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Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution

A Publisher Correction to this article was published on 29 January 2020

A Publisher Correction to this article was published on 21 November 2019

This article has been updated


The accurate computation of Hamiltonian ground, excited and thermal states on quantum computers stands to impact many problems in the physical and computer sciences, from quantum simulation to machine learning. Given the challenges posed in constructing large-scale quantum computers, these tasks should be carried out in a resource-efficient way. In this regard, existing techniques based on phase estimation or variational algorithms display potential disadvantages; phase estimation requires deep circuits with ancillae, that are hard to execute reliably without error correction, while variational algorithms, while flexible with respect to circuit depth, entail additional high-dimensional classical optimization. Here, we introduce the quantum imaginary time evolution and quantum Lanczos algorithms, which are analogues of classical algorithms for finding ground and excited states. Compared with their classical counterparts, they require exponentially less space and time per iteration, and can be implemented without deep circuits and ancillae, or high-dimensional optimization. We furthermore discuss quantum imaginary time evolution as a subroutine to generate Gibbs averages through an analogue of minimally entangled typical thermal states. Finally, we demonstrate the potential of these algorithms via an implementation using exact classical emulation as well as through prototype circuits on the Rigetti quantum virtual machine and Aspen-1 quantum processing unit.

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Fig. 1: Physical foundations of the QITE algorithm.
Fig. 2: Classical simulation and experimental implementation of QITE and QLanczos algorithms.
Fig. 3: Application of QITE to long-range spin and fermionic models, and a combinatorial optimization problem.
Fig. 4: Classical simulation and experimental implementation of the QMETTS algorithm.

Data availability

The data that support the findings of this study are available from the corresponding authors on reasonable request

Code availability

The code used to generate the data presented in this study can be publicly accessed on GitHub at

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  1. 1.

    Feynman, R. P. Simulating physics with computers. Int. J. Theor. Phys. 21, 467–488 (1982).

    MathSciNet  Article  Google Scholar 

  2. 2.

    Abrams, D. S. & Lloyd, S. Simulation of many-body Fermi systems on a universal quantum computer. Phys. Rev. Lett. 79, 2586–2589 (1997).

    ADS  Article  Google Scholar 

  3. 3.

    Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996).

    ADS  MathSciNet  Article  Google Scholar 

  4. 4.

    Aspuru-Guzik, A., Dutoi, A. D., Love, P. J. & Head-Gordon, M. Simulated quantum computation of molecular energies. Science 309, 1704–1707 (2005).

    ADS  Article  Google Scholar 

  5. 5.

    Kandala, A. et al. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature 549, 242–246 (2017).

    ADS  Article  Google Scholar 

  6. 6.

    Kandala, A. et al. Error mitigation extends the computational reach of a noisy quantum processor. Nature 567, 491–495 (2019).

    ADS  Article  Google Scholar 

  7. 7.

    Kempe, J., Kitaev, A. & Regev, O. The complexity of the local Hamiltonian problem. SIAM J. Comput. 35, 1070–1097 (2006).

    MathSciNet  Article  Google Scholar 

  8. 8.

    Farhi, E., Goldstone, J., Gutmann, S. & Sipser, M. Quantum Computation by Adiabatic Evolution Report No. MIT-CTP-2936 (MIT, 2000).

  9. 9.

    Kitaev, A. Y. Quantum measurements and the Abelian stabilizer problem. Preprint at (1995).

  10. 10.

    Farhi, E., Goldstone, J., Gutmann, S. & Sipser, M. A quantum approximate optimization algorithm. Report No. MIT-CTP/4610 (MIT, 2014).

  11. 11.

    Otterbach, J. S. et al. Unsupervised machine learning on a hybrid quantum computer. Preprint at (2017).

  12. 12.

    Moll, N. et al. Quantum optimization using variational algorithms on near-term quantum devices. Quantum Sci. Technol. 3, 030503 (2018).

    ADS  Article  Google Scholar 

  13. 13.

    Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 5, 4213 (2014).

    ADS  Article  Google Scholar 

  14. 14.

    McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New J. Phys. 18, 023023 (2016).

    ADS  Article  Google Scholar 

  15. 15.

    Grimsley, H. R., Economou, S. E., Barnes, E. & Mayhall, N. J. An adaptive variational algorithm for exact molecular simulations on a quantum computer. Nat. Commun. 10, 3007 (2019).

    ADS  Article  Google Scholar 

  16. 16.

    McClean, J. R., Boixo, S., Smelyanskiy, V. N., Babbush, R. & Neven, H. Barren plateaus in quantum neural network training landscapes. Nat. Commun. 9, 4812 (2018).

    ADS  Article  Google Scholar 

  17. 17.

    McArdle, S. et al. Variational ansatz-based quantum simulation of imaginary time evolution. npj Quantum Inf. 5, 75 (2019).

    ADS  Article  Google Scholar 

  18. 18.

    Lanczos, C. An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J. Res. Natl Bur. Stand. B 45, 255–282 (1950).

    MathSciNet  Article  Google Scholar 

  19. 19.

    Uhlmann, A. The ‘transition probability’ in the state space of a *-algebra. Rep. Math. Phys. 9, 273–279 (1976).

    ADS  MathSciNet  Article  Google Scholar 

  20. 20.

    Hastings, M. B. & Koma, T. Spectral gap and exponential decay of correlations. Commun. Math. Phys. 265, 781–804 (2006).

    ADS  MathSciNet  Article  Google Scholar 

  21. 21.

    Bravyi, S. B. & Kitaev, A. Y. Fermionic quantum computation. Ann. Phys. (N. Y.) 298, 210–226 (2002).

    ADS  MathSciNet  Article  Google Scholar 

  22. 22.

    Verstraete, F. & Cirac, J. I. Mapping local Hamiltonians of fermions to local Hamiltonians of spins. J. Stat. Mech. 2005, P09012 (2005).

    MathSciNet  Article  Google Scholar 

  23. 23.

    Berry, D. W., Childs, A. M. & Kothari, R. Hamiltonian simulation with nearly optimal dependence on all parameters. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science 792–809 (IEEE, 2015).

  24. 24.

    Vidal, G. Efficient simulation of one-dimensional quantum many-body systems. Phys. Rev. Lett. 93, 040502 (2004).

    ADS  Article  Google Scholar 

  25. 25.

    Schollwöck, U. The density-matrix renormalization group in the age of matrix product states. Ann. Phys. (N. Y.) 326, 96–192 (2011).

    ADS  MathSciNet  Article  Google Scholar 

  26. 26.

    Schuch, N., Wolf, M. M., Verstraete, F. & Cirac, J. I. Computational complexity of projected entangled pair states. Phys. Rev. Lett. 98, 140506 (2007).

    ADS  MathSciNet  Article  Google Scholar 

  27. 27.

    Haferkamp, J., Hangleiter, D., Eisert, J. & Gluza, M. Contracting projected entangled pair states is average-case hard. Preprint at (2018).

  28. 28.

    O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016).

    Google Scholar 

  29. 29.

    Lamm, H. & Lawrence, S. Simulation of nonequilibrium dynamics on a quantum computer. Phys. Rev. Lett. 121, 170501 (2018).

    ADS  Article  Google Scholar 

  30. 30.

    Rigetti Computing: Quantum Cloud Services;

  31. 31.

    McClean, J. R., Kimchi-Schwartz, M. E., Carter, J. & de Jong, W. A. Hybrid quantum-classical hierarchy for mitigation of decoherence and determination of excited states. Phys. Rev. A 95, 042308 (2017).

    ADS  Article  Google Scholar 

  32. 32.

    Colless, J. I. et al. Computation of molecular spectra on a quantum processor with an error-resilient algorithm. Phys. Rev. X 8, 011021 (2018).

    Google Scholar 

  33. 33.

    Terhal, B. M. & DiVincenzo, D. P. Problem of equilibration and the computation of correlation functions on a quantum computer. Phys. Rev. A 61, 022301 (2000).

    ADS  Article  Google Scholar 

  34. 34.

    Temme, K., Osborne, T. J., Vollbrecht, K. G., Poulin, D. & Verstraete, F. Quantum Metropolis sampling. Nature 471, 87 (2011).

    ADS  Article  Google Scholar 

  35. 35.

    Chowdhury, A. N. & Somma, R. D. Quantum algorithms for Gibbs sampling and hitting-time estimation. Quantum Inf. Comput. 17, 41–64 (2017).

    MathSciNet  Google Scholar 

  36. 36.

    Brandão, F. G. & Kastoryano, M. J. Finite correlation length implies efficient preparation of quantum thermal states. Commun. Math. Phys. 365, 1–16 (2019).

    ADS  MathSciNet  Article  Google Scholar 

  37. 37.

    White, S. R. Minimally entangled typical quantum states at finite temperature. Phys. Rev. Lett. 102, 190601 (2009).

    ADS  MathSciNet  Article  Google Scholar 

  38. 38.

    Stoudenmire, E. M. & White, S. R. Minimally entangled typical thermal state algorithms. New J. Phys. 12, 055026 (2010).

    ADS  Article  Google Scholar 

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M.M., G.K.-L.C., F.G.S.L.B., A.T.K.T. and A.J.M. were supported by the US NSF via RAISE-TAQS CCF 1839204. M.J.O’R. was supported by an NSF graduate fellowship via grant No. DEG-1745301; the tensor network algorithms were developed with the support of the US DOD via MURI FA9550-18-1-0095. E.Y. was supported by a Google fellowship. C.S. was supported by the US DOE via DE-SC0019374. G.K.-L.C. is a Simons Investigator in Physics and a member of the Simons Collaboration on the Many-Electron Problem. The Rigetti computations were made possible by a generous grant through Rigetti Quantum Cloud Services supported by the CQIA–Rigetti Partnership Program. We thank G. H. Low, J. R. McClean and R. Babbush for discussions, and the Rigetti team for help with the QVM and QPU simulations.

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M.M., C.S. and G.K.-L.C. designed the algorithms. F.G.S.L.B. established the mathematical proofs and error estimates. E.Y. and M.J.O’R. performed classical tensor network simulations. M.M., C.S. and A.T.K.T. carried out classical exact emulations. A.T.K.T. and A.J.M. designed and carried out the Rigetti QVM and QPU experiments. All authors contributed to the discussion of results and writing of the manuscript.

Corresponding authors

Correspondence to Mario Motta or Garnet Kin-Lic Chan.

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The authors declare no competing interests.

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Motta, M., Sun, C., Tan, A.T.K. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nat. Phys. 16, 205–210 (2020).

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