Load-Balanced Fractional Repetition Codes

We introduce load-balanced fractional repetition (LBFR) codes, which are a strengthening of fractional repetition (FR) codes. LBFR codes have the additional property that multiple node failures can be sequentially repaired by downloading no more than one block from any other node. This allows for better use of the network, and can additionally reduce the number of disk reads necessary to repair multiple nodes. We characterize LBFR codes in terms of their adjacency graphs, and use this characterization to present explicit constructions LBFR codes with storage capacity comparable existing FR codes. Surprisingly, in some parameter regimes, our constructions of LBFR codes match the parameters of the best constructions of FR codes

Replication based storage systems with local repair

We consider the design of regenerating codes for distributed storage systems that enjoy the property of local, exact and uncoded repair, i.e., (a) upon failure, a node can be regenerated by simply downloading packets from the surviving nodes and (b) the number of surviving nodes contacted is strictly smaller than the number of nodes that need to be contacted for reconstructing the stored file. Our codes consist of an outer MDS code and an inner fractional repetition code that specifies the placement of the encoded symbols on the storage nodes. For our class of codes, we identify the tradeoff between the local repair property and the minimum distance. We present codes based on graphs of high girth, affine resolvable designs and projective planes that meet the minimum distance bound for specific choices of file sizes

Graph Representation of Topological Stabilizer States

Topological quantum states, especially these in topological stabilizer quantum error correction codes, are currently the focus of intense activity because of their potential for fault-tolerant operations. While every stabilizer state maps to a graph state under local Clifford operations, the graphs associated with topological stabilizer codes remain unknown. In this thesis, I show that the toric code graph is composed of only two kinds of subgraphs: star graphs and half graphs. The topological order of the toric code is identified with the existence of multiple star graphs, which reveals a nice connection between repetition codes and the toric code. The graph structure readily yields a log-depth and a constant-depth (including ancillae) circuit for state preparation. Next, I derive the necessary and sufficient conditions for a family of graph states to be in TQO-1, a class of quantum error correction code states whose code distance scales macroscopically with the number of physical qubits. Using these criteria, I consider a number of specific graph families, including the star and complete graphs, and the line graphs of complete and completely bipartite graphs, and discuss which are topologically ordered and how to construct the codewords. The formalism is then employed to construct several codes with macroscopic distance, including a three-dimensional topological code generated by local stabilizers that also has a macroscopic number of encoded logical qubits. Last, the connection between the characterization of topological order using graph theory and the hierarchy of topological order is analyzed