Abstract
This paper presents a method based on orthogonal arrays of constructing pure quaternary quantum error-correcting codes. As an application of the method, some infinite classes of quantum error-correcting codes with distances 2, 3, and 4 can be obtained. Moreover, the infinite class of quantum codes with distance 2 is optimal. The advantage of our method also lies in the fact that the quantum codes we obtain have less items for a basis quantum state than existing ones.
1 Introduction
Quantum systems are more fragile than classical systems. When quantum information travels across a noisy channel, errors are unavoidable [1–3]. The primary tool to deal with different types of quantum noises is quantum error-correcting codes (QECCs) [1, 2, 4, 5]. They play an important role in quantum information tasks, such as in entanglement purification, quantum key distribution, fault-tolerant quantum computation, and so on [6–8]. Since its discovery, code construction has come a long way [9–16]. Plenty of binary QECCs have been obtained, some of which from classical error-correcting codes (CECCs) [16–19]. Relatively speaking, there are still less studies on quaternary QECCs. We are motivated by the fact that CECCs are one-to-one connected to orthogonal arrays (OAs) [20]. It would be interesting to see if OAs can reciprocate and help QECCs, especially, quaternary ones. Therefore, the main aim of this work is to construct quaternary QECCs from OAs.
If L is an r × N array with elements from S = {0, 1, …, s − 1} and every r × k subarray of L contains each k-tuple based on S as a row with same frequency, then the array is said to be an orthogonal array of strength k (for some k in the range 0 ≤ k ≤ N). We will use OA (r, N, s, k) to denote such an array [21]. The theory of OAs has been developed significantly since the seminal work of Rao [22]. In particular, in recent years many new methods for constructing strength k OAs have been proposed, and a lot of new classes of OAs have been presented [23–29]. An OA (r, N, s, k) is said to be an irredundant orthogonal array (IrOA), if every row in any r × (N − k) subarray is unique [20]. If all of a pure quantum state’s reductions to k qudits are maximally mixed, it is said to be k-uniform. And this state consists of N subsystems with d levels. A connection between a k-uniform state and an irredundant orthogonal array (IrOA) was established by Goyeneche et al. [20]. For simplicity, the normalization factors are omitted from this paper.
Lemma 1[20] Ifis an IrOA(r, N, s, k), then the superposition ofrproduct states,is ak-uniform state.By using this connection in Lemma 1, a lot of k-uniform states have been constructed from OAs [20, 30–37]. This kind of k-uniform states is closely related to QECCs [12, 20]. Usually, quantum information theory benefits from OAs [38–43]. These new developments in OAs and uniform states provide a higher possibility to construct infinite classes of QECCs from OAs [35, 36].In this work, we present a method based on OAs of constructing pure quaternary QECCs. As an application of the method, some infinite classes of QECCs with distances 2, 3, and 4 can be obtained. We know that quantum bound reflects the optimality of QECCs and is a key parameter to judge whether a construction method is effective or not. Moreover, the resulting infinite class of quantum codes with distance 2 is optimal. The advantage of our method also lies in the fact that the constructed QECCs have less items in each basis state than existing ones.This paper is organized as follows. After introducing symbols, definitions, and required lemmas in Section 2, Section 3 presents the main results. The conclusion is drawn in Section 4.
2 Preliminaries
We introduce several symbols, definitions and lemmas used in this paper.
Let denote the n dimensional space over a Galois field , ω2 = ω + 1. For the convenience of codeword expression, we use . AT is the transposition of matrix A. , and 0r and 1r represent the r × 1 vector of 0s and 1s, respectively. We define the Kronecker product A ⊗ B and the Kronecker sum A ⊕ B as and , respectively, if and with entries from a finite field with binary operations (+ and ⋅). Here, aij + B denotes the u × v matrix with elements aij + brs (1 ≤ r ≤ u, 1 ≤ s ≤ v). And if necessary, matrix A can always be viewed as a set of its row vectors. The strength of an orthogonal array L is denoted by t(L). We also use a k-strength OA to denote an OA of strength k for k ≥ 0. Let .
Definition 1[44] SupposeSl = {(u1, …, ul)|ui ∈ S, i = 1, 2, …, l}. The number of positions in which two vectorsv = (v1, …, vl),u = (u1, …, ul) ∈ Sldiffer from one another is defined as the Hamming distanceHD(u, v) between them. Let HD(L) represents all possible values of the Hamming distance between two distinct rows of an OA L. The minimal distance of a matrixAmeans the minimal Hamming distance between its distinct rows and denoted byMD(A).Let k ≥ 1 and represent the additive group of order sk which consists all k-tuples of elements from . The typical vector addition is used as the binary operation. If , is a subgroup of of order s, and , i = 1, …, sk−1–1 will be used to denoted its cosets.
Definition 2[44] LetDbe anr × cmatrix with elements from. For everyr × ksubmatrix ofD, its rows are seen as entries of. If in the submatrix each set,i = 0, 1, …, sk−1–1, is represented equally frequently, then theDis said to be a difference scheme of strengthk. We useDk(r, c, s) to denote such a matrix. Whenk = 2, we denoteDk(r, c, s) byD(r, c, s).
Definition 3[29] LetLbe anOA(r, N, s, k). Suppose the rows ofLcan be partitioned intousubmatrices {L1, L2, …, Lu} such that eachLiis anwithk1 ≥ 0. Then the set {L1, L2, …, Lu} is called an orthogonal partition of strengthk1ofL. In particular, {L1, L2, …, Lu} is called a strengthk1orthogonal partition of a spaceif.
Definition 4LetD = Dk(r, c, s). A set of difference schemes {D1, D2, …, Du} is called ak1-strength orthogonal partition ofD, ifDi⋂Dj = ø fori ≠ jand.
Lemma 2[44] Ifs≢2 (mod 4), ands ≤ k, then the difference schemeDk(sk−1, k + 1, s) exists.
Lemma 3[34] If L =OA(sk, N, s, k), then MD(L) =N − k + 1.
Lemma 4[45] (1) LetD = Dk(r, c, s). ThenD ⊕ (s) = OA(rs, c, s, k).(2) LetD = Dk (m, n, s) andL = OA (r, N, s, k) fork = 2, 3. ThenD ⊕ L = OA (mr, nN, s, k).
Lemma 5[36] (Expansive replacement method) Assume thatLAis ak-strength OA withslevels in factor 1 and thatLBis ak-strength OA withsrows. After building a one-to-one mapping between the levels of factor 1 inLAand the rows ofLB, we may construct an OA of strength k by substituting each level of factor 1 inLAwith the matching row fromLB.
Lemma 6[44] For a prime powers ≥ 2, an OA(sk, s + 1, s, k) exists ifs ≥ k − 1 ≥ 0.
Lemma 7[12] If the reductions of all states in a subspaceQofto any givenkparties are equal, thenQis an ((N,K,k + 1))sQECC, and vice versa. Furthermore, if any state inQisk-uniform, thenQis pure, and vice versa.We can also define a QECC ((N,K,k + 1))s according to Lemma 7, where N denotes the code length, K is the dimension of the encoding state, k + 1 denotes the distance, and s denotes the levels number. For s = 2, it is simply ((N, K, k + 1)).
Lemma 8[46] (quantum Singleton bound) IfK > 1 in an ((N,K,k + 1))sthenK ≤ sN−2k. Similarly, a pure ((N,1,k + 1))ssatisfies 2k ≤ N.
Definition 5A QECC such that the equality inLemma 8holds is called optimal.
3 Construction of ((N,K,k + 1))4 QECC
This section provides a construction method of quaternary quantum error-correcting codes (QECCs) from orthogonal arrays (OAs). In Theorem 1, we use Lemma 4 (2) to construct QECCs with distance 2. Theorems 2 and 3 produce QECCs with distances 3 and 4 from the OAs with orthogonal partitions. In Theorem 4, we study the existence of QECCs with any distance by using a special construction of OAs.
For everyN ≥ 2, there is a QECC ((N,K,2))4for each integer 1 ≤ K ≤ 4N−2where thecode is optimal.
Proof. When N ≥ 5, a difference scheme D = DN−1(4N−2, N, 4) exists by Lemma 2. Let L = D ⊕ (4) = OA (4N−1, N, 4, N − 1). By Lemma 3, MD(L) = N − (N − 1) + 1 = 2. Set . Let Li = di ⊕ (4) = OA (4, N, 4, 1) for i = 1, 2, …, 4N−2. Then t (Li) = 1, MD (Li) = N.
From Lemma 1, can generate 4N−2 1-uniform states . They can be used as a set of orthogonal basis to generate a subspace Q of . Thus Q is an optimal code by Lemma 7 and Definition 5.
In addition, for any integer 1 ≤ K ≤ 4N−2–1, if QK is the subspace spanned by |φ1⟩, …, |φK⟩, then it is a ((N,K,2))4 code.
When 1 < N < 5, we can construct the following QECCs .
When N = 2, an optimal ((2,1,2))4 code can be generated with a basis |φ⟩ = |01⟩ + |12⟩ + |23⟩ + |30⟩.
When N = 3, take . Let Ai = di ⊕ (4) and . Obviously, A and Ai are OAs for i = 1, 2, 3, 4. From Lemma 3, MD(A) = 2, and by Lemma 7, an optimal ((3,4,2))4 QECC can be obtained from A1, …, A4.
When N = 4, take . Then B4(i−1)+j = di ⊕ ((4) (j − 1) ⊕ (4)) is an OA (4, 4, 4, 1) for i, j = 1, 2, 3, 4 and is an OA (64, 4, 4, 2). By simple calculation, we have MD(B) = 2. By Lemma 7, an optimal ((4,16,2))4 QECC can be obtained from B1, …, B16.
Remark. The quantum codes obtained by Theorem 1 have less items in a basis state than existing ones. For example, every basis states of the ((3,4,2))4 code has four items. It has far less number of items for a basis state than the ((3,4,2))4 in [13]. Compared with the codes [[N,N − 2,2]]4 in [47] for N = 9 + 6m with 0 ≤ m ≤ 165, we have the codes for all N ≥ 2.
SupposeLis anOA(r, N, 4, 2) withMD(L) ≥ 3. A QECC ((N,K,3))4exists, if there are vectorsb1, b2, …, bKinthat fulfillHD(bu, bv) ≥ 3 and |HD(bu, bv) − HD(L)|≥ 3 foru ≠ v.
Proof: Let
, where
Xu= 1
r⊗
bu+
Lfor 1 ≤
u≤
K. Both
Xand
Xuare 2-strength OAs. Let
x1=
bu+
l1,
x2=
bv+
l2∈
Xfor
l1,
l2∈
L. Then we can compute the Hamming distance (HD) between
x1and
x2and the minimum distance (MD) of
X.
(1) HD (x1, x2) = MD(L) ≥ 3, if u = v, l1 ≠ l2.
(2) HD (x1, x2) = HD (bu, bv) ≥ 3, if u ≠ v, l1 = l2.
(3) If u ≠ v and l1 ≠ l2, we have HD (x1, x2) ≥HD (bu + l2, x2) − HD (bu + l2, x1) or HD (x1, x2) ≥HD (bu + l2, x1) − HD (bu + l2, x2), hence HD (x1, x2) ≥|HD (bu, bv) − HD(L)|≥ 3.
Therefore, MD(X) ≥ 3. We can obtain K states from {X1, X2, …, XK} and Lemma 1. Let Q be a subspace of with the K states to be an orthogonal basis. Thus Q is a QECC ((N,K,3))4 by Lemma 7.
There exists a QECCwithforn ≥ 3 and with 3 ≤ p ≤ 5 forn = 2.
Proof. Let {D1, D2, D3, D4} be orthogonal partition of the difference scheme D (16, 3, 4) = (016, (4) ⊕ 04, 04 ⊕ (4)) and be an orthogonal partition of strength two of . Let Yi denote the ith row of with for n ≥ 3. Takewhere Li = for i = 1, 2, …, 4p−n and (a1, a2, …, ap) is an OA (4n, p, 4, 2).
Because Dj is a 2-strength difference scheme and Li is a 2-strength OA, it follows from Lemma 4 that Mk = Dj⊕ Li is a 2-strength OA for k = 1, 2, …, 4p−n+1. Let m1 = d1 ⊕ l1, m2 = d2 ⊕ l2 ∈ Mk for d1, d2 ∈ Dj, l1, l2 ∈ Li. Then we have
Therefore, MD (Mk) ≥ 3 and Mk is an IrOA for any k. Furthermore, M is an OA and has strength two because it is equal to after row permutations. Similarly, we can obtain MD(M) ≥ 3. From Lemma 1, M1, M2, …, can generate 4p−n+1 states. They can be used as a basis to form a subspace Q of . From Lemma 7, Q is a QECC .
Similarly, when 3 ≤ p ≤ 5 and n = 2, we can construct a QECC.
Example 1Let the following + be the operation in. Let,,,. In Theorem 3, we take 3 ≤ p ≤ 5 andn = 2. Let (a1, a2, …, ap) be anOA (16, p, 4, 2) andLi=(a1, a2, (a3, …, ap) + 116 ⊗ Yi), whereYidenotes theith row offori = 1, 2, …, 4p−2. Thenis an orthogonal partition of strength 2 of. We can obtain QECCs,and. With 6 ≤ p ≤ 21 forn = 3, Theorem 3 produces QECCs.
If anOA(4n, p, 4, 3) exists forp > n ≥ 3, then there is aQECC.
Proof. This can be proved in the same way as Theorem 3.
Example 2LetD1 = (016, (4) ⊕ 04, 04 ⊕ (4), (4) ⊕ (4)),D2 = (016, (4) ⊕ 04, 04 ⊕ (4), 1 + (4) ⊕ (4)),D3 = (016, (4) ⊕ 04, 04 ⊕ (4), 2 + (4) ⊕ (4)),D4 = (016, (4) ⊕ 04, 04 ⊕ (4), 3 + (4) ⊕ (4)). Then the difference schemeD3(64, 4, 4) = (064, (4) ⊕ 016, 04 ⊕ (4) ⊕ 04, 016 ⊕ (4)) has a 3-strength orthogonal partition {D1, D2, D3, D4}. Takep = 5, 6 andn = 3 in Theorem 4. Let (a1, a2, …, ap) be anOA(64, p, 4, 3) andLi=(a1, a2, a3, (a4, …, ap) + 164 ⊗ Yi), whereYidenotes theith row offori = 1, 2, …, 4p−3. Thenis an orthogonal partition strength 3 of. By Theorem 4, two new QECCsandcan be obtained.
LetLNdenote anOA(r, N, 4, k). Letfors ≤ 4 andN1 + N2 ≤ N. IfMD(Y) ≥ k + 1, then there exists anQECC.
Proof. Let for i = 1, 2, …, s. Since Yi is isomorphic to Y1, Yi is an OA and t (Yi) = k. And we have . If MD(Y) ≥ k + 1, then Yi is an IrOA (r, N1 + N2, 4, k). By Lemma 7, an QECC exists.
Example 3As illustrations for small size codes, we obtain ((6,2,3))4, ((7,4,3))4and ((5,4,3))4.Take anOA (32, 7, 4, 2) = (a1, a2, …, a7) in [48]. For the cases = 2, takeY = (02 ⊕ (a5, a6), (2) ⊕ (a2, a3, a4, a7)). ThenMD(Y) = 3. Application of Theorem 5 yields a new ((6,2,3))4code.Lets = 4 andY = (04 ⊕ (a4, a5, a6), (4) ⊕ (a1, a2, a3, a7)). ThenMD(Y) = 3. By Theorem 5, we can construct a ((7,4,3))4code in [47].LetL5 = (a1, a2, …, a5) be anOA (16, 5, 4, 2) andY = (04 ⊕ (a2, a3), (4) ⊕ (a1, a4, a5)). ThenMD(Y) = 3 and we obtain an optimal ((5,4,3))4code from Theorem 5. Every basis states of the ((5,4,3))4code has 64 items. Compared to ((5,4,3))4in [14], it includes less items for its base states.
LetL = OA(r, N, 4, k) with MD(L) ≥ k + 1. We can construct a QECC ((N,K,k + 1))4if there are vectorsb1, b2, …, bKinsuch that.
Proof: Let . Evidently, MD(M) ≥ k + 1 and Mi is an OA (r, N, 4, k). By Lemma 7, there is a QECC ((N,K,k + 1))4.
Example 4ForN = 7 andr = 32, takeL = OA(32, 7, 4, 2) in [48]. We can getwhich meet the requirements in Theorem 6 whereb1 = (0000000),b2 = (0001103),b3 = (0011332),b4 = (0012030),b5 = (0013200),b6 = (0020021),b7 = (0022113),b8 = (0023323),b9 = (0030210),b10 = (0031313). Then we can construct a new ((7,10,3))4QECC, which is better than the code ((7,4,3))4in [47].
Ifmis an integer satisfying 4m−1 + 3 < 2d ≤ 4m + 3, then there exists a QECCfor 2m(d − 1) ≤ nd ≤ (4m + 1)m.
Proof: Let q = 4m. From Lemma 6, there exists LB = OA (qd−1, q + 1, q, d − 1). By Lemma 3, MD (LB) = q − d + 3. When the q levels, 0, 1, …, q − 1, are replaced respectively by distinct rows of , we can construct LC = OA (qd−1 (q + 1)m, 4, d − 1). Removing the last 0, 1, 2, …, (q − 2d + 3)m columns from LC, an L = OA (qd−1, nd, 4, d − 1) for 2m (d − 1) ≤ nd ≤ (4m + 1)m can be obtained and MD(L) ≥ d. By Lemma 7, the desired QECC exists.
Remark. When m = 1, two optimal QECCs ((2,1,2))4 and ((4,1,3))4 can be obtained.
Example 5By giving different values todin Theorem 7, some new QECCs with larger distances can be obtained, which are listed inTable 1.
TABLE 1
| d | m | QECC | nd |
|---|---|---|---|
| 5 | 2 | 16 ≤ nd ≤ 34 | |
| 6 | 2 | 20 ≤ nd ≤ 34 | |
| 7 | 2 | 24 ≤ nd ≤ 34 | |
| 8 | 2 | 28 ≤ nd ≤ 34 | |
| 9 | 2 | 32 ≤ nd ≤ 34 | |
| 10 | 3 | 54 ≤ nd ≤ 195 | |
| 32 | 3 | 186 ≤ nd ≤ 195 | |
| 33 | 3 | 192 ≤ nd ≤ 195 | |
| 34 | 4 | 264 ≤ nd ≤ 1028 | |
| 100 | 4 | 792 ≤ nd ≤ 1028 | |
| 109 | 4 | 864 ≤ nd ≤ 1028 | |
| 127 | 4 | 1008 ≤ nd ≤ 1028 | |
| 128 | 4 | 1016 ≤ nd ≤ 1028 | |
| 129 | 4 | 1024 ≤ nd ≤ 1028 | |
| 130 | 5 | 1290 ≤ nd ≤ 5120 |
Some new QECCs with larger distance by Theorem 7.
Construction of new codes ((16,1,6))4, ((24,1,8))4, ((23,81,5))4, ((15,4,5))4, ((14,16,4))4, ((23,4,7))4, ((20,256,4))4and ((6,1,4))4fromLemma 7.
Proof: An IrOA (48, 16, 4, 5) with MD = 6 obtained by using product of two OA (28, 16, 2, 5)s in [49] and an IrOA (412, 24, 4, 7) with MD = 8 obtained by using product of two OA (212, 24, 2, 7)s in [49] can generate two new QECCs ((16,1,6))4 and ((24,1,8))4 respectively. By using product of two OA (4608,23,2,4)s obtained from the ((23,9,5)) QECC in Example 7 in [15], we can get an OA (46082, 23, 4, 4) with an orthogonal partition {C1, C2, …, C81} of strength 4 which can generate a new QECC ((23,81,5))4.
An IrOA (48, 15, 4, 5) with an orthogonal partition {A1, A2, A3, A4} of strength 4, an IrOA (48, 14, 4, 5) with an orthogonal partition {B1, B2, …, B16} of strength 3, an IrOA (412, 23, 4, 7) with an orthogonal partition {E1, E2, E3, E4} of strength 6 and an IrOA (412, 20, 4, 7) with an orthogonal partition {F1, F2, …, F256} of strength 3 produce four new QECCs ((15,4,5))4, ((14,16,4))4, ((23,4,7))4 and ((20,256,4))4 respectively. In particular, an IrOA (64,6,4,3) in [48] yields an optimal QECC ((6,1,4))4 in [50].
4 Conclusion
Binary QECCs have been widely studied, but the research on quaternary QECCs is still rare. In the study, from OAs we construct a large number of pure quaternary QECCs, some of which are optimal. The advantage of the method presented is that the quantum codes we obtain have fewer items for a basis quantum state compared with the existing ones. In future, we intend to construct more optimal QECCs with the distance and investigate the q-ary QECCs for other prime powers and non-primes q from OAs.
Statements
Data availability statement
The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.
Author contributions
Supervision, SP; conceptualization, SP and FY; investigation, SP, FY, RY, JD, and TW; methodology, FY and RY; validation, FY, RY, JD, and TW; writing—original draft, FY; writing—review and editing, SP and FY All authors have read and agreed to the published version of the manuscript.
Funding
This work is supported by National Natural Science Foundation of China (Grant Nos. 11971004, 622722 08, 62172196); Open Foundation of State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications (Grant No. SKLNST-2022-1-01); Science and Tech-nology Research Project of Henan Province (202102210163); The Open Foundation of Guangxi Key Laboratory of Trusted Software (Grant No. KX202040).
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
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Summary
Keywords
quantum error-correcting codes, orthogonal arrays, k-uniform states, orthogonal partition, difference scheme
Citation
Pang S, Yang F, Yan R, Du J and Wang T (2023) Construction of quaternary quantum error-correcting codes via orthogonal arrays. Front. Phys. 11:1148398. doi: 10.3389/fphy.2023.1148398
Received
20 January 2023
Accepted
15 February 2023
Published
24 February 2023
Volume
11 - 2023
Edited by
Nanrun Zhou, Shanghai University of Engineering Sciences, China
Reviewed by
Mingxing Luo, Southwest Jiaotong University, China
Ma Hongyang, Qingdao University of Technology, China
Updates
Copyright
© 2023 Pang, Yang, Yan, Du and Wang.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Shanqi Pang, shanqipang@126.com; Jiao Du, jiaodudj@126.com; Tianyin Wang, wangtianyin79@163.com
This article was submitted to Quantum Engineering and Technology, a section of the journal Frontiers in Physics
Disclaimer
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