I have just upgraded the computer I use with my turbo trainer to an Acer Revo One (A PC equivilant of a Mac Mini) and it works fine with Zwift. Core i5 processor with embedded Intel HD5500 graphics. I was lucky to get one at half price from ebuyer, but they look to have sold out of that model. It comes with a remote that combines a keyboard and mouse – while all the reviews critise the remote it is surprisingly useful when on the trainer. Going down this route you would also need a screen, and obviously it isn’t as portable as a laptop.
My old laptop couldn’t run Zwift due to the graphics not being up to it.