Quantum approximation algorithms It is generally considered unlikely that quantum computers will be able to solve NP-complete problems efficiently. In the classical case one approach to tackle such problems is to use approximation algorithms. Has there been any research on approximation algorithms using quantum computing where quantumness gives significant speedup over classical approximation methods?

By "significant" I mean not necessarily exponential, but greater than for corresponding exact algorithms. In other words, I'm interested if relaxing the requirement that our algorithm yields the exact solution gives a significant advantage to quantum 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