In this paper, we study nonbinary toric codes and color codes. While the relationships between the qubit color codes and toric codes have been studied extensively from various perspectives, similar studies for qudit toric codes and color codes are missing. We show that just as in the binary case, qudit color codes can be mapped to qudit toric codes. Qudit color codes are known to have the capability for implementing the generalized Clifford group transversally. The proposed mapping of qudit color codes to toric codes would allow us to port the computational capabilities of the qudit color codes to qudit toric codes. Another application would be to decode qudit color codes via qudit toric codes. Along the way, we show an important structural result for higher-dimensional qudit toric codes. Specifically, we show that the toric codes over prime power dimension can be viewed as a direct sum of copies of codes over the prime dimension. This result simplifies the study of codes over prime power dimension by reducing them to the study of codes over the prime dimension. Furthermore, we provide explicit circuits for the transformation between qudit color codes and toric codes.
Toric codes and color codes are two important classes of topological codes. Kubica et al. [A. Kubica et al., New J. Phys. 17, 083026 (2015)] showed that any
D-dimensional color code can be mapped to a finite number of toric codes in
D dimensions. We propose an alternate map of three-dimensional (3D) color codes to 3D toric codes with a view to decoding 3D color codes. Our approach builds on Delfosse's result [N. Delfosse, Phys. Rev. A 89, 012317 (2014)] for 2D color codes and exploits the topological properties of these codes. Our result reduces the decoding of 3D color codes to that of 3D toric codes. Bit-flip errors are decoded by projecting on one set of 3D toric codes, while phase-flip errors are decoded by projecting onto another set of 3D toric codes. We use these projections to study the performance of a class of 3D color codes called stacked codes.
The topological color code and the toric code are two leading candidates for realizing fault-tolerant quantum computation. Here we show that the color code on a d -dimensional closed manifold is equivalent to multiple decoupled copies of the d -dimensional toric code up to local unitary transformations and adding or removing ancilla qubits. Our result not only generalizes the proven equivalence for d = 2, but also provides an explicit recipe of how to decouple independent components of the color code, highlighting the importance of colorability in the construction of the code. Moreover, for the d -dimensional color code with boundaries of distinct colors, we find that the code is equivalent to multiple copies of the d -dimensional toric code which are attached along a -dimensional boundary. In particular, for d = 2, we show that the (triangular) color code with boundaries is equivalent to the (folded) toric code with boundaries. We also find that the d -dimensional toric code admits logical non-Pauli gates from the d th level of the Clifford hierarchy, and thus saturates the bound by Bravyi and König. In particular, we show that the logical d -qubit control- Z gate can be fault-tolerantly implemented on the stack of d copies of the toric code by a local unitary transformation.