Section: New Results

Quantum LDPC codes

For some time it was feared that quantum computers could not be built because of distortions of quantum states due to interaction with the environment. This issue could be addressed by the use of quantum codes. Quantum LDPC codes are very interesting candidates here, because their very fast decoding algorithm allows high error correction rates. But the design of good quantum LDPC codes is far more complicated than for their classical counterparts, and cannot be done by random generation. The best-known constructions come from algebraic topology and simplicial homology, but their limits were unknown. Nicolas Delfosse used Riemannian geometry theorems of Gromov to prove that an $\left[\right[n,k,d\left]\right]$-quantum code constructed from the homology of a simplicial surface satisfies $k{d}^2\le C{(logk)}^{2}n$ for some constant $C$ [21] .

Color codes are quantum LDPC codes constructed from 3–regular surface tilings whose set of faces is 3–colorable. Delfosse used morphisms of chain complexes to prove that the decoding of a color code can be reduced to the decoding of three associated surface codes; hence, every decoding algorithm for surface codes yields a decoding algorithm for color codes. From this result, Delfosse obtained theoretical lower bounds on the error threshold of a family of color codes [20] .