Quantum codes with low density parity checks (LDPC) enable reliable storage of encoded quantum states by monitoring syndromes of simple low-weight check operators. Such codes have recently found applications in fault-tolerant quantum computing and as toy models of quantum memory Hamitonians. In this talk I will review some of the recent developments and open problems concerning quantum LDPC codes that are of relevance for quantum Hamiltonian complexity and the theory of topological quantum order.
In the second part of the talk I will present a new construction of quantum LDPC codes based on a tensor product of chain complexes and the Künneth theorem. Given a pair of stabilizer codes [[n,k,d]] and [[n',k',d']] with check operators of weight w and w', their homological product is a stabilizer code [[O(nn'),kk',dd']] with check operators of weight w+w'. Our proof requires one of the two input codes to obey a technical condition that we call composability, whereas the second input code can be arbitrary. We show that the Steane [[7,1,3]] code and its concatenated versions are composable. Using this result we construct the first example of quantum LDPC codes with check operators of logarithmic weight and the minimum distance scaling faster than the square root of the code size.