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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The data that support the findings of this study are available from the corresponding authors on reasonable request
The code used to generate the data presented in this study can be publicly accessed on GitHub at https://github.com/mariomotta/QITE.git
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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.
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). https://doi.org/10.1038/s41567-019-0704-4
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