Section: New Results

Quantum LDPC codes

Quantum codes are the analogous of error correcting codes for a quantum computer. A well known family of quantum codes are the CSS codes due to Calderbank, Shor and Steane can be represented by a pair of matrices $(H_X,H_Z)$ such that $H_X{H}_Z^T=0$. As in classical coding theory, if these matrices are sparse, then the code is said to be LDPC. An open problem in quantum coding theory is to get a family of quantum LDPC codes whose asymptotic minimum distance is in $\Omega \left(n^{\alpha }\right)$ for some $\alpha >1/2$. No such family is known and actually, only few known families of quantum LDPC codes have a minimum distance tending to infinity.

In an article in preparation, Benjamin Audoux (I2M, Marseille) and A. Couvreur investigate a problem suggested by Bravyi and Hastings. They studied the behaviour of iterated tensor powers of CSS codes and prove in particular that such families always have a minimum distance tending to infinity. They propose also 3 families of LDPC codes whose minimum distance is in $\Omega \left(n^{\beta }\right)$ for all $\beta <1/2$.