There are many different kinds of cat litter, so it really depends what type of kitty litter you have got and want to compost.
For clay-based cat litters or sand-based cat litters, I would say don’t try to compost them. The same goes for various crystalline types of cat litters. They are not natural products made from living materials. They are manufactured, often synthetics which will not break down like normal compostable material. They could potentially leech chemicals into your compost which could actually be detrimental to the composting process or harmful to the plants you would ultimately feed with your compost.
So probably a good rule of thumb is that if the cat litter is not made from an entirely natural product, don’t compost it.
Natural cat litters might include products made from sawdust, pine or cedar chips, recycled newspaper or wheat husks. All of these products are originally sourced from natural products, living plant based material, and ultimately they will break down into compost.
The problem becomes what is in the cat litter, your cat’s waste. Cat faeces are a nasty, smelly by-product of owning a cat. Like dog waste, they can contain dangerous bacteria or viruses, and special care should be taken when handling.
Personally I would advise against trying to compost these waste products. But if you insist on doing this, make a special compost pile or use a separate compost bin. Pile up alternate layers of used cat litter and clean cat litter or sawdust.
Turn or aerate the compost every couple of months.
Due to the waste products contained in the cat litter, I would recommend leaving it for around 18 months to fully rot down before attempting to use the composted cat litter. And even then, I would only suggest using it as a top dressing for trees and shrubs. I would not mix it in to growing material for potatoes or other vegetables that may touch the compost in the ground.