I want to understand the form of the intrinsic Hamiltonian of the IBM machines, such as ibmq_16_melbourne, and how this affects the data I get out when waiting using identity gates.

From the ZZ-characterisation tutorial in the documentation, it seems like the general form of the intrinsic Hamiltonian is $$ \mathcal{H} = \sum_i^N \frac{2\pi\nu_i}{2} (1-\sigma_i^z)+\sum_{ij}^N\frac{\xi_{ij}}{4}\left(1-\sigma_i^z\right)\left(1-\sigma_j^z\right),$$ with sums in theory over all $N$ qubits. Here $\nu_i\approx 5\,\text{GHz}$ and $\xi_{ij}$ seem to take values up to $\sim 100\,\text{kHz}$, but only significantly above $0$ for nearest neighbours. Is this correctly understood?

Now, if I prepare an initial state $\frac{1}{\sqrt2}\left(|0\rangle +|1\rangle\right)\otimes|0\rangle^{N-1}$ I do not see oscillations with a frequency anywhere near $5\,\text{GHz}$, but rather something closer to $100\,\text{kHz}$. Does this mean that the measurements are being taken in the co-moving frame with frequencies for each qubit being $\nu_i$? If so, why do I have these residual oscillations?


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