If we assume that both computers and the human mind are merely physical, does it
As Richard points out, logically, no, it does not follow. Just because two things are both (merely) physical, it does not follow that one of them can do anything that the other can do, not even if both of the (merely) physical things are brains. My pencil is a physical thing, but it can't do everything that my brain can. A cat's brain is physical, but it can't do everything that mine can. (Of course, mine can't do everything a cat's brain can either: I don't usually land on my feet when I jump from a height, and I'm pretty bad at catching mice.)
But I think your question really is simply whether a sufficiently advanced computer can do anything that a human brain can. Even so, we need to be a bit more precise. By "anything", I'm guessing that you really mean "anything cognitive"; so, I think your real question is a version of: Can computers think?
Philosophers, cognitive scientists, and computer scientists disagree on the answer to that question. I think that one of the best ways to think about how to answer it is this: How much of (human) cognition is computable? In other words, how much of the cognitive things we do (like think, reason, use language, plan, learn, remember, and so on) can be done by (or, more weakly, simulated by) a computer?
If the answer is that all of it is computable, then the answer to your question (as I'm reinterpreting it) is "yes". If the answer is that only some of it is computable, then it will be interesting to see which things are not computable, and why they aren't. But a very great deal of human cognitive activity has been shown to be computable (at least in part), so we can be hopeful that the real answer is that all of it is.
There has been a lot written on this topic. A good place to start is with Alan Turing's classic paper, "Computing Machinery and Intelligence"; there's an online version here
To find out what researchers in artificial intelligence (AI) have accomplished, visit the AI Topics website