Perhaps in qubit computing (speculating as I am not an expert), it is harder to detect the difference between the states which are somewhere in between 1 and 0, leading to the needs for error correction, but I also thought that, that particular property of qubit computing was interesting for its unpredictability, if it could be harnessed. Since a lot of machine learning models (example, Monte Carlo) use randomness, I always wondered how lack of precision might be an advantage. Out of my level of knowledge here a bit if anyone has any thoughts.
I had to look it up and, I'm not a quantum expert, but it seems my gut feeling was basically right:
"Similar to the qubit, the qutrit is the unit of quantum information that can be realized in suitable 3-level quantum systems. This is analogous to the unit of classical information trit of ternary computers. Note, however, that not all 3-level quantum systems are qutrits. The term "qu-d-it" (quantum d-git) denotes the unit of quantum information that can be realized in suitable d-level quantum systems." 
As for Monte Carlo with respect to ML, are you referring to the “random walk” aspect when you say “use randomness.” This refers to the fact that the method samples one possible sequence of events and uses that sequence to update its model. As opposed to dynamic programming methods where the value of all possible sequences is estimated and used to update the model. Not sure that the randomness from quantum is useful here. Where it could be useful is to encourage exploration of the state space. So, normally Monte Carlo methods have various tricks to make sure every sequence is sampled infinitely many times. These tricks could be implemented by the error in quantum computing, but I don’t know enough about the field to really be sure of that. And of course it would help in that you could compute the results of many sequences at once which is critical bottleneck in dynamic programming.
I just always observed that quantum computing was interesting because certain states seemed unpredictable, which is why scientists studying quantum computing struggle when they don't get the expected states. If that unpredictability is actually random due to the function of quantum states, then it would provide a unique tool since true randomness really is not easily found.
And the ability to potentially generate many random/expected states quickly - yes, I was thinking about being able to generate/explore the space faster. Or to use that in logical processing somehow as an advantage.
I have always been fascinated by a simple concept in logic - "maybe". I'm borrowing from "fuzzy logic" a bit here and twisting it the way I wish for my thought process.
If one formalizes "yes","no" and "maybe" (i.e. 0, 1 and something else), can one then make complex logical statements involving probabilities, randomness, etc. by exploiting the "maybe" case.
For example, if I have one of those magic eight-balls, but it represents quantum states - if I can clearly detect 1 or 0 but sometimes I get a 0.5 or a 0.8 unpredictably, how could I use that to my advantage...
Another line of thinking for this is neural networks, another thing which I wonder could be accelerated by quantum computing. If I make the analogy that, as a human with a brain that is like a neural network, when I am faced with inputs that I have not processed before, my brain has to come up with a solution to process it from my various experiences and it might product an output that makes sense, or it might not.
Knowing my own fallibility here, if I know that sometimes I am faced with inputs that I don't know how to process or didn't expect, I can find a logic for how to handle those situations that might let me work around them.
'Three logicians walk into a bar.
The bartender asks: "Do you all want a drink?"
The first logician says: "I don't know."
The second logician says: "I don't know."
The third logician says: "Yes."'
The name quantum teleportation brings to mind a technology out of Star Trek, where “transporters” can “beam” macroscale objects—even living humans—between far-distant points in space. Reality is less glamorous. In quantum teleportation, the states of two entangled particles are what is transported—for instance, the spin of an electron. Even when far apart, entangled particles share a mysterious connection; in the case of two entangled electrons, whatever happens to one’s spin influences that of the other, instantaneously.
This is a very lazy way of saying what happens in the experiment and what eventually leads to confusion. "influences" is the most problematic word here.
A non-incorrect statement would be, "the outcome of measurements are found to be correlated when the two distant observers meet at a future time and compare notes."
I think at our current understanding of the philosophy of physics, saying more is not something most people should be attempting.
Of course, quantum teleportation works, it's just not FTL, and the spooky connection between entangled particles can't be used to send information either.
Your parent thought that there could be one (“it seems like information would travel faster than light in this case”) and your response (“specifically, it violates ...”) was somewhat confusing.