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We provide a numerical investigation of two families of subsystem quantumcodes that are related to hypergraph product codes by gauge-fixing. The firstfamily consists of the Bravyi-Bacon-Shor (BBS) codes which have optimal codeparameters for subsystem quantum codes local in 2-dimensions. The second familyconsists of the constant rate "generalized Shor" codes of Bacon and Cassicino\cite{bacon2006quantum}, which we re-brand as subsystem hypergraph product(SHP) codes. We show that any hypergraph product code can be obtained byentangling the gauge qubits of two SHP codes. To evaluate the performance ofthese codes, we simulate both small and large examples. For circuit noise, a$[[21,4,3]]$ BBS code and a $[[49,16,3]]$ SHP code have pseudthresholds of$2\times10^{-3}$ and $8\times10^{-4}$, respectively. Simulations forphenomenological noise show that large BBS and SHP codes start to outperformsurface codes with similar encoding rate at physical error rates $1\times10^{-6}$ and $4\times10^{-4}$, respectively.

A Numerical Study of Bravyi-Bacon-Shor and Subsystem Hypergraph Product Codes