In this paper, we utilize a concatenation scheme to construct new families of quantum error correction codes achieving the quantum Gilbert-Varshamov (GV) bound asymptotically. Weconcatenate alternant codes with any linear code achievingthe classical GV bound to construct Calderbank-Shor-Steane (CSS) codes. We show that the concatenated code can achieve the quantum GV bound asymptotically and can approach the Hashing bound for asymmetric Pauli channels. By combing Steane’s enlargement construction of CSS codes, we derive a family of enlarged stabilizer codes achieving the quantum GV bound for enlarged CSS codes asymptotically. Asapplications, we derive two families of fast encodable and decodable CSS codes with parameters $\mathscr {Q}_{1}=[[N,\Omega (\sqrt {N}),\Omega (\sqrt {N})]]$ , and $\mathscr {Q}_{2}=[[N,\Omega (N/\log N),\Omega (N/\log N)/\Omega (\log N)]]$ . We show that $\mathscr {Q}_{1}$ can be encoded very efficiently by circuits of size $O(N)$ and depth $O(\sqrt {N})$ . For an input error syndrome, $\mathscr {Q}_{1}$ can correct any adversarial error of weight up to half the minimum distance bound in $O(N)$ time. $\mathscr {Q}_{1}$ can also be decoded in parallel in $O(\sqrt {N})$ time by using $O(\sqrt {N})$ classical processors. For an input error syndrome, we proved that $\mathscr {Q}_{2}$ can correct a linear number of ${X}$ -errors with high probability and an almost linear number of ${Z}$ -errors in $O(N)$ time. Moreover, $\mathscr {Q}_{2}$ can be decoded in parallel in $O(\log (N))$ time by using $O(N)$ classical processors.In this paper, we utilize a concatenation scheme to construct new families of quantum error correction codes achieving the quantum Gilbert-Varshamov (GV) bound asymptotically. Weconcatenate alternant codes with any linear code achievingthe classical GV bound to construct Calderbank-Shor-Steane (CSS) codes. We show that the concatenated code can achieve the quantum GV bound asymptotically and can approach the Hashing bound for asymmetric Pauli channels. By combing Steane’s enlargement construction of CSS codes, we derive a family of enlarged stabilizer codes achieving the quantum GV bound for enlarged CSS codes asymptotically. Asapplications, we derive two families of fast encodable and decodable CSS codes with parameters$\mathscr {Q}_{1}=[[N,\Omega (\sqrt {N}),\Omega (\sqrt {N})]]$, and$\mathscr {Q}_{2}=[[N,\Omega (N/\log N),\Omega (N/\log N)/\Omega (\log N)]]$. We show that$\mathscr {Q}_{1}$can be encoded very efficiently by circuits of size$O(N)$and depth$O(\sqrt {N})$. For an input error syndrome,$\mathscr {Q}_{1}$can correct any adversarial error of weight up to half the minimum distance bound in$O(N)$time.$\mathscr {Q}_{1}$can also be decoded in parallel in$O(\sqrt {N})$time by using$O(\sqrt {N})$classical processors. For an input error syndrome, we proved that$\mathscr {Q}_{2}$can correct a linear number of${X}$-errors with high probability and an almost linear number of${Z}$-errors in$O(N)$time. Moreover,$\mathscr {Q}_{2}$can be decoded in parallel in$O(\log (N))$time by using$O(N)$classical processors.

arXiv:2107.05174IEEE Transactions on Information TheoryAug 24, 20225 citations

New constructions of entanglement-assisted quantum codes

Adel N. Alahmadi, Ahmad N. Al‐Kenani, Rola Hijazi +3 more

Cryptography and CommunicationsJun 26, 202125 citations

Asymmetric Entanglement-Assisted Quantum Error-Correcting Codes and BCH Codes

Carlos Galindo, Fernando Hernando, Ryutaroh Matsumoto +1 more

The concept of asymmetric entanglement-assisted quantum error-correcting code (asymmetric EAQECC) is introduced in this article. Codes of this type take advantage of the asymmetry in quantum errors since phase-shift errors are more probable than qudit-flip errors. Moreover, they use pre-shared entanglement between encoder and decoder to simplify the theory of quantum error correction and increase the communication capacity. Thus, asymmetric EAQECCs can be constructed from any pair of classical linear codes over an arbitrary field. Their parameters are described and a Gilbert-Varshamov bound is presented. Explicit parameters of asymmetric EAQECCs from BCH codes are computed and examples exceeding the introduced Gilbert-Varshamov bound are shown.The concept of asymmetric entanglement-assisted quantum error-correcting code (asymmetric EAQECC) is introduced in this article. Codes of this type take advantage of the asymmetry in quantum errors since phase-shift errors are more probable than qudit-flip errors. Moreover, they use pre-shared entanglement between encoder and decoder to simplify the theory of quantum error correction and increase the communication capacity. Thus, asymmetric EAQECCs can be constructed from any pair of classical linear codes over an arbitrary field. Their parameters are described and a Gilbert-Varshamov bound is presented. Explicit parameters of asymmetric EAQECCs from BCH codes are computed and examples exceeding the introduced Gilbert-Varshamov bound are shown.

arXiv:1911.10031IEEE AccessJan 1, 202028 citations

Entanglement-assisted quantum codes from arbitrary binary linear codes

Jianfa Qian, Lina Zhang

Designs Codes and CryptographyJul 25, 201454 citations

Entanglement-Assisted Quantum Codes From Algebraic Geometry Codes

Francisco Revson F. Pereira, Ruud Pellikaan, Giuliano G. La Guardia +1 more

Quantum error-correcting codes play the role of suppressing noise and decoherence in quantum systems by introducing redundancy. Some strategies can be used to improve the parameters of these codes. For example, entanglement can provide a way for quantum error-correcting codes to achieve higher rates than the one obtained by means of the traditional stabilizer formalism. Such codes are called entanglement-assisted quantum error-correcting (EAQEC) codes. In this paper, we utilize algebraic geometry codes to construct several families of EAQEC codes derived from the Euclidean and the Hermitian construction. Three families constructed here consist of codes whose quantum Singleton defect is equal to zero, one, or two. We also construct families of EAQEC codes with an encoding rate exceeding the quantum Gilbert-Varshamov bound. Additionally, asymptotically good towers of linear complementary dual codes are used to obtain asymptotically good families of EAQEC codes consuming maximal entanglement. Furthermore, a simple comparison with the quantum Gilbert-Varshamov bound demonstrates that, by utilizing the proposed construction, it is possible to generate an asymptotically family of EAQEC codes that exceeds this bound.Quantum error-correcting codes play the role of suppressing noise and decoherence in quantum systems by introducing redundancy. Some strategies can be used to improve the parameters of these codes. For example, entanglement can provide a way for quantum error-correcting codes to achieve higher rates than the one obtained by means of the traditional stabilizer formalism. Such codes are called entanglement-assisted quantum error-correcting (EAQEC) codes. In this paper, we utilize algebraic geometry codes to construct several families of EAQEC codes derived from the Euclidean and the Hermitian construction. Three families constructed here consist of codes whose quantum Singleton defect is equal to zero, one, or two. We also construct families of EAQEC codes with an encoding rate exceeding the quantum Gilbert-Varshamov bound. Additionally, asymptotically good towers of linear complementary dual codes are used to obtain asymptotically good families of EAQEC codes consuming maximal entanglement. Furthermore, a simple comparison with the quantum Gilbert-Varshamov bound demonstrates that, by utilizing the proposed construction, it is possible to generate an asymptotically family of EAQEC codes that exceeds this bound.

arXiv:1907.06357IEEE Transactions on Information TheorySep 16, 20213 citations

New Entanglement-Assisted Quantum Constacyclic Codes

Ya-Jing Zhou, Xiaoshan Kai

International Journal of Theoretical PhysicsSep 18, 20230 citations

\mathscr {Q}_{1}=[[N,\Omega (\sqrt {N}),\Omega (\sqrt {N})]]

, and

\mathscr {Q}_{2}=[[N,\Omega (N/\log N),\Omega (N/\log N)/\Omega (\log N)]]

. We show that

\mathscr {Q}_{1}

can be encoded very efficiently by circuits of size

O(N)

and depth

O(\sqrt {N})

. For an input error syndrome,

\mathscr {Q}_{1}

can correct any adversarial error of weight up to half the minimum distance bound in

O(N)

time.

\mathscr {Q}_{1}

can also be decoded in parallel in

O(\sqrt {N})

time by using

O(\sqrt {N})

classical processors. For an input error syndrome, we proved that

\mathscr {Q}_{2}

can correct a linear number of

{X}

-errors with high probability and an almost linear number of

{Z}

-errors in

O(N)

time. Moreover,

\mathscr {Q}_{2}

can be decoded in parallel in

O(\log (N))

time by using

O(N)

classical processors.