I've been doing a Back to Eden style wood chip garden for the last six years, and I had a rough start for the first five years. I laid a layer of cardboard covered with 6-8" of wood chips on top. The cardboard and chips slowed the quack grass, but I ended up spending many hours on my knees rooting out quack runners while listening to podcasts. The quack roots had enough energy to push through the soggy card board and and the chips. It WAS easier to root it out.
A year ago I finally found an approach that has stopped the quack grass for me. I started by staking large sheets of lumber wrap that I pick up from our local lumber yard to a grassy area as soon as the snow melted. After giving the quack grass a good solar toasting for three months, I then covered the roasted grass with cardboard, leaves, and 3-4" of wood chips. I haven't seen a blade of grass in that area since then. I did learn that the lumber wrap MUST have the black face up facing the sun. Not all lumber wrap has a black surface, so I only selected black ones at my lumber yard.