# Are there interesting applications of estimating the energy for generalizations of the Heisenberg model?

I consider that $$H$$ is a Hamiltonian describing a quantum system of $$n$$ spin-1/2 particles (or qubits). I assume it can be written as (the $$\alpha_k^i$$ are real coefficients):

$$\tag{1}H=\sum_{i=1}^3 \sum_{k=1}^{\textrm{Poly}(n)} \alpha^{i}_k P_i^{(k)},$$

where each $$P_i^{(k)}$$ for $$i=1,2,3$$, is a tensor product of $$\sigma_0\equiv\mathbb{I}$$ and $$\sigma_i$$ single spin Pauli operators. For instance, for $$n=3$$, $$P_1^{(1)}=X \otimes X \otimes \mathbb{I}$$ and $$P_1^{(2)}=X \otimes X \otimes X$$ are allowed but $$X \otimes \color{red}Z\color{black} \otimes X$$ is not (because it mixes $$X$$ and $$Z$$). $$\textrm{Poly}(n)$$ means that the number of terms in my sum is a polynomial in $$n$$.

Let's assume that a quantum computer can find a good estimation of $$\langle \psi | H | \psi \rangle$$ for an arbitrary $$| \psi \rangle$$ in a short amount of time. What are some interesting applications I could do with this?

More precisely, I am aware of the Heisenberg models which belong to this category for instance. But I am interested in particular in models where the $$P_i^{(k)}$$ ideally acts on all the $$n$$ qubits (the Heisenberg model has the $$P_i^{(k)}$$ that acts on $$2$$ qubits maximum). If they do not act on all, they could act on a set of maybe $$\log(n)$$ qubits or $$\textrm{Poly}(n)$$ qubits for instance (I basically want it to be the case that when we increase the size of the Hamiltonian, the Pauli operators act on an increasing portion of the qubits). Are there interesting problems (i.e. that would have applications for quantum chemistry or simulation of materials) that one could find in such case?

It is probably worth mentioning that, in the context of strongly correlated electron system / antiferromagnetism / superexchange, the spin-spin interactions arise from perturbation theory. In the textbook case we consider a Hubbard Hamiltonian, $$H = t \sum_{\langle i,j\rangle,\sigma} \left( c_{i\sigma}^\dagger c_{j\sigma} + \mathrm{H.c.} \right) + U\sum_i n_{i\uparrow}n_{i\downarrow},$$ at half-filling and near the $$U\rightarrow\infty$$ limit. To second order in $$t/U$$, one obtains a Heisenberg model with strength $$J\propto t^2/U$$. However, if the expansion is continued to higher order in $$t/U$$, one also finds terms that are products of more than two spin operators. Although such terms are often neglected in practice, there are cases where they cause interesting physics. In particular this tends to happen with cyclic ring- or plaquette exchanges, where the product involves all sites around some lattice motif, e.g. triangular or rectangular plaquettes. See e.g. A. A. Nersesyan and A. M. Tsvelik, Phys. Rev. Lett. 78, 3939 (1997) [arXiv version], T. Cookmeyer, J. Motruk, and J. E. Moore, Phys. Rev. Lett. 127, 087201 (2021) [arXiv version] for some inspiration.

Another direction could be fracton models, which tend to involve products of relatively large (but intensive) numbers of spin operators. The so-called X-cube model features terms with 12 operators, for example. Fracton models (especially those where the products feature many sites) are more interesting from a theoretical physics perspective than a materials one, however. One reference is S. Vijay, J. Haah, and L. Fu, Phys. Rev. B 94, 235157 (2016) [arXiv version].

But I am interested in particular in models where the $$P_i^{(k)}$$ ideally acts on all the $$n$$ qubits (the Heisenberg model has the $$P_i^{(k)}$$ that acts on $$2$$ qubits maximum). If they do not act on all, they could act on a set of maybe $$\log(n)$$ qubits or $$\textrm{Poly}(n)$$ qubits for instance (I basically want it to be the case that when we increase the size of the Hamiltonian, the Pauli operators act on an increasing portion of the qubits). Are there interesting problems (i.e. that would have applications for quantum chemistry or simulation of materials) that one could find in such case?

Unfortunately, that just is not a very realistic scenario in materials where spin interactions are local and ultimately come about due to 2-body Coulomb interactions that decay with distance. Of course, there is some theoretical literature on fully connected (i.e. featuring interactions connecting all sites to all) spin systems with $$p$$-body interactions, see e.g. T. Jörg et al., Europhys. Lett. 89 40004 (2010) [arXiv version]. This presents many mathematical physics questions, but unless $$p\ll n$$ it seems unlikely to me that such Hamiltonians would be used to model realistic materials.

When you have more than two Pauli operators in a single term, you can model multipolar terms, as described in this paper:

... multipolar correlators of order $$p=1$$ (spin, dipolar), $$p=2$$ (quadrupolar), $$p=3$$ (octupolar), and $$p=4$$ (spin, hexadecupolar) $$\langle S_0^+S_r^-+\text{h.c.}\rangle=2\langle S_0^x S_r^x\rangle+2\langle S_0^y S_r^y\rangle\to 2S^2\cos^2(\nu)\cos(qr)$$

$$\langle S_0^+S_1^+S_r^-S_{r+1}^-+\text{h.c.}\rangle\to 2S^4\cos^4(\nu)\cos(qr)$$

$$\langle S_0^+S_1^+S_2^+S_r^-S_{r+1}^-S_{r+2}^-+\text{h.c.}\rangle\to 2S^6\cos^6(\nu)\cos(qr)$$

$$\langle S_0^+S_1^+S_2^+S_3^+S_r^-S_{r+1}^-S_{r+2}^-S_{r+3}^-+\text{h.c.}\rangle\to 2S^8\cos^8(\nu)\cos(qr)$$

Several examples of compounds for which multipolar terms are sometimes used (usually due to there being especially strong spin-orbit coupling) are listed here.

However, most, if not all of these "uniform" k-local terms can be quadratized efficiently, meaning that your model might not provide any additional value compared to an ordinary Heisenberg model, from a computational complexity perspective.

If you have a quantum computer that's powerful enough to do this type of calculation fast, then I would recommend spending energy on figuring out how to use it for solving more realistic models such as the molecular Hamiltonian which quantum computing advocates have been promising to do for a long time now!