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IEEE International Symposium on Information Theory (ISIT)

CSS codes and QSCs from Whiteman’s Generalized Cyclotomy of order four

Varsha Tiwari, Pramod Kumar Kewat

Published Jun 25, 2023• 2 citations
Publisher

Abstract

In this paper, Calderbank-Shor-Steane (CSS) quantum codes and Quantum synchronizable codes (QSCs) are obtained from cyclic codes, which are constructed using Whiteman’s generalized cyclotomy of order four. These CSS and QSCs have good error-correcting capability against phase and bit errors because many of the cyclic codes used to design them are optimal or nearly optimal. Moreover, QSCs obtained in this article achieve maximum tolerance against misalignment.

DOI
10.1109/isit54713.2023.10206544

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In this paper, Calderbank-Shor-Steane (CSS) quantum codes and Quantum synchronizable codes (QSCs) are obtained from cyclic codes, which are constructed using Whiteman’s generalized cyclotomy of order four. These CSS and QSCs have good error-correcting capability against phase and bit errors because many of the cyclic codes used to design them are optimal or nearly optimal. Moreover, QSCs obtained in this article achieve maximum tolerance against misalignment.
Quantum synchronizable codes (QSCs) are special quantum error-correcting codes (QECCs) that correct the effects of both quantum noise on qubits and misalignment in block synchronization. In this letter, we construct two classes of QSCs from the cyclic codes which are obtained by the cyclotomic classes of order four. The QSCs constructed are Calderbank-Shor-Steane QECCs which can tolerate maximum number of misalignment errors. Moreover, the QSCs usually possess good error-correcting capability towards bit error and phase error, due to many cyclic codes employed to construct QSCs are optimal or almost optimal.
Quantum synchronizable codes are kinds of quantum error-correcting codes that can not only correct the effects of quantum noise on qubits but also the misalignment in block synchronization. In this paper, the quantum synchronizable codes constructed are CSS quantum error-correcting codes whose synchronization capabilities reach the upper bound. And we use cyclic codes gained by sextic cyclotomic classes to construct two classes of quantum synchronizable codes. Moreover, the quantum synchronizable codes are posses good error-correcting capability towards bit error and phase error, since the cyclic codes we used are optimal or almost optimal.
Quantum synchronizable codes (QSCs) are designed to correct both Pauli errors and synchronization misalignment. In this letter, we introduce a new framework for constructing QSCs based on classical coset codes. This construction is versatile, yielding codes for both random and, notably, burst errors, thereby extending the scope of existing QSC designs. The resulting codes exhibit two notable properties: they fully preserve the phase-error-correction capability of their parent Calderbank-Shor-Steane (CSS) codes, and in the absence of misalignment, they revert to the full error-correcting power of the parent codes.
Quantum synchronizable codes (in short QSCs) are a special type of quantum error correcting codes (ECCs) which are useful to correct bit error, phase error, and misalignment in the block of synchronization. In this review paper, two classes of cyclic codes are converted into classes of QSCs. The QSCs with the greatest resistance to misalignment are Calderbank–Shor–Steane QECCs. Quantum information necessitates error correction, and quantum technologies are employed for safe communication, machine learning, and chemical analysis. Due to the non-cloning theorem, decoherence and the difficulty of measuring quantum states, quantum information processing is more challenging than classical information processing. Due to the high frequency of faults, a quantum computer needs an error detection and correction mechanism. To identify quantum information unit bounds in the absence of block synchronization, quantum information boundaries must be located using coding theory in quantum computing and communication systems and this prevents decoherence and correct errors while synchronizing qubits. Also, this method generates synchronizable and error-correcting quantum codes.
Quantum synchronizable codes (QSCs) are special quantum error-correcting codes that correct the effects of both quantum noise on qubits and misalignment in block synchronization. In this letter, we construct a family of QSCs from cyclic codes. The best attainable misalignment tolerance can also be achieved under certain constraints. And we give two classes of specific constructions of QSCs, which achieve the best attainable misalignment tolerance. Moreover, the second class of QSCs possess good error-correcting capability towards bit errors and phase errors.
Quantum synchronizable codes are quantum error correcting codes that can correct not only Pauli errors but also errors in block synchronization. The code can be constructed from two classical cyclic codesC\mathcal {C}C,D\mathcal {D}DsatisfyingC⊥⊂C⊂D\mathcal {C}^{\perp } \subset \mathcal {C} \subset \mathcal {D}C⊥⊂C⊂Dthrough the Calderbank-Shor-Steane (CSS) code construction. In this work, we establish connections between quantum synchronizable codes, subsystem codes, and hybrid quantum-classical codes constructed from the same pair of classical cyclic codes. We also propose a method to construct a synchronizable hybrid subsystem code which can correct both Pauli and synchronization errors, is resilient to gauge errors by virtue of the subsystem structure, and can transmit both classical and quantum information, all at the same time. The trade-offs between the number of synchronization errors that the code can correct, the number of gauge qubits, and the number of logical classical bits of the code are also established. In addition, we propose general methods to construct hybrid and hybrid subsystem codes of CSS type from classical codes, which cover relevant codes from our main construction.