The four-dimensional Chamon code

Abstract

Fracton models have attracted considerable interest as candidates for quantum memories because of their unconventional ground-state degeneracy (GSD) and restricted-mobility excitations. The four-dimensional (4D) Chamon code introduced in our previous work is constructed via the 4D XYZ product of two two-dimensional (2D) toric codes. Its GSD grows exponentially with the system size, similar to that of the three-dimensional (3D) Chamon code, suggesting that it may be regarded as a 4D generalization of the 3D Chamon code. However, the excitation properties of the 4D Chamon code have not been studied in depth, and a high-performance decoding strategy is still lacking. In this work, we first establish the correspondence between the algebraic structure of the 4D Chamon code and the 4D lattice, thereby characterizing the geometric distributions of qubits and stabilizers. Second, we show that the 4D Chamon code supports three types of restricted-mobility excitations analogous to those of the 3D Chamon code, further supporting its interpretation as a 4D generalization of the 3D Chamon code. Finally, we uncover two structural properties relevant to decoding: a hyperplane symmetry and a projection-induced 2D toric-code structure. By exploiting these properties, we develop a two-layer decoding strategy that decomposes the original decoding problem into multiple independent and parallelizable subproblems. Numerical simulations show that the proposed decoder substantially outperforms BP-OSD in decoding accuracy, demonstrating the benefit of incorporating the intrinsic geometric and algebraic structures of the code into decoder design.