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Several notions of code products are known in quantum error correction, suchas hyper-graph products, homological products, lifted products, balancedproducts, to name a few. In this paper we introduce a new product codeconstruction which is a natural generalisation of classical product codes toquantum codes: starting from a set of component Calderbank-Shor-Steane (CSS)codes, a larger CSS code is obtained where both $X$ parity checks and $Z$parity checks are associated to classical product codes. We deduce severalproperties of product CSS codes from the properties of the component codes,including bounds on the code distance, and show that built-in redundancies inthe parity checks result in so-called meta-checks which can be exploited tocorrect syndrome read-out errors. We then specialise to the case ofsingle-parity-check (SPC) product codes which in the classical domain are acommon choice for constructing product codes. Logical error rate simulations ofa SPC $3$-fold product CSS code having parameters $[[512,174,8]]$ are shownunder both a maximum likelihood decoder for the erasure channel and beliefpropagation decoding for depolarising noise. We compare the results with othercodes of comparable block length and rate, including a code from the family ofasymptotically good quantum Tanner codes. We observe that our reference productCSS code outperforms all other examined codes.

Classical product code constructions for quantum Calderbank-Shor-Steane codes