When large scale quantum computers come along, algorithms based off of the new principles will have an impact on how we encrypt data, with Shor's algorithm rendering RSA vulnerable and Grover's algorithm halving the effective keysizes for AES.
What I'm wondering is how having an actual quantum computer will affect the computation thereafter.
Using Grover's algorithm, AES-256 will be rendered as secure as AES-128 is today. But will a quantum computer be able to search that new space substantially quicker too? As such, would we need to more than double keysizes as a defence?