Landscape Analysis of Excited States Calculation over Quantum Computers

TL;DR

This paper analyzes and compares three variational quantum eigensolver models designed for excited state calculations, focusing on their landscape properties, orthogonality constraints, and resource requirements, advancing quantum chemistry computations on NISQ devices.

Contribution

It introduces and theoretically analyzes three VQE models with orthogonality constraints that ensure global optimality, providing insights into their landscape and resource efficiency.

Findings

All models have no local minima other than the global minimum.

The models effectively maintain orthogonality without external enforcement.

Resource and optimization complexities vary among the models.

Abstract

The variational quantum eigensolver (VQE) is one of the most promising algorithms for low-lying eigenstates calculation on Noisy Intermediate-Scale Quantum (NISQ) computers. Specifically, VQE has achieved great success for ground state calculations of a Hamiltonian. However, excited state calculations arising in quantum chemistry and condensed matter often requires solving more challenging problems than the ground state as these states are generally further away from a mean-field description, and involve less straightforward optimization to avoid the variational collapse to the ground state. Maintaining orthogonality between low-lying eigenstates is a key algorithmic hurdle. In this work, we analyze three VQE models that embed orthogonality constraints through specially designed cost functions, avoiding the need for external enforcement of orthogonality between states. Notably, these formulations possess the desirable property that any local minimum is also a global minimum, helping address optimization difficulties. We conduct rigorous landscape analyses of the models' stationary points and local minimizers, theoretically guaranteeing their favorable properties and providing analytical tools applicable to broader VQE methods. A comprehensive comparison between the three models is also provided, considering their quantum resource requirements and classical optimization complexity.