This is intended as a complement to Conifold's and Jobermark' answers
Penrose's argument can be broken down to two parts:
- Based on Lucas's Gödelian argument against mechanism, he argues that the human mind is more than just a Turing machine.
- The part of the human mind that is more than Turing machines can be explained by quantum phenomena in the brain.
If we do ignore the Gödelian argument and only pay attention to the latter, how does Penrose defeat the computational theory of mind with this argument?
So strictly speaking, the part of his argument that "defeats" the computational theory of mind has nothing to do with quantum phenomena. Nor is it really a new argument, he's only reviving an argument made by Lucas in 1959 (which can be simplified as: humans can see the truth of Gödel sentences but machines can't).
"Quantumness" comes into play to fill the gap between human minds and Turing machines, only after the gap has been already been established using Lucas' reasoning.
He states that quantum processes in the microtubules of neurons are the progenitors of consciousness, but how is this a non-computational process? The Church-Turing thesis tells us the limits of effective computation.
Here you have stepped into an ongoing controversy regarding the interpretation of the Church-Turing thesis: Is the Church-Turing thesis a broad result that puts a limit on any discrete and finite algorithmic process (a)? Or is it a very strict result about a specific class of models covered by Universal Turing Machines and Church's Lambda calculus - and its totally plausible that other models of computation might "break the Church-Turing barrier" or what is now called "Hypercomputation" (b)?
The majority opinion among mainstream computer scientists (i.e. those who specialize in computational complexity, the theory of algorithms, etc...) seems to be (a). In his thesis, Turing himself wrote that "It was stated ... that 'a function is effectively calculable if its values can be found by some purely mechanical process.' We may take this literally, understanding that by a purely mechanical process one which could be carried out by a machine. The development ... leads to ... an identification of computability† with effective calculability.". As you pointed out, Bernstein and Vazirani have shown that even quantum computers can't go beyond the Church-Turing limit and start solving undecidable problems.
To the theoretical CS crowd, talk of super-Turing computation is the their field's equivalent of perpetual motion machines and cold fusion. See Aaronson's lecture notes, as well as this article by Martin Davis - who was Church's student, and this article, by Andrew Hodges, a British mathematician who wrote a book on Turing.
On the other hand you will find that many outside of mainstream theoretical CS, subscribe to (b). As Aaronson himself says, google "Hypercomputation", and you will get dozens of articles and even entire conferences on the topic. In my own field of AI and Machine Learning, you frequently heard claims of some novel fuzzy logic or neural network approach which could potentially solve problems a Universal Turing Machine couldn't. Notably, Hava Siegelmann published in Science in 1995 that neural network based analogue computation could decide Turing-undecidable languages. Her results were disputed by Peter Shor, among others, and her result doesn't seem to have gained any traction beyond her own research.
In philosophy, it gets worse, with many philosophers arguing that Church-Turing is just a thesis and that non-Turing models of computation are entirely plausible, see for example here. In particular, Jack Copeland, a British philosopher who is also supposedly a Turing specialist, wrote an entry in the SEP called misunderstandings of the thesis, which purportedly refutes (a) (which Copeland calls 'Thesis M').
It doesn't help the confusion that another SEP article on Turing, written by the aforementioned Andrew Hodges, states the opposite (i.e. (a) is true).
Personally, as someone who started out in grad school working with Neural Networks and Fuzzy Logic, I bought into the various possibilities of hypercomputation, if only because I was attracted to the notion that my chosen field might be where the next big breakthrough was going to happen. As I progressed in my Ph.D, my understanding of computability and complexity theory got deeper, and I started to see why people like Aaronson and Davis don't take the idea of Hypercomputation seriously, and I am now firmly in the (a) camp. See this post in the theoretical CS SE.
What does this mean for Penrose's argument?
If you subscribe to (b), then his argument is totally plausible, but he doesn't even have to invoke anything quantum in order to defeat CTM, there are all sorts of ways that a human mind might be more powerful than a Turing machine, we just haven't worked out the details yet. See for example Paul Churchland's analysis of Penrose's argument in Chapter 9 of "The Engine of Reason, the Seat of the Soul", and his explanation of why quantum effects don't really matter to the argument. It should be noted that Churchland's argument as to why neural nets are capable of non-Turing computing is sketchy at best, and I don't agree with it. I mention it only as an illustration of possible response to Penrose.
If you subscribe to (a), then his argument is significantly more far fetched: At the very least, his speculations on non-Turing computable quantum processes, if true, would have serious consequences for physics and even math, since he would be effectively refuting Vazirani et al.
More likely, he wants to go further, and is implying a type of non-materialist dualism, a quantum gravity based argument that leads to a modern Platonism.