New bounds and constructions for error control codes

E. Brion, G. Harel, Nouari Kébaïli +2 more

We show that multidimensional Zeno effect allows one to efficiently protect quantum systems from coherence loss by a method similar to classical coding. Contrary to the conventional approach, our method is applicable to arbitrary error-induced Hamiltonians and general quantum systems. We propose algorithms for finding the coherence-protecting codes along with a method of practical encoding by non-holonomic control. Two new single-error–correcting codes are constructed that surpass the bounds valid for classical quantum codes.

W. W. Peterson

O.Y. Takeshita

A new construction is proposed for low density parity check (LDPC) codes using quadratic permutation polynomials over finite integer rings. The associated graphs for the new codes have both algebraic and pseudo-random nature, and the new codes are quasi-cyclic. Graph isomorphisms and automorphisms are identified and used in an efficient search for good codes. Graphs with girth as large as 12 were found. Upper bounds on the minimum Hamming distance are found both analytically and algorithmically. The bounds indicate that the minimum distance grows with block length. Near-codewords are one of the causes for error floors in LDPC codes; the new construction provides a good framework for studying near-codewords in LDPC codes. Nine example codes are given, and computer simulation results show the excellent error performance of these codes. Finally, connections are made between this new LDPC construction and turbo codes using interleavers generated by quadratic permutation polynomials.

Robert Stephen Cunningham

Ian F. Blake

E. Brion, V. M. Akulin, Ilya I. Dumer +2 more

We show that multidimensional Zeno effect combined with non-holonomic control allows to efficiently protect quantum systems from decoherence by a method similar to classical coding.Contrary to the conventional approach, our method is applicable to arbitrary error-inducing Hamiltonians and general quantum systems.We also propose algorithms of finding encoding that approaches the Hamming upper bound along with methods of practical realizations of the encodings.Two new codes protecting 2 information qubits out of 7 and 4 information qubits out of 9 against a single error with arbitrarily small probability of failure are constructed as an example.