Construction of Binary Quantum Error-Correcting Codes from Orthogonal Array.

By using difference schemes, orthogonal partitions and a replacement method, some new methods to construct pure quantum error-correcting codes are provided from orthogonal arrays. As an application of these methods, we construct several infinite series of quantum error-correcting codes including some optimal ones. Compared with the existing binary quantum codes, more new codes can be constructed, which have a lower number of terms (i.e., the number of computational basis states) for each of their basis states.

1. Introduction

Errors are inevitable in quantum information processing [1], so quantum error-correcting codes (QECCs) are very important for quantum communication and quantum computing. In 1995, Shor [1] gave the simplest quantum simulation of a classical coding plan and then constructed the first QECC. In 1998, Calderbank et al. provided a close connection between QECCs and classical error correction codes [2], which leads to constructing QECCs from known classical error correction codes. In recent years, the research on QECCs especially on binary QECCs has made great progress. Feng and Ma made a way to obtain good pure stabilizer quantum codes, binary or nonbinary [3]. Li and Li obtained quantum codes of minimum distance three which are optimal or near optimal, and some quantum codes of minimum distance four which are better than previously known codes [4]. Feng and Xing presented a characterization of (binary and non-binary) quantum codes. Based on this characterization, they derived a method to construct pure p-ary quantum codes with dimensions not necessarily equal to powers of p [5]. Some other constructions of non-stabilizer codes, such as CWS codes [6], the codes in [7], and permutation-invariant codes such as in [8,9,10,11] have been studied. However, the majority of binary QECCs constructed so far are stabilizer codes [12,13,14]. The main goal of this work is to link between orthogonal arrays and binary QECCs and to construct more families of new codes.

Orthogonal arrays (OAs) play a more and more important role in quantum information theory [15,16,17,18,19,20,21,22]. An array A with entries from a set is said to be an orthogonal array with s levels, strength t (for some t in the range if every subarray of A contains each t-tuple based on S as a row with the same frequency. We will denote such an array by OA. Recently, many new methods of constructing OAs, especially high strength OAs, have been presented, and many new classes of OAs have been obtained [23,24,25,26,27,28,29,30,31,32,33]. An OA is said to be an irredundant orthogonal array (IrOA) if, in any subarray, all of its rows are different [18]. A link between an IrOA with d levels and a t-uniform state was established by Goyeneche et al. [18], i.e., every column and every row of the array correspond to a particular qudit and a linear term of the state, respectively.

([18]). If is an IrOA, then the superposition of r product states, is a t-uniform state.

More and more attention has been paid to the construction and characterization of t-uniform states from OAs [15,16,17,18,34,35,36,37,38,39]. Very interestingly, uniform states are closely related to QECCs. Goyeneche and Życzkowski stated QECCs are one-to-one connected to k-uniform states of N qudits [18]. Shi et al. also presented the relation between a pure QECC and t-uniform state [40]. It is these new developments in OAs and uniform states that raise the possibility of constructing QECCs from OAs.

In this paper, the Hamming distance and minimal distance (MD) of OAs are applied to the theory of quantum information. By using difference schemes, orthogonal partitions and a replacement method, some new methods to construct pure quantum error-correcting codes are provided from orthogonal arrays. As an application of these methods, we construct several infinite series of quantum error-correcting codes including some optimal ones. Compared with the corresponding binary quantum error-correcting codes in [12,41], more new codes can be constructed, which have fewer terms for each of their basis states.

2. Preliminaries

First, the following concepts and lemmas are needed.

Let be the transposition of matrix A and . Let and denote the vectors of and , respectively. If and with elements from a Galois field with binary operations (+ and ·), the Kronecker product is defined as , where represents the matrix with entries , and the Kronecker sum is defined as where represents the matrix with entries  [23,24]. Let . Let over ring .

A matrix A can often be identified with a set of its row vectors if necessary.

([26]). Let A be an

Let be an abelian group of order s. , , denotes the additive group of order consisting of all t-tuples of entries from with the usual vector addition as the binary operation. Let . Then, is a subgroup of of order s, and its cosets will be denoted by , .

([42]). An

Let D be a difference scheme

([42]). Let HD

([43]). (quantum Singleton bound) Let Q be an

([42]). If

([37]). The minimal distance of an OA

([40]). Let Q be a subspace of

Lemma 3 can also be viewed as the definition of a QECC. Q is denoted as , where N is the length of the code, K is the dimension of the encoding state, d is the minimum Hamming distance, and s is the alphabet size. When , it is simply written as .

([44]). (1) Let D be a difference matrix

(2) Let D be a difference matrix

([36]). (Expansive replacement method). Suppose A is an OA of strength t with column 1 having s levels and that B also is an OA of strength t with s rows. After making a one-to-one mapping between the levels of column 1 in A and the rows of B, if each level of column 1 in A is replaced by the corresponding row from B, we can obtain an OA of strength t.

([42]). If OA

3. Main Results

This section presents some new methods for the construction of QECCs. We begin with a link between OAs and QECCs. There exists a perfect match between the parameters of an OA, A, with an orthogonal partition of strength and the parameters of an QECC, which is listed in Table 1.

Table 1

Correspondence between parameters of OAs and QECCs.

	OAs	QECCs
N	Number of factors	Length of code
K	Number of partitioned blocks	Dimension of code
d	min{t_1+1,MD(A)}	MD of code
s	Number of levels	alphabet size

The construction method for a QECC Q with parameter is summarized in the following Algorithm 1.

If

Step 1. Find an OA A with minimal distance and an orthogonal partition of strength by a difference scheme.

By Lemma 1, a difference scheme exists for any odd integer . Take . Due to Lemma 4, A is an OA. Let . Then is also an IrOA for . It follows from Lemma 2 that and ;

Step 2. Let . Give logical codewords , where is a -uniform state, generated by and Connection 1 in the Introduction.

Let . By the relation between irredundant orthogonal arrays and uniform states (Connection 1), can generate one-uniform states ;

Step 3. The uniform states are just the logical codewords of a QECC .

By Lemma 3 and Definition 5, Q is an optimal code.

Furthermore, if we take to be the subspace spanned by for integer , then it is a code.

In particular, for , taking as a basis state, we have a QECC.

Compared with the binary QECCs in [12], the QECCs obtained from Theorem 1 for have fewer terms for each basis state and more dimensions K not necessarily equal to powers of 2. The comparison is put in Table 2, where “K” denotes the dimension of QECCs and “No.” represents the number of terms for each basis state.

Table 2

Comparison of the obtained QECCs with those in [12].

	The QECCs in [12]			The QECCs by Theorem 1
	((4,K,2))	((6,K,2))	((8,K,2))	((4,K,2))	((6,K,2))	((8,K,2))
K	1,  2,  4	2^2, 2^3, 2^4	2^4, 2^5, 2^6	1, 2, 3, 4	1, 2, 3, …, 2^4	1, 2, 3, …, 2^6
No.	4,  4,  2	8,  4,  2	8,  4,  2	2, 2, 2, 2	2, 2, 2, …,2	2, 2, 2, …, 2

The following is about construction of QECCs with odd length N and minimum distance 2. □

(1) When

(2) When

(1) has vectors with weight 0, vectors with weight 2, vectors with weight 4, ⋯, vectors with weight , and vectors (with the first component equal to 1) with weight . The above vectors are denoted by , where . Let for . Take . Then and A are strength 1 orthogonal arrays and . By Connection 1, can generate K one-uniform states, which form an orthogonal basis of a subspace Q of . By Lemma 3, Q is an QECC;

(2) By arguments similar to those used in the proof of (1), we can obtain the desired QECC. □

Let L be an

Let for . Take . Both M and are OAs of strength two. Any two rows of M can be written as , where , .

(1) When , , ;

(2) When , , ;

(3) When and , we have or , so

So . By Connection 1, can generate K states, which form an orthogonal basis of a subspace Q of . By Lemma 3, Q is an QECC. □

There exists a

Let be a difference scheme of strength 2. Take is an OA for with and where is the ith row of for . Then is an orthogonal partition of strength 2 of . Let

By Lemma 4, is an OA of strength 2. Any two rows of can be written as , where , .

(1) When , ;

(2) When , ;

(3) When and , we have So .

Since M can be written as after row permutation, M is an OA of strength 2. Similarly, we also have . By Connection 1, can generate states, which form an orthogonal basis of a subspace Q of . By Lemma 3, Q is a QECC.

Especially, when and , a QECC exists with logical codewords: , .

The code is pure, but neither the 9 qubit Shor code in [1] nor the 9 qubit Ruskai code in [11] are pure. □

There exists a

Take and . Then is a partition of strength 2 of the difference scheme . For and , let where are as in Theorem 5. Similar arguments in Theorem 2 apply to M, we can obtain the desired QECCs.

Especially, when and , a code can be attained. □

There exists a

Let and . Then is a partition of strength 2 of the difference scheme . Take is an OA for with and where is the ith row of for . Then is an orthogonal partition of strength 3 of . Let Similar arguments in Theorem 5 apply to M, we can obtain the desired QECCs.

Especially, when and , a code exists. □

Suppose

Let for . Thus . Obviously, is an OA and Y is an OA. If , then . From Lemma 3, there exists an QECC. □

Let L be an

Let . Obviously, is an OA and . From Lemma 3, there exists an QECC. □

There exists a

Let . From Lemma 6, an OA exists. Obviously, , then an OA exists and is denoted by A. From Lemma 2, MD. Replacing the s levels, , by distinct rows of respectively, we can get an IrOA. By Lemma 3, a QECC exists.

Especially, when , by using Lemma 3 and IrOA, IrOA, and IrOA, three QECCs , , can be obtained. □

For any

Let . From Lemma 6, an OA exists. Obviously, B=OA, exists since . From Lemma 2, MD. By using the replacement method in Theorem 9, we can get C=OA. Removing the last columns from C, we can get an OA with MD for . By Lemma 3, the desired QECC exists.

Similarly, from the OA, we can obtain an OA. Then removing the last columns, we can have the desired result by Lemma 3. □

4. Examples

In this section, we use examples to illustrate applications of theorems.

Construction of a

Let

Furthermore, if taking

The QECCs in Example 1 are different from and particularly when , have less number of items for every basis state than those codes in [12]. To be self-contained, the QECCs for in [12] are provided as follows.

: .

: , .

: , , , .

Comparison of the method of code construction with [7].

Both methods can take any classical code to a quantum code. The method proposed in [7] can make it by solving for the amplitudes in the superposition. Since any classical code is an OA, the method in this paper can produce a quantum code which is also a -uniform state where from Connection 1. Moreover, if the OA with an orthogonal partition of strength , this method can produce a quantum code where . The amplitudes in the superposition for each logical codeword are all equal to . For example, the code in Example 1 after it is normalized is the same as the one constructed using the method proposed in [7]. It is noteworthy that in Example 1 if taking , then we can construct a stabilizer code with parameter whose logical codewords are , , , .

(1) For

(2) For

Construction of a

Let

This is in fact equivalent to the Steane code. It can correct one error such as

Construction of a

(1) Let

(2) Let

Construction of a

For the case

For the case

For

For

Comparison of the

The new quantum state in the QECC

Compared with the above two codes, it is clear that our construction method has the advantage of a small number of terms.

Some new QECCs with larger minimum distance by Corollary 1.

Let

Let

Let

Let

Let

Let

5. Conclusions

In the work, by using OAs, we study the relation between uniform states and binary QECCs. Several methods for constructing QECCs from OAs are presented. Some optimal QECCs are obtained. Our methods have three advantages. The first is to be able to construct an QECC from each QECC we construct for arbitrary integer . The second is that Theorems 1 and 7–9 can be generalized to construct QECCs for arbitrary d and a prime power q. The third is that for the constructed QECCs, their every basis state has less than or equal to terms compared with the existing binary QECCs in [12,41]. A link between an IrOA and the uniform state is established by Connection 1. In fact, from Theorem 1 to Theorem 9 we always make quantum codes by using uniform states generated by orthogonal partitions. On the other hand, when a quantum code is pure we can easily obtain uniform states. For example, each of the logical codewords in the quantum code in [7] is a one-uniform state. When it is not pure it is worth studying how to use quantum codes to make uniform states. In the future, we will also investigate constructing more optimal QECCs with .