Study Of Quantum Correcting Codes Based On Finite Rings

Quantum computation becomes a hot topic in modern information science because it has computational powers exceeding classical computation.Quantum computation has two re-markable superiorities.First,quantum computation can take advantage of quantum par-allelism,which accelerates the computation and increases the storage capacity.Second,quantum computation is capable of simulating quantum communication that classical com-puter cannot accomplish.Both quantum parallelism and simulating quantum communica-tion essentially make use of quantum coherence.However,qubits in a quantum computer are not individual and the interactions of environment lead to decoherence.Quantum error-correction is a significant method to alleviate the detrimental effects of decoherence and ensures quantum computation and quantum communication process normally.Unlike clas-sical error-correction,quantum error-correction should adapt to quantum physical systems of arbitrary order.Since the order of a finite field must be a prime power,quantum error-correcting codes over finite fields have limitation.Finite rings are more suitable for describ-ing quantum physical systems.Quantum error-correcting codes over finite rings can adapt to any quantum physical system.In this dissertation,several contributions based on quantum error-correcting codes over finite rings have been presented as follows.?1?The definition of quantum error-correcting codes over the ring of congruence classes modulo m is proposed,and a one-to-one correspondence between quantum error-correcting codes and additive codes over Z_m is made.Furthermore,additive codes over Z_m can be related to additive codes over an extended ring of Z_m,which simplifies the procedure of constructing quantum error-correcting codes.The conjugate operation over the extended ring is defined and it is equivalent to the Hermitian operation when m is a prime number.Based on the selection of polynomials for generating the extended rings,conjugate dual con-taining codes correspond to quantum error-correcting codes over Z_m,which further optimize the procedure of constructing quantum codes.?2?The weight enumerators of quantum codes over Z_m and the MacWilliams identity are dis-cussed.The Hamming bound,the Singleton bound and the Gilbert-Varshamov bound over Z_m are presented,respectively.For non-degenerate quantum codes,an enhanced Gilbert-Varshamov bound is provided by analyzing the ideals in Z_m.The asymptotic form of the enhanced Gilbert-Varshamov bound is derived and the existence of quantum asymptotically good codes over arbitrary quantum physical system is proved.?3?The structures of conjugate dual containing cyclic codes over the extended ring are dis-cussed.Based on the selection of extended rings,the generator polynomials and the defining sets of conjugate dual codes have good properties.Several conditions based on the generator polynomial and the defining set are presented for judging whether a cyclic code is conjugate dual containing or not.Quantum cyclic codes over finite rings are proved to have the abil-ity of constructing codes with new parameters that quantum cyclic codes over finite fields cannot obtain.This phenomenon illustrates the significance and practicability of quantum error-correction over finite rings.?4?Quantum BCH codes over F_q with length???are constructed,respectively.Compared with the quantum BCH codes in the literature,these new codes have a larger minimum distance when the code lengths and the dimensions are fixed.For primer number p=4r+1 or p=4r+3,quantum quadratic residue codes with length p are defined over Z₄ when r is odd.Quantum BCH codes and quantum RS codes over Z_m are derived and the bounds of maximum designed distance for them are proposed.Finally,several quantum BCH codes and RS codes over Z_m are listed.