On Bounds for Quantum Error Correcting Codes over EJ-Integers

Eda Yıldız

Sixia Yu, Lai, C. H., Changhun Oh

We report two analytical bounds for quantum error-correcting codes that do not have preexisting classical counterparts. Firstly the quantum Hamming and Singleton bounds are combined into a single tighter bound, and then the combined bound is further strengthened via the well-known Lloyd's theorem in classical coding theory, which claims that perfect codes, codes attaining the Hamming bound, do not exist if the Lloyd's polynomial has some non-integer zeros. Our bound characterizes quantitatively the improvement over the Hamming bound via the non-integerness of the zeros of the Lloyd's polynomial. In the case of 1-error correcting codes our bound holds true for impure codes as well, which we conjecture to be always true, and for stabilizer codes there is a 1-logical-qudit improvement for an infinite family of lengths.

Ryuji Takagi, Theodore J. Yoder, Isaac L. Chuang

Matthew Coudron, C. A. Pasternak

We examine the performance of quantum error correcting codes subjected to random Haar distribution transformations of weight t. Rather than requiring correction of all errors, we require some high probability that a random error is corrected. We find that, for any integer i and arbitrarily high probability p < 1, there are codes which perfectly correct errors up to weight t and can correct errors up to weight t + i with probability at least p. We also find an analog to the quantum Hamming bound for the new error model. Lastly, we prove that codes generated from classical Reed-Muller codes can correct errors of weight up to 3d/4 with a probability approaching 1 as the length of the code increases, whereas they can only correct up to weight d/2 perfectly.

Markus Grassl

황규호