In ANY weight loss program that you follow, hair loss often occurs at the same time. This is a natural result and the hair will grow back in once you reach your target weight. Often the hair is even thicker than before!
In fact, compared to other diets, the low carb diet is BEST for your hair. Three key vitamins to take to help your hair are protein, iron and zinc - and these are in high supply on a low carb diet. Note that these vitamins will not STOP the temporary hair loss while you diet. The hair loss is a normal part of your body's reaction to losing the weight. However it will help the hair grow in more fully when you have reached your target.
Two things that impede hair growth are an excess of Vitamin A, and stress. So a way to reduce the hair loss situation is to find ways to relax in your life, and to make sure that your vitamin A intake is not overabundant.
First, It's good to note that a hair doesn't simply grow forever. Each hair on your head grows for 4-5 years, then falls out. So your hair is always renewing itself. The long growth stage is called anagen and the short stopping phase that follows is called catagen. Then the hair falls out, which is telogen. The hair follicle then goes back into anagen and makes a long hair for the next 4-5 years.
What happens during weight loss is Telogen effluvium. It also happens after any stress to the body. Some of your hair follicles go into catagen to conserve body resources. So the hairs stop growing and "rest". Once your body feels safe again, those hairs go back into anagen - but because there was the pause in the middle, the brand new growing hair pushes the old hair out of the follicle. So in reality you have just as much hair as you did before - and it's probably far healthier hair. But the new hairs are short :)
Normally around 10% of your head of hair is in the catagen or telogen phase, in essence resetting itself to prepare for new hair to grow in. Because you have so much hair on your head, you don't even notice. During weight loss that percentage gets higher, but this is only temporary. Your body starts creating new hair in its normal cycle once it settles into a healthy weight and those hairs start growing again. Remember that hair doesn't reach a long state overnight! So all those new, healthy hairs need time to grow out to whatever length of hair you have chosen.
Usually it takes 6 months for your head of hair to have grown back out to typical fullness. Hair grows around 1/2" per month, so at the 6 month mark, the hair is around 3" long which is enough to seem thick on most heads. Again, it's not that the hair is MISSING during this intermediate time. It's just that it is brand new and growing from scratch.
DermNetNZ.org on Telogen effluvium
InteliHealth.com on Telogen effluvium
Lisa Shea's Library of Low Carb Books