A model problem consisting of two isolated particles with a mutual interaction depending upon retarded and advanced positions and constrained to one-dimensional motion is analyzed herein. This system has a number of features in common with a previously considered one-particle double-delay model problem. The equations of motion have closed-form solutions and the systems are both deterministic and causal. In both cases, standard low-order approximations to the exact problem, in general, have spurious solutions, which must be recognized and removed in order to extract the physically meaningful parts and to proceed reasonably with quantization. In addition, the two-particle system has specific characteristics. The center-of-momentum motion separates out. The dependence of the solutions on the mass ratio of the two particles can be examined. Further, not only does the relative motion have the ordinary solution, in which the two particles move out of phase, it can also have an extraordinary solution, in which they move in phase with each other. Finally, the total generalized linear momentum and Hamiltonian can be evaluated and seen to be constants of the motion.
We simulate the excited states of the Lipkin model using the recently proposed quantum equation of motion (qEOM) method. The qEOM generalizes the EOM on classical computers and gives access to collective excitations based on quasiboson operators ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{O}}_{n}^{\ifmmode\dagger\else\textdagger\fi{}}(\ensuremath{\alpha})$ of increasing configuration complexity $\ensuremath{\alpha}$. We show, in particular, that the accuracy strongly depends on the fermion to qubit encoding. Standard encoding leads to large errors, but the use of symmetries and the Gray code reduces the quantum resources and improves significantly the results on current noisy quantum devices. With this encoding scheme, we use IBM quantum machines to compute the energy spectrum for a system of
N=2, 3, and 4 particles, and compare the accuracy against the exact solution. We found that the results of the approach with $\ensuremath{\alpha}=2$, an analog of the second random phase approximation (SRPA), are, in principle, more accurate than with $\ensuremath{\alpha}=1$, which corresponds to the random phase approximation (RPA), but the SRPA is more amenable to noise for large coupling strengths. Thus, the proposed scheme shows potential for achieving higher spectroscopic accuracy by implementations with higher configuration complexity, if a proper error mitigation method is applied.