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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

Abstract

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 https://github.com/mariomotta/QITE.git

Change history

  • 29 January 2020

    An amendment to this paper has been published and can be accessed via a link at the top of the paper.

  • 21 November 2019

    An amendment to this paper has been published and can be accessed via a link at the top of the paper.

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Acknowledgements

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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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). https://doi.org/10.1038/s41567-019-0704-4

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