I'd go for Quality Center as TD is no longer supported.
Take some time and read up on BPT from the HP site, and see if it will give you an advantage.
You don't have to use a test case repository tool to automate, but it depends on your overall situation if it is a good step for you or not to incorporate a tool of that manner. If you are using QTP already, then QC is the most logical choice.
Have you taken the time to determine your specific needs of a test case repository tool. Have you researched the pros and cons of integrating QC into your processes?
basically ask your self these questions:
What will happen if I do?
What will happen if I don't?
What won't happen if I do?
What won't happen if I don't?
Just to add to Tony's response, Quality Center is entirely optional when it comes to automation. You can fully automate your applications with QuickTest alone. Quality Center provides a central repository for you to store your tests and organize test sets for execution, but that is not a requirement to implement automation with QTP. The only time you need QC is if you want to use the BPT feature for component-based design.