators (matrices) I, σ X , σ Y , and σ Z are The transformations of quantum states by Pauli operators is as Therefore, operators σ X , σ Z , and σ Y are regarded as a bit flip, a phase flip, and a combination of bit and phase flips, respectively. Multiplications between Pauli operators are defined as The Pauli group P 1 on a qubit is a group composed of Pauli operators and their multiplications with the factor ±1, ±j. Then, P 1 = ±{I, σ X , jσ X , σ Y , jσ Y , σ Z , jσ Z }. The Pauli group on n qubits P n is defined as n tensor product of the Pauli operators. Then, the elements of P n are either commutative or anti-commutative. The commutative operator "•" for two operators A and B is defined as Quotient group P n /C where C = {±I, ±jI} is defined as the center of P n [24]. Therefore, the notation X ↔ σ X , Y ↔ −jσ Y , Z ↔ σ Z [25] are used in the rest of the paper.

Quantum Error Correction Code
QECCs are used in quantum computing to protect quantum information from errors due to decoherence and other quantum noises. QECCs are essential to achieve fault-tolerant quantum computation [6]. In classical error correcting code, it is easy to make the copy of information. In contrast, it is impossible to make the copy of quantum information due to the non-cloning theorem [3]. Therefore, quantum information can be extended to highly entangled quantum state with the help of ancillary qubits and Unitary transforms. Classical error correcting codes use a syndrome measurement to diagnose errors which corrupt an encoded state. QECC also employs the syndrome detection with the help of quantum stabilizers operators. A block diagram of the QECC process is shown in Figure 1. The quantum information can be protected from noisy quantum channel with the help of ancillary qubits, the quantum stabilizer operators, and syndrome measurement.
Therefore, operators σ X , σ Z , and σ Y are regarded as a bit flip, a phase flip, and a combination of bit and phase flips, respectively. Multiplications between Pauli operators are defined as and ; and .
The Pauli group P1 on a qubit is a group composed of Pauli operators and their multiplications with the factor ±1, ±j. Then, 1 . The Pauli group on n qubits Pn is defined as n tensor product of the Pauli operators. Then, the elements of Pn are either commutative or anti-commutative. The commutative operator "•" for two operators A and B is defined as Quotient group Pn/C where C = {±I, ±jI} is defined as the center of Pn [24]. Therefore, the notation [25] are used in the rest of the paper.

Quantum Error Correction Code
QECCs are used in quantum computing to protect quantum information from errors due to decoherence and other quantum noises. QECCs are essential to achieve fault-tolerant quantum computation [6]. In classical error correcting code, it is easy to make the copy of information. In contrast, it is impossible to make the copy of quantum information due to the non-cloning theorem [3]. Therefore, quantum information can be extended to highly entangled quantum state with the help of ancillary qubits and Unitary transforms. Classical error correcting codes use a syndrome measurement to diagnose errors which corrupt an encoded state. QECC also employs the syndrome detection with the help of quantum stabilizers operators. A block diagram of the QECC process is shown in Figure 1. The quantum information can be protected from noisy quantum channel with the help of ancillary qubits, the quantum stabilizer operators, and syndrome measurement. A quantum state ψ is stabilized by operator . The quantum states, which are stabilized by all elements of any subgroup S of the Pauli group Pn form a subspace CS of H ⊗n . The subspace CS is defined as, Figure 1. Quantum error correction operating process.
A quantum state |ψ is stabilized by operator g ∈ P n if g|ψ = |ψ . The quantum states, which are stabilized by all elements of any subgroup S of the Pauli group P n form a subspace C S of H ⊗n . The subspace C S is defined as, If C S is non-trivial subspace, S is an abelian subgroup which is closed under multiplication. The subgroup generated by elements g 1 , g 2 , . . . , g m is denoted as S = g 1 , g 2 , . . . , g m . Then, any two operators on S are commutative. Since g 1 , g 2 , . . . , g m are m (=n − k) independent Pauli operators, S forms subspace Cs to be [[n, k, d min ]] quantum stabilizer code [7] which encodes k logical qubits into n physical qubits and can correct t = (d min − 1)/2 errors [6]. For example, the quantum stabilizer code [ [5,1,3]] can correct one error and four generators in Table 1 produce the full quantum stabilizer set S. Table 1. Generators of [ [5,1,3]] quantum stabilizer code.

Generators
Operators Let {E} ⊂ P n be the error set which makes the state |Ψ to the corrupted state E|Ψ . Since elements of Pauli operators are either commutative or anti-commutative, a vector on error set is either commutative or anti-commutative with elements of stabilizer group S. Therefore, the corrupted state E|Ψ is identified by the elements of stabilizer group S and the error detection is defined as The operator E i is correctable by stabilizer group S if the following condition is satisfied.
where E i † is the conjugate transpose of E i and N(S) is the normalizer of S in P n . Then, normalizer of S is defined as N(S) is the collection of all operators in Pauli group which is commutative with elements in S. Therefore, the minimum distance d min is determined as where W(A) is defined as the number of positions not equal to Pauli operators I in A and min(x) is the minimum number in set x.

Binary Formalism of Quantum Stabilizer Codes
In classical error correcting codes, the parity-check matrices give the constraint that the codewords must have vanishing scalar product with every vector of the parity-check matrices. In quantum error correcting codes, binary expression of quantum stabilizer operators also remains the parity-check constraint to quantum codeword.
Any Pauli operators can be expressed as the product of X-containing and Z-containing operators such as XYYZI = XXXII × IZZZI. Therefore, a simple but useful mapping exists between elements of Pauli operators and binary vector as I → (0, 0), X → (1, 0), Z → (0, 1), Y → (1, 1). Consequently, the n − k generators of an [[n, k]] quantum stabilizer code can be formed by a parity-check matrix H which is a concatenation of H X , H Z as follows, where H X , H Z are the binary matrices of size (n − k) × n. For example, the quantum stabilizer code [ [5,1,3]] in Table 1 has corresponding parity-check matrices as Since there exists the requirement that quantum stabilizer operators must be commutative, the constraint known as the symplectic inner product (SIP) is applied to H. We assume that m-th row of parity-check matric H, r m is expressed as r m = [x m |z m ], where z m and x m are binary strings for Z and X, respectively. Hence, the symplectic product of the m 1 -th row and m 2 -th row is given as x ki × z li . This product will give us zero if the number of different positions in X and Z are even. Hence, for a given parity-check matrix H = [H X |H Z ] with size (n − k) × 2n, the SIP formulation is defined as where 0 a is the a × a zero matrix. The constraint in (3) The parity-check matrix in (1) has the rank (n − k). Hence, the dual space of H has the dimension 2n − m (=m + 2k). Then, the normalizer group N(S) can be generated by an (m + 2k) × 2n binary matrix. The first m rows are the parity-check matrix and the last 2k row are the logical operators denoted as X, Z. Logical operators satisfy the conditions as Using Gaussian elimination, we can transform the parity check matrix into standard form as Therefore, logical operators are in standard form as Finally, the codewords of the quantum stabilizer code are given as where c i ∈ {0, 1}. For the quantum stabilizer code [ [5,1,3]] in Table 1, the standard form of parity-check matrix is investigated as,

Circulant Matrices Based on DS and QECC Construction
In this section, the definition, properties of DS, and circulant permutation matrices will be first introduced. Then, the QECC construction from circulant matrices based on parameters of DS are discussed with two examples.

Circulant Permutation Matrices
Let I n be the identity matrix of size n × n. Then, I n (x) is the shift of I n where the rows of I n are circularly shifted to the right by x positions (0 ≤ x ≤ n − 1). Generally, we notice that I n (0) = I n and I n (x ± kn) = I n (x) for any integer k. Let I n (1) c be the c times of multiplying I n (1), we have I n (1) c = I n (c) (0 ≤ c ≤ n − 1).

Example 2.
With n = 4, we have: A n × n circulant permutation binary matrix P n is defined as where i k is the binary value. P n can be given as the linear combination of identity matrix and its shifted matrices.
It is assumed that i 0 + i 1 + . . . + i n−1 = k. Let t 0 <t 1 < . . . < t k−1 be the position index of nonzero elements in the sequence set {i 0 , i 1 , . . . , i n−1 }. For example, if the sequence set {i 0 , i 1 , . . . , i n−1 } is {1, 1, 0, 0, 1, 0, 1}, then t 0 = 0, t 1 = 1, t 2 = 4, and t 3 = 6. The matrix P n can also be expressed by using the Hall-polynomial form p n (x) [18] as Let T be the transpose operator. Then, the transpose matrix of P n is denoted as P n T . Let p n (x) T be the Hall-polynomial form of P n T . Then, the polynomial p n (x) T is expressed as where t 0 , t 1 , . . . , t k−1 are the values in (8). For a (n, k, λ) DS D = {d 1 , d 2 , . . . , d k }, the circulant permutation matrix P n in (7) is made where the element i j is 1 if j ∈ D and is 0 otherwise. Then, the Hall-polynomial form p n (x) D for the DS D is expressed as

Construction of Quantum Stabilizer Code Based on DS
With difference sets (n, k, λ) D, the product of the two circulant permutation matrices can be expressed as a function of parameter of DS and the shift values in the following theorem. Theorem 1. Let h 1 (x) and h 2 (x) be the Hall-polynomials of D(s 1 ) and D(s 2 ), which are defined as h 1 (x) = p n D(s 1 ) and h 2 (x) = p n D(s 2 ) , respectively. Let the circulant permutation matrices H 1 and H 2 correspond to h 1 (x) and h 2 (x), respectively. Then, the product of the two polynomials h 1 (x), h 2 (x) T and the product of the two matrices H 1 and H 2 T are given as where the size of matrix J n is n × n and whose entries are all one.
Proof. From the definition of the Hall-polynomial, h 1 (x) and h 2 (x) can be expressed as Then, the Hall-polynomial h 2 (x) T for (9) is given as Therefore, the product of the two polynomials h 1 (x) and h 2 (x) T is given as (11) can be expressed as Hence, Equation (11) is expressed as Since the circulant permutation matrices corresponding to the polynomials x s 1 −s 2 and n−1 ∑ l=0 x l are I n (s 1 − s 2 ) and J n , respectively, the product of H 1 and H 2 T is expressed as Therefore, the expressions in (12) and (13) prove Theorem 1.
Since the product of H 1 and H 2 T in Theorem 1 is expressed as the function of k, λ, s 1 , and s 2 , the constraint on parameter of DSs to satisfy the SIP condition of parity-check matrix is explained in the following theorem. Proof. From Theorem 1, we have: The summation of (14) and (15) is If k − λ is even, all elements of the matrix (k − λ) × [I n (s 1 − s 2 ) + I n (s 2 − s 1 )] in (16) are even. Moreover, all elements of the matrix 2λ × J n in (16) are also even. Then, all elements of the matrix (k − λ) × [I n (s 1 − s 2 ) + I n (s 2 − s 1 )] + 2λ × J n in (16) are even. Therefore, if k ≡ λ modulo 2, the equation and H 2 which is made from the parameter of DS with the constraint k ≡ λ modulo 2 satisfies the SIP condition.
In Table 2, eight DSs with the constraint k ≡ λ modulo 2 are listed among the DSs in [18,19]. For the practical applications of proposed construction, two DSs with parameters (7,4,2) and (15,7,3) are considered in Examples 3 and 4.
Among the seven operators, there are a maximum of three linearly independent operators. If g 1 , g 2 and g 3 are chosen as the maximum of three linearly independent operators, the other operators are expressed as g 4 = g 1 × g 3 ; g 5 = g 1 × g 2 × g 3 ; g 6 = g 1 × g 2 ; g 7 = g 2 × g 3 . With S = g 1 , g 2 , g 3  Then, as Equation (5) The codewords of the quantum stabilizer code [ [7,4]] are expressed as The minimum distance d min of the [ [7,4]] code is determined by the smallest weight of N(S)\S. One of the smallest weights is X 1 × I I I I I I I. Since W( X 1 × I I I I I I I) = 2, the minimum distance d min is 2. Therefore, the quantum stabilizer code from the DS with parameter (7,4,2) is [[7,4,2]]. As shown in Table 3, the parameter constraints for difference sets in proposed construction are different from the ones in [23]. Since 2p − 1 ≡ p − 1 modulo 2 where p is an even number, DSs which are used in [23] can be also used in the proposed construction. In contrast, DSs in the proposed construction are not always used in [23] because 4p − 1 must be a prime number. As a result, the proposed construction is more general than [23]'s construction and the proposed construction enlarges the results of using DSs for quantum stabilizer code construction. In addition, in comparison to the proposed codes with existing quantum codes, quantum codes with length 7 and 15 are discussed. It is known that existing quantum stabilizer codes with length 7 have code parameters [ [7,3,2]] from quadratic residue sets in [26], or [[7,3,2]] and [ [7,4,2]] constructed over the quaternary alphabet, listed in [27]. To compare to the proposed codes and codes in [26], the number of information bits of the proposed codes is 1 bit larger than the referenced code. As referenced in the list in [27], a stabilizer with length 15 and the same parameters of [ [15,10,2]] that were constructed over quaternary alphabet are found. Table 3. Comparison of our proposed method and [23]'s method.

Conclusions
In this paper, the conditions of a DS are examined to satisfy the SIP condition and a new construction method of quantum stabilizer codes from the DS is proposed. The condition of a DS to satisfy the SIP constraint is equivalent to determine a DS with k ≡ λ modulo 2. Quantum stabilizer codes [ [7,4,2]] and [ [15,10,2]] are presented from the proposed construction with DS (7, 4, 2) and DS (15,7,3), respectively, for practical applications. Moreover, since there are many DSs with parameters that satisfy k ≡ λ modulo 2, it is possible to produce new quantum stabilizer codes with greater length. In comparison with the referenced construction, the proposed construction provides more candidates for the quantum stabilizer code based on DSs.