Simulating excited states of the Lipkin model on a quantum computer

TL;DR

This paper demonstrates simulating excited states of the Lipkin model on quantum computers using the qEOM method, highlighting the importance of encoding schemes and symmetry utilization for accuracy on noisy devices.

Contribution

It introduces a quantum simulation approach for Lipkin model excited states using qEOM, emphasizing encoding strategies and symmetry use to improve results on current quantum hardware.

Findings

Encoding schemes significantly affect accuracy.

Symmetry and Gray code improve quantum resource efficiency.

Higher configuration complexity can yield more accurate spectra.

Abstract

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 quasi-boson operators $\hat{O}^\dagger_n(\alpha)$ of increasing configuration complexity $\alpha$. 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 $\alpha = 2$, an analog of the second random phase approximation (SRPA), are, in principle, more accurate than with $\alpha = 1$, 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.