I will only answer to the part of the question regarding how quantum mechanics can be useful for analysis of classical data via machine learning.
There are also works related to "quantum AI", but that is a much more speculative (and less defined) kind of thing, which I do not want to go into.
So, can quantum computers be used to speed-up data analysis via machine learning algorithms? Quoting Scott Aaronson's Read the fine print paper, that’s a simple question with a complicated answer.
It should first of all be noted that trying to answer this kind of question is a big part of what the research area of Quantum Machine Learning is about (more recently, the terms quantum-enhanced machine learning or quantum assisted machine learning seem to be preferred to refer to this merging of QM and ML, to distinguish it from the use of ML to help solve problems in QM).
As you can see from the Wikipedia page, there are many things going on in the field, and it would be pointless to try and give a comprehensive list of relevant papers here, as it would get outdated quickly.
Quoting from Schuld et al. 2014, the idea behind Quantum-Assisted Machine Learning (QAML) is the following:
Since the volume of globally stored data is growing by around 20%
every year (currently ranging in the order of several hundred exabytes
), the pressure to find innovative approaches to machine learning
is rising. A promising idea that is currently investigated by academia
as well as in the research labs of leading IT companies exploits the
potential of quantum computing in order to optimise classical machine
Going back to your question, a first seemingly positive answer was provided by Harrow et al. 2009, which gave an efficient quantum algorithm to invert linear system of equations (under a number of conditions over the system), working when the data is stored in quantum states. Being this a fundamental linear algebra operation, the discovery led to many proposed quantum algorithms to solve machine learning problems by some of the same authors (1307.0401, 1307.0411, 1307.0471), as well as by many others.
There are now many reviews that you can have a look at to get more comprehensive lists of references, like 1409.3097, 1512.02900, 1611.09347, 1707.08561, 1708.09757, Peter Wittek's book, and likely more.
However, it is far from established how this would work in practice. Some of the reasons are well explained in Aaronson's paper: Read the fine print (see also published version: nphys3272).
Very roughly speaking, the problem is that quantum algorithms generally handle "data" as stored in quantum states, often encoding vectors into the amplitudes of the state.
This is, for example, the case for the QFT, and it is still the case for HHL09 and derived works.
The big problem (or one of the big problems) with this is that it is far from obvious how you can efficiently load the "big" classical data into this quantum state for processing. The typical answer to this is "we just have to use a qRAM", but that also comes with many caveats, as this process needs to be very fast to maintain the exponential speed-up that we now can be achieved once the data is in quantum form.
I again refer to Aaronson's paper for further details on the caveats.