Fault-Tolerant Gates On Hypergraph Product Codes

Quantum computers have the capacity to change the landscape of computing. To make these devices robust to experimental imperfections, we require some form of quantum error correction. Current approaches focus on topological codes, as they appear to be in the realm of experimental capabilities. These techniques will likely lead to the first proof of principle of fault tolerance in the next 5 yr. In the long run, however, these approaches require a very large number of physical qubits to construct a fault-tolerant quantum circuit that can run interesting algorithms. As an alternative, it is proposed that quantum low-density parity-check (LDPC) codes could serve as an architecture that circumvents this problem. The best candidate class of quantum LDPC codes were discovered by Tillich and Zemor and are called hypergraph product codes. These codes have a constant encoding rate and were recently shown to have a constant fault-tolerant error threshold. With these features, they asymptotically offer a smaller overhead compared to topological codes. Here, we demonstrate how to perform Clifford gates on this class of codes using code deformation. To this end, we introduce wormhole defects on hypergraph product codes. Together with state injection, we can perform a universal set of gates within a single block of the class of hypergraph product codes.