Simulating excited states of the Lipkin model on a quantum computer

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 Ô † n (α) of increasing configuration complexity α. 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 α = 2, an analog of the second random phase approximation (SRPA), are, in principle, more accurate than with α = 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.

Quantum Simulations in Effective Model Spaces (I): Hamiltonian Learning-VQE using Digital Quantum Computers and Application to the Lipkin-Meshkov-Glick Model

“…The separation of scales in realistic systems means that the ideas developed within the LMG model have also been helpful in those systems. Its rich phenomenology from a simple Hamiltonian provides sufficient complexity that has led to a number of previous studies that explore quantum correlations and entanglement [33][34][35][40][41][42], quantum algorithms [43][44][45][46], and more, to develop understanding and techniques that can be applied to quantum simulations of nuclei and multi-nucleon systems. Previous quantum simulations of the LMG model [43][44][45][46] have directly mapped the elementary SU(2)-spaces associated with each fermion to qubits in the quantum computer (or classical simulator).…”

“…Its rich phenomenology from a simple Hamiltonian provides sufficient complexity that has led to a number of previous studies that explore quantum correlations and entanglement [33][34][35][40][41][42], quantum algorithms [43][44][45][46], and more, to develop understanding and techniques that can be applied to quantum simulations of nuclei and multi-nucleon systems. Previous quantum simulations of the LMG model [43][44][45][46] have directly mapped the elementary SU(2)-spaces associated with each fermion to qubits in the quantum computer (or classical simulator). In this way VQE has been used to determine the ground state of few-nucleon systems in the model [43] using IBM's quantum computers [47], and ADAPT-VQE [48,49] has been used to examine systems of up to N = 12 nucleons using a classical simulator [45].…”

“…The authors also applied their techniques to valence model spaces of nuclei in the sd-and pf-shells, with interesting results, concluding that there could be substantial benefits to applying such quantum algorithms to nuclear structure calculations. One of these studies [46] also considered a number of other mappings, including the J-space mapping (that we utilize in this work) and binary encoding of the occupation number with consideration of the efficient Gray code, with good results obtained for the ground states, and also excited states, for small systems.…”

A quantum-classical co-processing protocol towards simulating nuclear reactions on contemporary quantum hardware

“…Furthermore, we exclusively focus on the Hamiltonian propagation and not on the initial state preparation problem. Different techniques [20][21][22][23][24][25][26][27][28] can be preliminarily employed to achieve a desired physical initial state.…”