Incorrect weak-coupling justification in LSH hadron dynamics tutorial

[kcmccormibm]

URL to the relevant tutorial

https://quantum.cloud.ibm.com/docs/en/tutorials/loop-string-hadron-dynamics

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The tutorial incorrectly states:

In the weak-coupling regime ($x\gg1$), the dynamics is dominated by the electric term $H_E$, which favors states with large $n_l$.

From the Hamiltonian

$$ W=H_E+\mu H_M+xH_I, $$

we see that large $x$ instead makes the off-diagonal term $xH_I$ dominant. This agrees with the original paper. It is in the strong-coupling regime that $H_E$ dominates; there, flux is energetically expensive and the vacuum has $n_l=0$.

Large $n_l$ is a separate modeling choice motivated by a weak-coupling vacuum ansatz. The authors choose a large incoming boundary flux, producing a large, approximately uniform flux across the lattice. This allows flux-dependent hopping prefactors such as

$$ \sqrt{\frac{n_l}{n_l+1}} $$

to be approximated by one.

Suggested correction

In the weak-coupling regime $x\gg1$, the off-diagonal interaction term $xH_I$ dominates. The weak-coupling vacuum is modeled using a large incoming boundary flux, giving $n_l\gg1$ across the lattice. The resulting $n_l$-dependent hopping prefactors approach one, simplifying $H_I$ to nearest-neighbor hopping on the fermionic qubits.

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