Variational Quantum Computation of Excited States

Oscar Higgott; Daochen Wang; Stephen Brierley

doi:10.22331/q-2019-07-01-156

Variational Quantum Computation of Excited States

Oscar Higgott^1,2, Daochen Wang^1,3, and Stephen Brierley¹

¹Riverlane, 3 Charles Babbage Road, Cambridge CB3 0GT
²Department of Physics and Astronomy, University College London, London, WC1E 6BT
³Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD 20742

Published:	2019-07-01, volume 3, page 156
Eprint:	arXiv:1805.08138v5
Doi:	https://doi.org/10.22331/q-2019-07-01-156
Citation:	Quantum 3, 156 (2019).

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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.

Popular summary

Small quantum computers will soon be available with the capability of performing some computational tasks that are beyond the reach of even the largest classical supercomputers. However, these devices will be noisy, limiting the complexity of the algorithms that they can implement correctly. We present a new quantum algorithm that can calculate the spectrum of complex quantum systems, and show how it may be used to perform useful computations even on the imperfect quantum devices available in the near- future.
The algorithm we have developed - variational quantum deflation - uses a hybrid of both quantum and classical resources to determine the excited state energies of quantum systems. We achieve this at almost no extra cost over the most promising hybrid method for determining ground state energies - the variational quantum eigensolver - by first finding the ground state and then energetically exciting it, allowing the first excited state to be found as the ground state of the new modified system. By repeating this procedure and incorporating techniques to mitigate device errors, our method can determine multiple excited states even on imperfect hardware. Our algorithm is therefore a promising candidate for useful near-term quantum-enhanced computation.

► BibTeX data

@article{Higgott2019variationalquantum,
  doi = {10.22331/q-2019-07-01-156},
  url = {https://doi.org/10.22331/q-2019-07-01-156},
  title = {Variational {Q}uantum {C}omputation of {E}xcited {S}tates},
  author = {Higgott, Oscar and Wang, Daochen and Brierley, Stephen},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {3},
  pages = {156},
  month = jul,
  year = {2019}
}

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