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.
Subscribe to Journal
Get full journal access for 1 year
only $4.92 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Tax calculation will be finalised during checkout.
Rent or Buy article
Get time limited or full article access on ReadCube.
All prices are NET prices.
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
Feynman, R. P. Simulating physics with computers. Int. J. Theor. Phys. 21, 467–488 (1982).
Abrams, D. S. & Lloyd, S. Simulation of many-body Fermi systems on a universal quantum computer. Phys. Rev. Lett. 79, 2586–2589 (1997).
Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996).
Aspuru-Guzik, A., Dutoi, A. D., Love, P. J. & Head-Gordon, M. Simulated quantum computation of molecular energies. Science 309, 1704–1707 (2005).
Kandala, A. et al. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature 549, 242–246 (2017).
Kandala, A. et al. Error mitigation extends the computational reach of a noisy quantum processor. Nature 567, 491–495 (2019).
Kempe, J., Kitaev, A. & Regev, O. The complexity of the local Hamiltonian problem. SIAM J. Comput. 35, 1070–1097 (2006).
Farhi, E., Goldstone, J., Gutmann, S. & Sipser, M. Quantum Computation by Adiabatic Evolution Report No. MIT-CTP-2936 (MIT, 2000).
Kitaev, A. Y. Quantum measurements and the Abelian stabilizer problem. Preprint at https://arxiv.org/abs/quant-ph/9511026 (1995).
Farhi, E., Goldstone, J., Gutmann, S. & Sipser, M. A quantum approximate optimization algorithm. Report No. MIT-CTP/4610 (MIT, 2014).
Otterbach, J. S. et al. Unsupervised machine learning on a hybrid quantum computer. Preprint at https://arxiv.org/abs/1712.05771 (2017).
Moll, N. et al. Quantum optimization using variational algorithms on near-term quantum devices. Quantum Sci. Technol. 3, 030503 (2018).
Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 5, 4213 (2014).
McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New J. Phys. 18, 023023 (2016).
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).
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).
McArdle, S. et al. Variational ansatz-based quantum simulation of imaginary time evolution. npj Quantum Inf. 5, 75 (2019).
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).
Uhlmann, A. The ‘transition probability’ in the state space of a *-algebra. Rep. Math. Phys. 9, 273–279 (1976).
Hastings, M. B. & Koma, T. Spectral gap and exponential decay of correlations. Commun. Math. Phys. 265, 781–804 (2006).
Bravyi, S. B. & Kitaev, A. Y. Fermionic quantum computation. Ann. Phys. (N. Y.) 298, 210–226 (2002).
Verstraete, F. & Cirac, J. I. Mapping local Hamiltonians of fermions to local Hamiltonians of spins. J. Stat. Mech. 2005, P09012 (2005).
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).
Vidal, G. Efficient simulation of one-dimensional quantum many-body systems. Phys. Rev. Lett. 93, 040502 (2004).
Schollwöck, U. The density-matrix renormalization group in the age of matrix product states. Ann. Phys. (N. Y.) 326, 96–192 (2011).
Schuch, N., Wolf, M. M., Verstraete, F. & Cirac, J. I. Computational complexity of projected entangled pair states. Phys. Rev. Lett. 98, 140506 (2007).
Haferkamp, J., Hangleiter, D., Eisert, J. & Gluza, M. Contracting projected entangled pair states is average-case hard. Preprint at https://arxiv.org/abs/1810.00738 (2018).
O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016).
Lamm, H. & Lawrence, S. Simulation of nonequilibrium dynamics on a quantum computer. Phys. Rev. Lett. 121, 170501 (2018).
Rigetti Computing: Quantum Cloud Services; https://qcs.rigetti.com/dashboard
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).
Colless, J. I. et al. Computation of molecular spectra on a quantum processor with an error-resilient algorithm. Phys. Rev. X 8, 011021 (2018).
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).
Temme, K., Osborne, T. J., Vollbrecht, K. G., Poulin, D. & Verstraete, F. Quantum Metropolis sampling. Nature 471, 87 (2011).
Chowdhury, A. N. & Somma, R. D. Quantum algorithms for Gibbs sampling and hitting-time estimation. Quantum Inf. Comput. 17, 41–64 (2017).
Brandão, F. G. & Kastoryano, M. J. Finite correlation length implies efficient preparation of quantum thermal states. Commun. Math. Phys. 365, 1–16 (2019).
White, S. R. Minimally entangled typical quantum states at finite temperature. Phys. Rev. Lett. 102, 190601 (2009).
Stoudenmire, E. M. & White, S. R. Minimally entangled typical thermal state algorithms. New J. Phys. 12, 055026 (2010).
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.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
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
npj Quantum Information (2021)
Implementation of quantum imaginary-time evolution method on NISQ devices by introducing nonlocal approximation
npj Quantum Information (2021)
Quantum Machine Intelligence (2021)
Quantum algorithm for preparing the ground state of a physical system through multi-step quantum resonant transitions
Quantum Information Processing (2021)
Structural and Electronic Calculations of CdTe Using DFT: Exchange–Correlation Functionals and DFT-1/2 Corrections
Journal of Electronic Materials (2021)