Nonbinary quantum error-correcting codes from algebraic curves

Jon-Lark Kim, Gretchen L. Matthews

Artur Elezi

Pradeep Kiran Sarvepalli, Andreas Klappenecker

Aparna Kanungo

Quantum error correcting codes have been introduced to encode the data bits\nin extra redundant bits in order to accommodate errors and correct them. However,\ndue to the delicate nature of the quantum states or faulty gate operations, there is\na possibility of catastrophic spread of errors which might render the error correction\ntechniques ineffective. Hence, in this thesis we concentrate on how various operations\ncan be carried out fault-tolerantly so that the errors are not propagated in the same\nblock. We prove universal fault-tolerance for nonbinary CSS codes. This thesis is\nfocussed only on nonbinary quantum codes and all the results pertain to nonbinary\ncodes.\nEfficient error detection and correction techniques using fault-tolerant techniques\ncan help as long as we ensure that the gate error probability is below a certain\nthreshold. The calculation of this threshold is therefore important to see if quantum\ncomputations are realizable or not, even with fault-tolerant operations. We derive an\nexpression to compute the gate error threshold for nonbinary quantum codes and test\nthis result for different classes of codes, to get codes with best threshold results.

Tong Ningning -, WU Yu-ping

An algorithm for constructing nonbinary Quasi-cyclic LDPC (QC-LDPC) codes on the basis of masking and puncturing techniques in consideration of the shortage of algebraic construction algorithm and time-varying performance of wireless channels is proposed . This scheme can reduce the complexity of the hardware implementation, enhance the error correction performance of the LDPC codes constructed by structure algorithm based on masking technology, and improve the average effectiveness of the communication system by realizing a rate-compatible encoding based on puncturing technology. The simulation results show that the nonbinary LDPC codes constructed with masking and puncturing techniques can provide significantly stronger error correction capabilities as compared with those constructed with algebraic method which are used directly.

René Bødker Christensen, Carlos Munuera, Francisco Revson F. Pereira +2 more

We study entanglement-assisted quantum error-correcting codes (EAQECCs) arising from classical one-point algebraic geometry codes from the Hermitian curve with respect to the Hermitian inner product. Their only unknown parameter is $ c $, the number of required maximally entangled quantum states since the Hermitian dual of an AG code is unknown. In this article, we present an efficient algorithmic approach for computing $ c $ for this family of EAQECCs. As a result, this algorithm allows us to provide EAQECCs with excellent parameters over any field size.