Can Quantum Computers test modified quantum mechanics? I wonder if with the raise of the quantum computing era, we could test somehow non-linear quantum mechanics failures up to certain scales. That is, how could quantum computers test key assumptions of quantum mechanics?
1st. Linearity/superposition.
2nd. Entanglement.
3rd. Contextuality (QM is contextual, but how to test contextuality failures?)
4th. Uncertainty principles/modified uncertainty principles.
5th. Unitarity.
In other words, can modified by nature quantum computers behave non-linearly, unentangledly, non-contextually, certainly and non-unitarily? Are there experimental set-ups to probe those features or how could a quantum computer show one single non-stantard quantum computing behaviour?
Also, I wonder if the failure of success of any particular quantum algorithm could show hints of "ultra-quantum" theories beyond standard quantum mechanics, or to weak some hypotheses/axioms of current quantum mechanics postulates.
 A: Of course! Given that quantum computers assume standard quantum mechanics in their evolution, any evolution deviating from predictions (for a perfect quantum computer) would signify a deviation from standard quantum mechanics.
One example could be testing commutation relations such as $[\hat{x},\hat{p}]=i\mathbb{I}$ (in units of $\hbar=1$, with the identity operator $\mathbb{I}$) on a continuous-variable quantum computer (I take this example from the end of this paper). One can perform a series of displacements around a closed curve in phase space, which should just lead to the addition of a phase factor, such as
$$U=e^{iX\hat{p}}e^{iP\hat{x}}e^{-iX\hat{p}}e^{-iP\hat{x}}.$$ With canonical commutation relations, the unitary is simply a phase factor $U=e^{-iXP}$, which can be found using the Baker-Campbell-Hausdorff formula. With a modified quantum mechanics that leads to the non-canonical commutation relations
$$[\hat{x},\hat{p}]=i(\mathbb{I}+\beta\hat{p}^2),$$ we instead find
$$U=e^{-iXP}e^{-i\beta X(P\hat{p}^2+P^2\hat{p}+P^3/3)}+\mathcal{O}(\beta^2).$$
When the initial state is anything but a momentum eigenstate, standard quantum mechanics predicts no difference between the initial and final state; measuring a difference between them signifies the presence of modified quantum mechanics. Or, one could attempt to prepare a momentum eigenstate and interfere this unitary with some other unitary in a controlled operation such that two branches of the wavefunction acquire a relative, $\beta$-dependent phase that can then be measured.
In summary, the answer is always yes, but one has to be clever in designing a particular experiment to test a particular law of quantum mechanics. This is especially tricky because one always has to worry about decoherence and other effects that also force quantum computers to deviate from the predictions of standard quantum mechanics.
A: Absolutely! Unlike previous devices, a working quantum computers can test quantum mechanics in a fundamentally new regime: high fidelity wavefunctions with high complexity. High complexity means that the wavefunctions is not just highly entangled, but also requires deep quantum circuits (or a long amount of time evolution by a Hamiltonian) to prepare. High fidelity means that there is very little imprecision in the prepared wavefunction and the following measurements. Condensed matter experiments (e.g. on superconductors) can have high complexity, but the fidelity is low.
One can thus test quantum mechanics in this relatively-unexplored high-complexity regime using quantum computers. For example, in addition to showing evidence of a quantum advantage, the Google and USTC quantum supremacy/advantage experiments also tested quantum mechanics in the high-complexity regime (significantly but not completely verifying assumptions 1, 2, and 5 in the question) by applying deep quantum circuits and comparing the expected outcome to theory. However, present-day quantum computers usually don't actually measure the ideal output since there is a finite (~1%) chance that each operation will make an error, which means many operations will likely lead to errors. However, their experiments successfully checked that the number of correct measurements scaled as expected as the circuit depth increased.
As quantum computer gate fidelity and qubit count increases, better experiments will be possible. But different experimental protocols are necessary, since predicting the output of quantum circuits with many qubits (which was necessary for the quantum advantage experiments) is not feasible using classical computers. However, one can instead run deep quantum circuits on a quantum computer for which the outcome is easy to predict, and use this approach to test quantum mechanics using future quantum computers. This was proposed in this preprint, which also argued that such experiments can rule out the possibility that quantum mechanics emerges from classical mechanics. However, although this kind of experiment can be practically performed on quantum computers of the near-future, even more stringent tests of quantum mechanics will be possible with error-corrected quantum computers.
There is also a lot of research that addresses how one can verify if a quantum computer is working correctly; see this paper for a review. But verifying that a high-fidelity quantum computer with many qubits is working correctly also imposes stringent tests on quantum mechanics! As quantum error correction allows quantum computers to perform very deep circuits with high-fidelity, it will be possible to obtain cryptographically-strong evidence that quantum mechanics is correct using "interactive experiments". See this paper for a review, although there has been significant progress since.
To understand the basic idea of an interactive experiment in a classical (i.e. not quantum) example: suppose you want to verify that your friend can accurately count the number of leaves on a tree. But if you just ask how many leaves there are, how do you verify the result without counting all of the leaves yourself? But what if after asking your friend for a leaf count, you then pull off a few leaves from the tree while your friend isn't looking. Then you can ask your friend again and check that the leaf count decreased by the correct number.
