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?
