In Challenge 3 which investigates the many-body localization phenomenon, the tight-binding Hamiltonian used was $H_\text{tb}/\hbar = \sum_i(X_i X_{i+1} + Y_i Y_{i+1}) + \sum_i \epsilon_i Z_i$. However, this Hamiltonian does not include particle interaction. It should be evident by doing a Jordan-Wigner transformation to map it to a fermionic Hamiltonian, which leads to $H_\text{tb}/\hbar = \sum_i (c_i^\dagger c_{i+1} + \text{h.c.}) + \sum_i \epsilon_i c_i^\dagger c_i$. This is a free fermion Hamiltonian because there is no two-body interaction term like $n_i n_{i+1}$ present, where $n_i = c_i^\dagger c_i$.

In fact, the paradigmatic spin-chain Hamiltonian used to study many-body localization in the literature is the Heisenberg model. One example would be the XXZ model, where an additional term $\sum_i Z_i Z_{i+1}$ is included to give exactly the two-body interaction term stated above. Under the section "Many-body quantum dynamics" in the Challenge notebook, it is stated that "Under $H_\text{tb}$, each site can only be occupied by a single particle, resulting in particle repulsion interaction". But a priori, there is nothing forbidding two particles occupying the same site under such Hamiltonian (without interaction), I think.

So in summary, I think the Hamiltonian used in this problem does not include particle-particle interaction and hence is not suited for studying many-body localization. Everything we saw in this Challenge problem is still the result of Anderson localization. Please correct me if I am wrong.

Hi, are there solutions to the problems anywhere? Now that the challenge is over, it would be useful for educational purposes to have the solutions available.

The Qiskit Slack workspace link in README.md is broken (gives This link is no longer active To join this workspace, you’ll need to ask the person who originally invited you for a new link.)

Hello,
I am not sure if that is common to all participants, but I do not have the notebook for the exercise 4.