I'd forget about that, only version would be, if you could get a collective pitch quad and power it with one turbine.
Turbines in general are very slow on rpm-changes... with the quads manouverability relying alone on rpm changes, it will be very, very complicated to set a turbing quadro up...
Yes, four control vanes at arms extremity, and a central air exhaust for main thrust. This is certainly done like this inside some military drones.
Seems not really complicated to do.
Nevertheless there could be some differences and code adaptation.
Another interesting setup is collective pitch using 4 electric motors (or 6 for Hexas). Should give faster response time, less power consumption, and the possibility to reverse thrust for acro fly.
Again the code should be slightly adapted for this.
The central motor setup for collective pitch seems complicated to me, unreliable, and weight costly.
I thought about using a turbine to power a generator and then using the generator to chargo a lipo, which powers the quads brushless motors.
This way you get the reaction and burst power of the brushless, but the power from the turbine.
But I doubt there is any weight gain from carrying the turbine fuel over just plain batteries. Those turbines suck a lot of gas.
As previously stated... the transient response of a turbine is complete crap. They just don't spin up and down fast enough to make appreciable differences in thrust in a short amount of time.
If you were dead set on using 4 turbines... I'd instead look at variable geometry turbos for inspiration.
As you can see here, the area that the gases pass through can be radially changed by rotating the external collar.
Clearly you'd have to manufacture your own rig to accomplish this since turbine engines are inline and a turbo is designed to sling air perpendicular to the axis of rotation.
Maybe vectoring could be employed as an option too.
Food for thought.
You can do this with a simple methanol 2 strokes engine. Really less expensive.
Could be interesting for long fly, but quite noisy... I don't think that the fuel motor and methanol weight is a problem. Electric motors used on multi are very powerfull.
An advantage with methanol it that you can recharge in a few seconds :=) even during flight, perhaps with another multi :=)
I'm not sure you could keep lipo, they need to be cold to be charged again. The generator should be followed by a rectifier and a regulation circuit. Eventualy a lipo could be added for safety, if the main fuel engine stop.