A Caltech Library Service

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

Motta, Mario and Sun, Chong and Tan, Adrian T. K. and O’Rourke, Matthew J. and Ye, Erika and Minnich, Austin J. and Brandão, Fernando G. S. L. and Chan, Garnet Kin-Lic (2020) Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics, 16 (2). pp. 205-210. ISSN 1745-2473.

[img] PDF - Supplemental Material
See Usage Policy.


Use this Persistent URL to link to this item:


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.

Item Type:Article
Related URLs:
URLURL TypeDescription ReadCube access ItemCode Correction - 21 November 2019 ReadCube access - Publisher Correction - 21 November 2019 Correction - 29 January 2020 ReadCube access - Publisher Correction - 29 January 2020
Motta, Mario0000-0003-1647-9864
Sun, Chong0000-0002-8299-9094
O’Rourke, Matthew J.0000-0002-5779-2577
Minnich, Austin J.0000-0002-9671-9540
Brandão, Fernando G. S. L.0000-0003-3866-9378
Chan, Garnet Kin-Lic0000-0001-8009-6038
Additional Information:© 2019 Springer Nature Limited. Received 01 April 2019; Accepted 24 September 2019; Published 11 November 2019. 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 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. Author Contributions: 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. The authors declare no competing interests.
Group:UNSPECIFIED, Institute for Quantum Information and Matter
Funding AgencyGrant Number
NSF Graduate Research FellowshipDGE-1745301
Air Force Office of Scientific Research (AFOSR)FA9550-18-1-0095
Department of Energy (DOE)DE-SC0019374
Simons FoundationUNSPECIFIED
Rigetti Quantum Cloud ServicesUNSPECIFIED
CQIA-Rigetti Partnership ProgramUNSPECIFIED
Subject Keywords:Information theory and computation; Quantum information; Quantum simulation
Issue or Number:2
Record Number:CaltechAUTHORS:20190918-131942784
Persistent URL:
Official Citation: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).
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:98727
Deposited By: George Porter
Deposited On:15 Nov 2019 23:00
Last Modified:04 Jun 2020 10:14

Repository Staff Only: item control page