Modified group non-membership is in AWPP
  It is known that the group non-membership problem is in QMA relative to any group oracle and in ${\rm SPP}\cap{\rm BQP}$ relative to group oracles for solvable groups. We consider a modified version of the group non-membership problem where the order of the group is also given as an additional input. We show that the problem is in AWPP relative to any group oracle. To show the result, we use the idea of the postselected quantum computing. 
Tomoyuki Morimae, Harumichi Nishimura, Francois Le Gall
quant-ph cs.CC