Three-dimensional (3D) quantum XYZ product can construct a class of non-CSS quantum codes by using three classical codes. However, there has been limited study on their error-correcting performance 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 the 3D XYZ product of three repetition codes. Second, we show that the 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 quantum codes by using either four classical codes or two CSS quantum codes. Compared with the 4D homological product, we show that the 4D XYZ product can construct non-CSS codes with higher code dimension or code distance. Third, we consider two instances of the 4D XYZ product, to which we refer as the 4D Chamon code and the 4D XYZ product concatenated code, respectively. Our simulation results show that, the 4D XYZ product can construct non-CSS codes with better error-correcting performance against Pauli- $Z$ -biased noise than CSS codes constructed by the 4D homological product. Finally, we present the geometric arrangement of the 4D Chamon code within a 4D cubic lattice, demonstrating that it possesses two key characteristics of fracton models, which strongly suggest that it is a novel 4D fracton model.Three-dimensional (3D) quantum XYZ product can construct a class of non-CSS quantum codes by using three classical codes. However, there has been limited study on their error-correcting performance 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 the 3D XYZ product of three repetition codes. Second, we show that the 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 quantum codes by using either four classical codes or two CSS quantum codes. Compared with the 4D homological product, we show that the 4D XYZ product can construct non-CSS codes with higher code dimension or code distance. Third, we consider two instances of the 4D XYZ product, to which we refer as the 4D Chamon code and the 4D XYZ product concatenated code, respectively. Our simulation results show that, the 4D XYZ product can construct non-CSS codes with better error-correcting performance against Pauli-$Z$-biased noise than CSS codes constructed by the 4D homological product. Finally, we present the geometric arrangement of the 4D Chamon code within a 4D cubic lattice, demonstrating that it possesses two key characteristics of fracton models, which strongly suggest that it is a novel 4D fracton model.

arXiv:2408.03123Aug 6, 20240 citations

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-biased noise than CSS codes constructed by the 4D homological product. Finally, we present the geometric arrangement of the 4D Chamon code within a 4D cubic lattice, demonstrating that it possesses two key characteristics of fracton models, which strongly suggest that it is a novel 4D fracton model.