Variational Quantum Computation of Excited States

TLDR

Excited‑state energies of electronic‑structure Hamiltonians are crucial for optical spectra and reaction rates, yet current methods rely on high‑depth controlled‑unitaries or many samples, unlike low‑depth VQE used for ground states. The study demonstrates that overlap estimation can deflate found eigenstates to compute excited‑state energies and degeneracies. The authors propose a near‑term implementation using the same qubit count as VQE, at most twice the depth, robust to control errors, and compatible with error‑mitigation.

Abstract

The calculation of excited state energies of electronic structure Hamiltonians has many important applications, such as the calculation of optical spectra and reaction rates. While low-depth quantum algorithms, such as the variational quantum eigenvalue solver (VQE), have been used to determine ground state energies, methods for calculating excited states currently involve the implementation of high-depth controlled-unitaries or a large number of additional samples. Here we show how overlap estimation can be used to deflate eigenstates once they are found, enabling the calculation of excited state energies and their degeneracies. We propose an implementation that requires the same number of qubits as VQE and at most twice the circuit depth. Our method is robust to control errors, is compatible with error-mitigation strategies and can be implemented on near-term quantum computers.