Quantum Hamiltonian systems of lattice type are considered which are associated with classical finite-dimensional Lie algebras [like MH(2), SO(3), etc.], and possess complete integrability properties. The canonical Heisenberg
pn,
qn variables are found for these Hamiltonian systems and it is shown how to construct their
S matrices. These new Hamiltonians are generalizations of the quantum Toda lattice Hamiltonian as well as
XXX models of statistical mechanics.
Exactly solvable Hamiltonians are useful in the study of quantum many-body systems using quantum computers. In the variational quantum eigensolver, a decomposition of the target Hamiltonian into exactly solvable fragments can be used for the evaluation of the energies via repeated quantum measurements. In this work, we apply more general classes of exactly solvable qubit Hamiltonians than previously considered to address the Hamiltonian measurement problem. The most general exactly solvable Hamiltonians we use are defined by the condition that within each simultaneous eigenspace of a set of Pauli symmetries, the Hamiltonian acts effectively as an element of a direct sum of so(N) Lie algebras and can, therefore, be measured using a combination of unitaries in the associated Lie group, Clifford unitaries, and mid-circuit measurements. The application of such Hamiltonians to decomposing molecular electronic Hamiltonians via graph partitioning techniques shows a reduction in the total number of measurements required to estimate the expectation value compared to previously used exactly solvable qubit Hamiltonians.