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.
Highlights
Eigenvalue problems are ubiquitous in almost all fields of science and engineering
We have introduced a new method–variational quantum deflation (VQD)–for calculating low-lying excited state energies of quantum systems using a quantum computer
Our method requires the same number of qubits as the variational quantum eigensolver (VQE) for ground state methods, at most twice the maximum circuit depth and a negligible increase in the number of required measurements
Summary
Eigenvalue problems are ubiquitous in almost all fields of science and engineering. Google’s PageRank algorithm alone has had a significant impact on modern society, and at its core solves an eigenvalue problem associated with a stochastic matrix describing the World Wide Web [28]. Modifications have been suggested to enable VQE to find excited state energies: e.g. a folded spectrum method [30] which requires finding the expectation of the squared Hamiltonian with quadratically more terms, or symmetrybased methods which are non-systematic [23]. Such suggestions have been more recently superseded by two proposals: a method that minimises the von Neumann entropy [35] and the quantum subspace expansion method [5, 24]. Exploiting further the fact that VQE retains the classical parameters of ansatz states that enable their repreparation, low-depth quantum circuits can be readily used to calculate these overlap terms
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