# Can we easily find the ground states for one-dimensional ANNNI-like Ising models?

The Hamiltonian for a simple one-dimensional Ising model on a finite (linear) chain of $$L$$ spin-half particles might be:

$$H = -J \sum_{i=0}^{L-1} \sigma_i^z \sigma_{i+1}^z.\tag{1}$$

The interactions are between adjacent qubits in the lattice. Such a model is relatively boring; the interaction $$J$$ is constant for each pair of neighbors, and the ground states are pretty straightforward to describe.

But we can spice it up in several different ways:

• We can break some symmetry and add boundary conditions at $$i=0$$ and/or $$i=L-1$$;
• We can add external magnetic fields, either longitudinal or transverse (or both); or
• We can start to consider not just nearest neighbors $$i,i+1$$ but also next-nearest neighbors $$i,i+2$$ - and next-next nearest neighbors $$i,i+3$$, etc.

This last model is where my interest mostly lies. Naturally we can write the Hamiltonian as:

$$H(\sigma) = -J_1 \sum_{i=0}^{L-1} \sigma_i^z \sigma_{i+1}^z-J_2 \sum_{i=0}^{L-1} \sigma_i^z \sigma_{i+2}^z-J_3 \sum_{i=0}^{L-1} \sigma_i^z \sigma_{i+3}^z\cdots.\tag{2}$$

When the maximum interaction distance of two ($$J_2\ne 0$$ but $$J_3,\ldots, J_{L-}=0$$), this has been referred to as an ANNNI model, for anisotropic next-nearest neighbor Ising interaction. There might be some neat and dynamic frustration going on in such a model; indeed it's not clear to me what the behavior is when $$J_1\lt 0\lt J_2$$ for example, even absent any magnetic field.

Perhaps... such a problem may be amenable to a QAOA-like algorithm.

• Given $$J_1,J_2,J_3,\ldots, J_{L-1}$$, can we say anything about the conditions for which a quantum computer could efficiently find eigenstates/ground states of such a next-next-next... nearest neighbor spin-chain?

For example is a next-next-next... Ising interaction amenable to characterization with an ion-trap based quantum computer?