Dr. Alexey Kovaleve, University of California, Riverside
Abstract
In my talk, I will discuss various families of quantum low-density parity check (LDPC) codes and relate them to local on graphs Hamiltonians. I hope that this can lead to topological classification of such Hamiltonians, at least in some simple cases. Concrete examples of LDPC codes with explicitly known parameters become important in such a setting. To this end, I consider a large family of hyperbicycle codes. This family is constructed from two classical codes and leads to bicycle and hypergraph-product codes in limiting cases, hence the name hyperbicycle. Of particular interest are families of quantum LDPC codes with finite rate and distance scaling as square root of blocklength since this represents the best known exponent in distance scaling, even for codes of dimensionality 1. The
hyperbicycle construction also allows to construct LDPC codes with number of encoded qubits
scaling as logarithm of blocklength and the distance asymptotically reaching the distance of toric
codes at large blocklength.