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##### GRACE - 2015

Overall Objectives
New Software and Platforms
Bilateral Contracts and Grants with Industry
Partnerships and Cooperations
Bibliography

## 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 $\left({H}_{X},{H}_{Z}\right)$ 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 $\beta <1/2$.