scholarly journals Universal Framework for Quantum Error-Correcting Codes

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 937
Author(s):  
Zhuo Li ◽  
Lijuan Xing
Keyword(s):  
Group Algebra ◽  
Quantum Systems ◽  
Error Correcting Codes ◽  
Quantum Codes ◽  
Algebraic Notation ◽  
General Quantum ◽  
Quantum Error

We present a universal framework for quantum error-correcting codes, i.e., a framework that applies to the most general quantum error-correcting codes. This framework is based on the group algebra, an algebraic notation associated with nice error bases of quantum systems. The nicest thing about this framework is that we can characterize the properties of quantum codes by the properties of the group algebra. We show how it characterizes the properties of quantum codes as well as generates some new results about quantum codes.

scholarly journals ON OPTIMAL QUANTUM CODES

2004 ◽  
Vol 02 (01) ◽  
pp. 55-64 ◽  
Author(s):  
MARKUS GRASSL ◽  
THOMAS BETH ◽  
MARTIN RÖTTELER
Keyword(s):  
Minimum Distance ◽  
Prime Power ◽  
Quantum Systems ◽  
Error Correcting Codes ◽  
Mds Codes ◽  
Quantum Codes ◽  
Maximum Distance Separable ◽  
Shortened Codes ◽  
Quantum Error ◽  
Quantum Mds Codes

We present families of quantum error-correcting codes which are optimal in the sense that the minimum distance is maximal. These maximum distance separable (MDS) codes are defined over q-dimensional quantum systems, where q is an arbitrary prime power. It is shown that codes with parameters 〚n, n - 2d + 2, d〛q exist for all 3≤n≤q and 1≤d≤n/2+1. We also present quantum MDS codes with parameters 〚q2, q2-2d+2, d〛q for 1≤d≤q which additionally give rise to shortened codes 〚q2-s, q2-2d+2-s, d〛q for some s.

ON TWO PROBLEMS OF ASYMMETRIC QUANTUM CODES

2014 ◽  
Vol 28 (06) ◽  
pp. 1450017 ◽  
Author(s):  
RUIHU LI ◽  
GEN XU ◽  
LUOBIN GUO
Keyword(s):  
Cyclic Codes ◽  
Error Correcting Codes ◽  
Bch Code ◽  
Quantum Codes ◽  
Quantum Code ◽  
Asymmetric Quantum ◽  
Asymmetric Quantum Codes ◽  
Special Cases ◽  
Quantum Error ◽  
Better Than

In this paper, we discuss two problems on asymmetric quantum error-correcting codes (AQECCs). The first one is on the construction of a [[12, 1, 5/3]]2 asymmetric quantum code, we show an impure [[12, 1, 5/3 ]]2 exists. The second one is on the construction of AQECCs from binary cyclic codes, we construct many families of new asymmetric quantum codes with dz> δ max +1 from binary primitive cyclic codes of length n = 2m-1, where δ max = 2⌈m/2⌉-1 is the maximal designed distance of dual containing narrow sense BCH code of length n = 2m-1. A number of known codes are special cases of the codes given here. Some of these AQECCs have parameters better than the ones available in the literature.

Entanglement-assisted quantum codes achieving the quantum singleton bound but violating the quantum hamming bound

2014 ◽  
Vol 14 (13&14) ◽  
pp. 1107-1116
Author(s):  
Ruihu Li ◽  
Luobin Guo ◽  
Zongben Xu
Keyword(s):  
Error Correction ◽  
Infinite Family ◽  
Error Correcting Codes ◽  
Quantum Codes ◽  
Singleton Bound ◽  
Minimum Distances ◽  
Quantum Error

We give an infinite family of degenerate entanglement-assisted quantum error-correcting codes (EAQECCs) which violate the EA-quantum Hamming bound for non-degenerate EAQECCs and achieve the EA-quantum Singleton bound, thereby proving that the EA-quantum Hamming bound does not asymptotically hold for degenerate EAQECCs. Unlike the previously known quantum error-correcting codes that violate the quantum Hamming bound by exploiting maximally entangled pairs of qubits, our codes do not require local unitary operations on the entangled auxiliary qubits during encoding. The degenerate EAQECCs we present are constructed from classical error-correcting codes with poor minimum distances, which implies that, unlike the majority of known EAQECCs with large minimum distances, our EAQECCs take more advantage of degeneracy and rely less on the error correction capabilities of classical codes.

On quantum codes obtained from cyclic codes over A2

2015 ◽  
Vol 13 (03) ◽  
pp. 1550031 ◽  
Author(s):  
Abdullah Dertli ◽  
Yasemin Cengellenmis ◽  
Senol Eren
Keyword(s):  
Cyclic Codes ◽  
Error Correcting Codes ◽  
Gray Map ◽  
Quantum Codes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Arbitrary Length ◽  
Necessary And Sufficient ◽  
Quantum Error

In this paper, quantum codes from cyclic codes over A2 = F2 + uF2 + vF2 + uvF2, u2 = u, v2 = v, uv = vu, for arbitrary length n have been constructed. It is shown that if C is self orthogonal over A2, then so is Ψ(C), where Ψ is a Gray map. A necessary and sufficient condition for cyclic codes over A2 that contains its dual has also been given. Finally, the parameters of quantum error correcting codes are obtained from cyclic codes over A2.

scholarly journals A complete hierarchy for the pure state marginal problem in quantum mechanics

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Xiao-Dong Yu ◽  
Timo Simnacher ◽  
Nikolai Wyderka ◽  
H. Chau Nguyen ◽  
Otfried Gühne
Keyword(s):  
Quantum Mechanics ◽  
Quantum State ◽  
Error Correcting Codes ◽  
Quantum Codes ◽  
Semidefinite Programs ◽  
Marginal Problem ◽  
Separability Problem ◽  
Pure Quantum State ◽  
Maximally Entangled States ◽  
Quantum Error

AbstractClarifying the relation between the whole and its parts is crucial for many problems in science. In quantum mechanics, this question manifests itself in the quantum marginal problem, which asks whether there is a global pure quantum state for some given marginals. This problem arises in many contexts, ranging from quantum chemistry to entanglement theory and quantum error correcting codes. In this paper, we prove a correspondence of the marginal problem to the separability problem. Based on this, we describe a sequence of semidefinite programs which can decide whether some given marginals are compatible with some pure global quantum state. As an application, we prove that the existence of multiparticle absolutely maximally entangled states for a given dimension is equivalent to the separability of an explicitly given two-party quantum state. Finally, we show that the existence of quantum codes with given parameters can also be interpreted as a marginal problem, hence, our complete hierarchy can also be used.

scholarly journals Quantifying the Performance of Quantum Codes

2011 ◽  
Vol 18 (01) ◽  
pp. 1-31 ◽  
Author(s):  
Carlo Cafaro ◽  
Sonia L'Innocente ◽  
Cosmo Lupo ◽  
Stefano Mancini
Keyword(s):  
Error Correcting Codes ◽  
Quantum Error Correction ◽  
Quantum Codes ◽  
Repetition Code ◽  
Partial Correlations ◽  
Stabilizer Code ◽  
Error Probabilities ◽  
Quantum Error ◽  
Memory Channel ◽  
Noise Models

We study the properties of error correcting codes for noise models in the presence of asymmetries and/or correlations by means of the entanglement fidelity and the code entropy. First, we consider a dephasing Markovian memory channel and characterize the performance of both a repetition code and an error avoiding code ([Formula: see text] and [Formula: see text], respectively) in terms of the entanglement fidelity. We also consider the concatenation of such codes ([Formula: see text]) and show that it is especially advantageous in the regime of partial correlations. Finally, we characterize the effectiveness of the codes [Formula: see text], [Formula: see text] and [Formula: see text] by means of the code entropy and find, in particular, that the effort required for recovering such codes decreases when the error probability decreases and the memory parameter increases. Second, we consider both symmetric and asymmetric depolarizing noisy quantum memory channels and perform quantum error correction via the five-qubit stabilizer code [Formula: see text]. We characterize this code by means of the entanglement fidelity and the code entropy as function of the asymmetric error probabilities and the degree of memory. Specifically, we uncover that while the asymmetry in the depolarizing errors does not affect the entanglement fidelity of the five-qubit code, it becomes a relevant feature when the code entropy is used as a performance quantifier.

Quantum codes over rings

2014 ◽  
Vol 12 (04) ◽  
pp. 1450020 ◽  
Author(s):  
Kenza Guenda ◽  
T. Aaron Gulliver
Keyword(s):  
Linear Codes ◽  
Error Correcting Codes ◽  
Codes Over Rings ◽  
Quantum Codes ◽  
Matrix Product ◽  
Frobenius Rings ◽  
Linear Codes Over Rings ◽  
Generalized Gray Map ◽  
Css Construction ◽  
Quantum Error

This paper considers the construction of quantum error correcting codes from linear codes over finite commutative Frobenius rings. We extend the Calderbank–Shor–Steane (CSS) construction to these rings. Further, quantum codes are extended to matrix product codes. Quantum codes over 𝔽pk are also obtained from linear codes over rings using the generalized Gray map.

Entanglement-assisted quantum codes from quaternary codes of dimension five

2017 ◽  
Vol 15 (03) ◽  
pp. 1750017 ◽  
Author(s):  
Liangdong Lu ◽  
Ruihu Li ◽  
Luobin Guo
Keyword(s):  
Noise Suppression ◽  
Error Correcting Codes ◽  
Quantum Codes ◽  
Optimal Codes ◽  
Maximal Entanglement ◽  
Quaternary Codes ◽  
Almost All ◽  
Quantum Error ◽  
Better Than

Maximal-entanglement entanglement-assisted quantum error-correcting codes (EAQE-CCs) can achieve the EA-hashing bound asymptotically and a higher rate and/or better noise suppression capability may be achieved by exploiting maximal entanglement. In this paper, we discussed the construction of quaternary zero radical (ZR) codes of dimension five with length [Formula: see text]. Using the obtained quaternary ZR codes, we construct many maximal-entanglement EAQECCs with very good parameters. Almost all of these EAQECCs are better than those obtained in the literature, and some of these EAQECCs are optimal codes.

NEW CLASS OF QUANTUM CODES CONSTRUCTED FROM CYCLIC DIFFERENCE SET

2012 ◽  
Vol 10 (01) ◽  
pp. 1250015 ◽  
Author(s):  
SHENG-MEI ZHAO ◽  
YU XIAO ◽  
YAN ZHU ◽  
XIU-LI ZHU ◽  
MIN-HSIU HSIEH
Keyword(s):  
Iterative Decoding ◽  
Difference Sets ◽  
Difference Set ◽  
Error Correcting Codes ◽  
High Rate ◽  
Quantum Codes ◽  
New Class ◽  
Check Matrix ◽  
Approach Zero ◽  
Quantum Error

We introduce joint difference sets as a generalization of cyclic difference sets, and we construct a new class of quantum error-correcting codes (QECCs) from these joint difference sets. The main benefits of our method are as follows. First, we can construct quantum codes that are both high rate and with large block length, while maintaining good performance. Second, the density of constructed quantum parity check matrix can approach zero when the code length is very large. This allows us to use a simple iterative decoding algorithm. Interestingly, our method yields the well-known [5,1,3] QECC.

scholarly journals An explicit construction of quantum codes from one-generator generalized quasi-cyclic codes

2021 ◽  
Vol 336 ◽  
pp. 04001
Author(s):  
Yu Yao ◽  
Yuena Ma ◽  
Husheng Li ◽  
Jingjie Lv
Keyword(s):  
Cyclic Codes ◽  
Explicit Construction ◽  
Error Correcting Codes ◽  
Inner Product ◽  
Quantum Codes ◽  
Dual Codes ◽  
Index 2 ◽  
Quantum Error ◽  
Hermitian Inner Product ◽  
Quasi Cyclic Codes

In this paper, we take advantage of a class of one-generator generalized quasi-cyclic (GQC) codes of index 2 to construct quantum error-correcting codes. By studying the form of Hermitian dual codes and their algebraic structure, we propose a sufficient condition for self-orthogonality of GQC codes with Hermitian inner product. By comparison, the quantum codes we constructed have better parameters than known codes.

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