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

We provide a numerical investigation of two families of subsystem quantum codes that are related to hypergraph product codes by gauge-fixing. The first family consists of the Bravyi-Bacon-Shor (BBS) codes which have optimal code parameters for subsystem quantum codes local in 2-dimensions. The second family consists 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 by entangling the gauge qubits of two SHP codes. To evaluate the performance of these 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 for phenomenological noise show that large BBS and SHP codes start to outperform surface codes with similar encoding rate at physical error rates $1\times 10^{-6}$ and $4\times10^{-4}$, respectively.