On overhead reduction in hypergraph product codes

The hypergraph product is a method to produce a quantum stabilizer code from two input classical linear codes, a canonical example being the surface code as a hypergraph product of two classical repetition codes. Many properties of the hypergraph product code can be inherited from those of the classical codes such as the code dimension and minimum distance. We investigate strategies to reduce the number of physical qubits in hypergraph product codes while maintaining some of their useful properties for fault tolerance. When the classical input codes have no redundant parity checks, we show that the code dimension and minimum distance of the hypergraph product code remain unchanged through this transformation. We discuss approaches to mitigate the effects of hook errors with single-ancilla syndrome extraction and provide examples of reduced hypergraph product codes that admit distance-preserving syndrome measurement schedules. Finally, we show how our overhead reduction strategy can be made compatible with the Pauli-product-measurement scheme by Xu et al.