Quantum approximation algorithms
A comment upgraded to partial answer:

There is quite some work these days on a conjectured (or not) quantum version of the PCP theorem: see for example this blog post by Scott Aaronson
http://www.scottaaronson.com/blog/?p=139
or this answer by Peter Shor on MathOverflow
https://mathoverflow.net/questions/45106/quantum-pcp-theorem/45167#45167

Concerning quantum approximation alpgorithms, you can look at this reference "Approximation algorithms for QMA-complete problems" 
http://arxiv.org/abs/1101.3884