Pradeep Sarvepalli

The quantum erasure channel models phenomena such as loss or leakage of qubits. Using quantum codes, we can recover from such errors. In this paper, we are interested in studying the performance of two-dimensional color codes over the quantum erasure channel. Our approach makes use of the local equivalence between color codes and surface codes. We propose a variety of decoding algorithms for color codes over the erasure channel. First, we propose algorithms that decode by projecting the erasures on the color to surface codes. Then, instead of directly decoding on the color code or on the equivalent copies of surface codes, we decode jointly on the color code and the equivalent surface codes. We observe a threshold of 44.3% for the color code on the square octagonal lattice.

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.

Three dimensional (3D) toric codes are a class of stabilizer codes with local checks and come under the umbrella of topological codes. While decoding algorithms have been proposed for the 3D toric code on a cubic lattice, there have been very few studies on the decoding of 3D toric codes over arbitrary lattices. Color codes in 3D can be mapped to toric codes. However, the resulting toric codes are not defined on cubic lattice. They are arbitrary lattices with triangular faces. Decoding toric codes over an arbitrary lattice will help in studying the performance of color codes. Furthermore, gauge color codes can also be decoded via 3D toric codes. Motivated by this, we propose an efficient algorithm to decode 3D toric codes on arbitrary lattices (with and without boundaries).

Toric codes and color codes are two important classes of topological codes. Kubica et al. [A. Kubica, New J. Phys. 17, 083026 (2015)NJOPFM1367-263010.1088/1367-2630/17/8/083026] 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)PLRAAN1050-294710.1103/PhysRevA.89.012317] 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.