  Constructing quantum LDPC codes with a minimum distance that grows faster
than a square root of the length has been a major challenge of the field. With
this challenge in mind, we investigate constructions that come from
high-dimensional expanders, in particular Ramanujan complexes. These naturally
give rise to very unbalanced quantum error correcting codes that have a large
$X$-distance but a much smaller $Z$-distance. However, together with a
classical expander LDPC code and a tensoring method that generalises a
construction of Hastings and also the Tillich-Zemor construction of quantum
codes, we obtain quantum LDPC codes whose minimum distance exceeds the square
root of the code length and whose dimension comes close to a square root of the
code length. When the ingredient is a 3-dimensional Ramanujan complex, we show
that its 2-systole behaves like a square of the log of the complex size, which
results in an overall quantum code of minimum distance $n^{1/2}\log n$, and
sets a new record for quantum LDPC codes. When we use a 2-dimensional Ramanujan
complex, or the 2-skeleton of a 3-dimensional Ramanujan complex, we obtain a
quantum LDPC code of minimum distance $n^{1/2}\log^{1/2}n$. We then exploit the
expansion properties of the complex to devise the first polynomial time
algorithm that decodes above the square root barrier for quantum LDPC codes.
