Variational Quantum Computation of Excited States

Oscar Higgott1,2, Daochen Wang1,3, 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

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.

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.

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[1] Tyson Jones, Suguru Endo, Sam McArdle, Xiao Yuan, and Simon C. Benjamin, "Variational quantum algorithms for discovering Hamiltonian spectra", Physical Review A 99 6, 062304 (2019).

[2] Oleksandr Kyriienko, "Quantum inverse iteration algorithm for near-term quantum devices", arXiv:1901.09988.

[3] Joonho Lee, William J. Huggins, Martin Head-Gordon, and K. Birgitta Whaley, "Generalized Unitary Coupled Cluster Wavefunctions for Quantum Computation", arXiv:1810.02327.

[4] Ken M Nakanishi, Kosuke Mitarai, and Keisuke Fujii, "Subspace-search variational quantum eigensolver for excited states", arXiv:1810.09434.

[5] Kosuke Mitarai and Keisuke Fujii, "Methodology for replacing indirect measurements with direct measurements", arXiv:1901.00015.

[6] Ilya G. Ryabinkin and Scott N. Genin, "Iterative qubit coupled cluster method: A systematic approach to the full-CI limit in quantum chemistry calculations on NISQ devices", arXiv:1906.11192.

[7] Ryan LaRose, Arkin Tikku, Étude O'Neel-Judy, Lukasz Cincio, and Patrick J. Coles, "Variational Quantum State Diagonalization", arXiv:1810.10506.

[8] Suguru Endo, Ying Li, Simon Benjamin, and Xiao Yuan, "Variational quantum simulation of general processes", arXiv:1812.08778.

[9] Ken M. Nakanishi, Keisuke Fujii, and Synge Todo, "Sequential minimal optimization for quantum-classical hybrid algorithms", arXiv:1903.12166.

[10] Robert M. Parrish, Joseph T. Iosue, Asier Ozaeta, and Peter L. McMahon, "A Jacobi Diagonalization and Anderson Acceleration Algorithm For Variational Quantum Algorithm Parameter Optimization", arXiv:1904.03206.

[11] Earl Campbell, "A random compiler for fast Hamiltonian simulation", arXiv:1811.08017.

[12] Robert M. Parrish, Edward G. Hohenstein, Peter L. McMahon, and Todd J. Martinez, "Quantum Computation of Electronic Transitions using a Variational Quantum Eigensolver", arXiv:1901.01234.

[13] Robert M. Parrish, Edward G. Hohenstein, Peter L. McMahon, and Todd J. Martinez, "Hybrid Quantum/Classical Derivative Theory: Analytical Gradients and Excited-State Dynamics for the Multistate Contracted Variational Quantum Eigensolver", arXiv:1906.08728.

[14] Kosuke Mitarai, Yuya O. Nakagawa, and Wataru Mizukami, "Theory of analytical energy derivatives for the variational quantum eigensolver", arXiv:1905.04054.

[15] Sam McArdle, Suguru Endo, Alan Aspuru-Guzik, Simon Benjamin, and Xiao Yuan, "Quantum computational chemistry", arXiv:1808.10402.

[16] T. E. O'Brien, B. Senjean, R. Sagastizabal, X. Bonet-Monroig, A. Dutkiewicz, F. Buda, L. DiCarlo, and L. Visscher, "Calculating energy derivatives for quantum chemistry on a quantum computer", arXiv:1905.03742.

[17] Yudong Cao, Jonathan Romero, Jonathan P. Olson, Matthias Degroote, Peter D. Johnson, Mária Kieferová, Ian D. Kivlichan, Tim Menke, Borja Peropadre, Nicolas P. D. Sawaya, Sukin Sim, Libor Veis, and Alán Aspuru-Guzik, "Quantum Chemistry in the Age of Quantum Computing", arXiv:1812.09976.

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