A few issues spring to mind.
Can either of you afford to buy the house and get a mortgage on your own ?
If it is your intention that your girlfriend own the house outright then will
you be contributing to the deposit, You will have to gift her the money, there may be tax implications,
Also, I know that breaking up is not in your plans and an unpleasant thing to talk about, but if you pay half the mortgage on a house you do not own, and you do break up, will you be able to get your share back ?
I am sure there are other reasons why this could be messy.