The problem of recovering from qubit erasures has recently gained attention as erasures occur in many physical systems such as photonic systems, trapped ions, superconducting qubits and circuit quantum electrodynamics. While several linear-time decoders for error correction are known, their errorcorrecting capability is limited to half the minimum distance of the code, whereas erasure correction allows one to go beyond this limit. As in the classical case, stopping sets pose a major challenge in designing efficient erasure decoders for quantum LDPC codes. In this paper, we show through simulation, that an attractive alternative here, is the use of quantum expander codes in conjunction with the peeling decoder that has linear complexity. We also discuss additional techniques including small-set-flip decoding, that can be applied following the peeling operation, to improve decoding performance and their associated complexity.

Quantum low density parity-check (QLDPC) codes represent an attractive candidate for achieving fault-tolerant quantum computation. Erasure decoding of QLDPC codes has gained traction recently as many physical systems such as neutral-atom systems, photonic systems, trapped-ion systems etc. suffer from qubit erasures. Techniques for erasure decoding of QLDPC codes via belief propagation (BP) have been proposed in the literature, such as BP with guided decimation (BP-GD), where decimation refers to sequentially fixing hard values for the variable nodes. We provide an alternative decimation-based BP algorithm, which we term the BP with degree-based decimation (BP-DD) algorithm, that intelligently chooses a check node based on its degree and subsequently decimates a variable node in its neighborhood. We apply the BP-DD algorithm to two families of QLDPC codes, namely hypergraph product codes and lifted product codes, and show improved logical error performance over the BP-GD algorithm, without incurring additional complexity. Our simulations show that the BP-DD algorithm provides a versatile solution for erasure decoding of QLDPC codes, in contrast to some techniques in the literature that are applicable to only specific classes of QLDPC codes.

We consider replication-based distributed storage systems in which each node stores the same quantum of data and each data bit stored has the same replication factor across the nodes. Such systems are referred to as balanced distributed databases. When existing nodes leave or new nodes are added to this system, the balanced nature of the database is lost, either due to the reduction in the replication factor, or the non-uniformity of the storage at the nodes. This triggers a rebalancing algorithm, that exchanges data between the nodes so that the balance of the database is reinstated. The goal is then to design rebalancing schemes with minimal communication load. In a recent work by Krishnan et al., coded transmissions were used to rebalance a carefully designed distributed database from a node removal or addition. These coded rebalancing schemes have optimal communication load, however, require the file-size to be at least exponential in the system parameters. In this work, we consider a cyclic balanced database (where data is cyclically placed in the system nodes) and present coded rebalancing schemes for node removal and addition in such a database. These databases (and the associated rebalancing schemes) require the file-size to be only cubic in the number of nodes in the system. We bound the advantage of our node removal rebalancing scheme over the uncoded scheme, and show that our scheme has a smaller communication load. In the node addition scenario, the rebalancing scheme presented is a simple uncoded scheme, which we show has optimal load.Due to space restrictions, the current version of this paper contains only a subset of the results concerning the node removal scenario. The full version of this paper, including additional results and examples, is available online [1].