I visit AZ every summer and include some hikes in the desert when I'm there.
First, don't just take water. That can lead to electrolyte loss, as actually happened to me the first year I did this. Now I dissolve some gatorade powder in the water, and I haven't had that problem since. I usually mix it 1/2 to 2/3 strength relative to what the directions say. The first thing I do when I get to AZ is to buy one of those large tubs of gatorade powder. I go thru most of it in the 10 days or so I'm out there.
I don't like carrying more than 1 gallon of water, so I plan my hikes accordingly. Depending on where exactly and the weather that day, that can be good enough for a all-day hike of 8 hours or more, or only 4 hours in some cases. In cases where I know the conditions are severe, I either plan to be out no more than 4 hours or I'm going out and back along the same route and turn around when the container gets half full.
A good example of the severe case is when I went hiking in the southern White Tank mountains west of Phoenix. (I recommend avoiding the White Tanks, too many yahoos with guns, dirtbikes, and trash everywhere, but I didn't know that at the time.) It was a hot day, even for that location in August. I drank all I could stand before leaving, and took a gallon of diluted gatorade. When that got down to 1/2 gallon, I turned around and went back to the car. I only got a few miles in, and I think the whole hike lasted only about 3 hours. I had a little liquid left when I got back to the car, which I then drank and felt about right.
Other times at somewhat higher elevations and a bit cooler temperatures, I've gone the whole day on one gallon. One example was a whole-day hike around Juniper Mesa in the Prescott NF.
So in the end it matters a lot where you're going and what the conditions are in that location on that day. For the real serious desert in the heat of summer, I'd say a gallon per 3-4 hours, then derate according to weather from there.