High-dimensional quantum XYZ product codes for biased noise

Abstract: Three-dimensional (3D) quantum XYZ product can construct a class of non-CSS codes by using three classical codes. However, their error-correcting performance has not been studied in depth so far and whether this code construction can be generalized to higher dimension is an open question. In this paper, we first study the error-correcting performance of the 3D Chamon code, which is an instance of 3D XYZ product of three repetition codes. Next, we show that 3D XYZ product can be generalized to four dimension and propose four-dimensional (4D) XYZ product code construction, which constructs a class of non-CSS codes by using either four classical codes or two CSS codes. Compared with 4D homological product, we show that 4D XYZ product can construct non-CSS codes with higher code dimension or code distance. Finally, we consider two instances of 4D XYZ product, to which we refer as 4D Chamon code and 4D XYZ product concatenated code, respectively. Our simulation results show that, 4D XYZ product can construct non-CSS codes with better error-correcting performance for $Z$-biased noise than CSS codes constructed by 4D homological product.