Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Maximally localized dynamical quantum embedding for solving many-body correlated systems

A Publisher Correction to this article was published on 07 July 2021

This article has been updated

A preprint version of the article is available at arXiv.


Quantum computing opens new avenues for modeling correlated materials, which are notoriously challenging to solve due to the presence of large electronic correlations. Quantum embedding approaches, such as dynamical mean-field theory, provide corrections to first-principles calculations for strongly correlated materials, which are poorly described at lower levels of theory. Such embedding approaches are computationally demanding on classical computing architectures and hence remain restricted to small systems, limiting the scope of their applicability. Hitherto, implementations on quantum computers have been limited by hardware constraints. Here, we derive a compact representation, where the number of quantum states is reduced for a given system while retaining a high level of accuracy. We benchmark our method for archetypal quantum states of matter that emerge due to electronic correlations, such as Kondo and Mott physics, both at equilibrium and for quenched systems. We implement this approach on a quantum emulator, demonstrating a reduction of the required number of qubits.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: From AIM to MLDE.
Fig. 2: MLDE benchmark and application on a quantum computer.
Fig. 3: Kondo physics.
Fig. 4: Mott transition.
Fig. 5: Dynamical susceptibility.
Fig. 6: MLDE for multi-orbital correlated LaNiO3.

Data availability

Source data are available via Code Ocean56.

Code availability

The code is available via Code Ocean56.

Change history


  1. 1.

    Kotliar, G. et al. Electronic structure calculations with dynamical mean-field theory. Rev. Mod. Phys. 78, 865–951 (2006).

    Google Scholar 

  2. 2.

    Georges, A., Kotliar, G., Krauth, W. & Rozenberg, M. J. Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions. Rev. Mod. Phys. 68, 13 (1996).

    MathSciNet  Google Scholar 

  3. 3.

    Zgid, D. & Gull, E. Finite temperature quantum embedding theories for correlated systems. New J. Phys. 19, 023047 (2017).

    Google Scholar 

  4. 4.

    Fertitta, E. & Booth, G. H. Rigorous wave function embedding with dynamical fluctuations. Phys. Rev. B 98, 235132 (2018).

    Google Scholar 

  5. 5.

    Sun, Q. & Chan, G. K.-L. Quantum embedding theories. Acc. Chem. Res. 49, 2705–2712 (2016).

    Google Scholar 

  6. 6.

    Gubernatis, J. E., Jarrell, M., Silver, R. N. & Sivia, D. S. Quantum Monte Carlo simulations and maximum entropy: dynamics from imaginary-time data. Phys. Rev. B 44, 6011–6029 (1991).

    Google Scholar 

  7. 7.

    Gunnarsson, O., Haverkort, M. W. & Sangiovanni, G. Analytical continuation of imaginary axis data for optical conductivity. Phys. Rev. B 82, 165125 (2010).

    Google Scholar 

  8. 8.

    Mitchell, A. K. & Fritz, L. Kondo effect with diverging hybridization: possible realization in graphene with vacancies. Phys. Rev. B 88, 075104 (2013).

    Google Scholar 

  9. 9.

    Capone, M., de’ Medici, L. & Georges, A. Solving the dynamical mean-field theory at very low temperatures using the Lanczos exact diagonalization. Phys. Rev. B 76, 245116 (2007).

    Google Scholar 

  10. 10.

    Perroni, C. A., Ishida, H. & Liebsch, A. Exact diagonalization dynamical mean-field theory for multiband materials: effect of Coulomb correlations on the Fermi surface of Na0.3CoO2. Phys. Rev. B 75, 045125 (2007).

    Google Scholar 

  11. 11.

    Capone, M., Fabrizio, M., Castellani, C. & Tosatti, E. Strongly correlated superconductivity. Science 296, 2364–2366 (2002).

    Google Scholar 

  12. 12.

    Liebsch, A. & Ishida, H. Temperature and bath size in exact diagonalization dynamical mean field theory. J. Phys. Condens. Matter 24, 053201 (2012).

    Google Scholar 

  13. 13.

    Ishida, H. & Liebsch, A. Fermi-liquid, non-Fermi-liquid, and Mott phases in iron pnictides and cuprates. Phys. Rev. B 81, 054513 (2010).

    Google Scholar 

  14. 14.

    Arrigoni, E., Knap, M. & von der Linden, W. Nonequilibrium dynamical mean-field theory: an auxiliary quantum master equation approach. Phys. Rev. Lett. 110, 086403 (2013).

    Google Scholar 

  15. 15.

    Lu, Y., Höppner, M., Gunnarsson, O. & Haverkort, M. W. Efficient real-frequency solver for dynamical mean-field theory. Phys. Rev. B 90, 085102 (2014).

    Google Scholar 

  16. 16.

    Ganahl, M. et al. Efficient DMFT impurity solver using real-time dynamics with matrix product states. Phys. Rev. B 92, 155132 (2015).

    Google Scholar 

  17. 17.

    Bauernfeind, D., Zingl, M., Triebl, R., Aichhorn, M. & Evertz, H. G. Fork tensor-product states: efficient multiorbital real-time DMFT solver. Phys. Rev. X 7, 031013 (2017).

    Google Scholar 

  18. 18.

    Lu, Y., Höppner, M., Gunnarsson, O. & Haverkort, M. W. Efficient real-frequency solver for dynamical mean-field theory. Phys. Rev. B 90, 085102 (2014).

    Google Scholar 

  19. 19.

    Rungger, I. et al. Dynamical mean field theory algorithm and experiment on quantum computers. Preprint at (2019).

  20. 20.

    Sakai, S., Civelli, M. & Imada, M. Hidden fermionic excitation boosting high-temperature superconductivity in cuprates. Phys. Rev. Lett. 116, 057003 (2016).

    Google Scholar 

  21. 21.

    Sen, S., Wong, P. J. & Mitchell, A. K. The Mott transition as a topological phase transition. Phys. Rev. B 102, 081110(R) (2020).

  22. 22.

    Baldini, E. et al. Electron-phonon-driven three-dimensional metallicity in an insulating cuprate. Proc. Natl Acad. Sci. USA 117, 6409–6416 (2020).

  23. 23.

    Acharya, S. et al. Metal–insulator transition in copper oxides induced by apex displacements. Phys. Rev. X 8, 021038 (2018).

    Google Scholar 

  24. 24.

    Liu, J. et al. First observation of a Kondo resonance for a stable neutral pure organic radical, 1,3,5-triphenyl-6-oxoverdazyl, adsorbed on the Au(111) surface. J. Am. Chem. Soc. 135, 651–658 (2013).

    Google Scholar 

  25. 25.

    Zhang, Y.-h et al. Temperature and magnetic field dependence of a Kondo system in the weak coupling regime. Nat. Commun. 4, 2110 (2013).

    Google Scholar 

  26. 26.

    Frisenda, R. et al. Kondo effect in a neutral and stable all organic radical single molecule break junction. Nano Lett. 15, 3109–3114 (2015).

  27. 27.

    Droghetti, A. & Rungger, I. Quantum transport simulation scheme including strong correlations and its application to organic radicals adsorbed on gold. Phys. Rev. B 95, 085131 (2017).

    Google Scholar 

  28. 28.

    Appelt, W. H. et al. Predicting the conductance of strongly correlated molecules: the Kondo effect in perchlorotriphenylmethyl/Au junctions. Nanoscale 10, 17738–17750 (2018).

  29. 29.

    Balzer, M., Gdaniec, N. & Potthoff, M. Krylov-space approach to the equilibrium and nonequilibrium single-particle Green’s function. J. Phys. Condens. Matter 24, 035603 (2012).

    Google Scholar 

  30. 30.

    Lee, H., Plekhanov, E., Blackbourn, D., Acharya, S. & Weber, C. The Mott to Kondo transition in diluted Kondo superlattices. Commun. Phys. 2, 49 (2019).

    Google Scholar 

  31. 31.

    Surer, B. et al. Multiorbital Kondo physics of Co in Cu hosts. Phys. Rev. B 85, 085114 (2012).

    Google Scholar 

  32. 32.

    Žitko, R., Peters, R. & Pruschke, T. Splitting of the Kondo resonance in anisotropic magnetic impurities on surfaces. New J. Phys. 11, 053003 (2009).

    Google Scholar 

  33. 33.

    Granath, M. & Schött, J. Signatures of coherent electronic quasiparticles in the paramagnetic Mott insulator. Phys. Rev. B 90, 235129 (2014).

    Google Scholar 

  34. 34.

    Liebsch, A. & Ishida, H. Temperature and bath size in exact diagonalization dynamical mean field theory. J. Phys. Condens. Matter 24, 053201 (2011).

    Google Scholar 

  35. 35.

    Nowadnick, E. A. et al. Quantifying electronic correlation strength in a complex oxide: a combined DMFT and ARPES study of LaNiO3. Phys. Rev. B 92, 245109 (2015).

    Google Scholar 

  36. 36.

    Jones, T., Brown, A., Bush, I. & Benjamin, S. C. Quest and high performance simulation of quantum computers. Sci. Rep. 9, 10736 (2019).

    Google Scholar 

  37. 37.

    Fradkin, E. Jordan–Wigner transformation for quantum-spin systems in two dimensions and fractional statistics. Phys. Rev. Lett. 63, 322–325 (1989).

    MathSciNet  Google Scholar 

  38. 38.

    Wecker, D. et al. Solving strongly correlated electron models on a quantum computer. Phys. Rev. A 92, 062318 (2015).

    Google Scholar 

  39. 39.

    Bauer, B., Wecker, D., Millis, A. J., Hastings, M. B. & Troyer, M. Hybrid quantum-classical approach to correlated materials. Phys. Rev. X 6, 031045 (2016).

    Google Scholar 

  40. 40.

    Cini, M. Topics and Methods in Condensed Matter Theory. From Basic Quantum Mechanics to the Frontiers of Research (Springer, 2007).

  41. 41.

    Zgid, D. & Gull, E. Finite temperature quantum embedding theories for correlated systems. New J. Phys. 19, 023047 (2017).

    Google Scholar 

  42. 42.

    Schäfer, T., Toschi, A. & Tomczak, J. M. Separability of dynamical and nonlocal correlations in three dimensions. Phys. Rev. B 91, 121107 (2015).

    Google Scholar 

  43. 43.

    Neuhauser, D., Baer, R. & Zgid, D. Stochastic self-consistent second-order Green’s function method for correlation energies of large electronic systems. J. Chem. Theory Comput. 13, 5396–5403 (2017).

  44. 44.

    Wolf, F. A., Go, A., McCulloch, I. P., Millis, A. J. & Schollwöck, U. Imaginary-time matrix product state impurity solver for dynamical mean-field theory. Phys. Rev. X 5, 041032 (2015).

    Google Scholar 

  45. 45.

    Toschi, A., Katanin, A. A. & Held, K. Dynamical vertex approximation: a step beyond dynamical mean-field theory. Phys. Rev. B 75, 045118 (2007).

    Google Scholar 

  46. 46.

    Hafermann, H. et al. Superperturbation solver for quantum impurity models. Europhys. Lett. 85, 27007 (2009).

    Google Scholar 

  47. 47.

    Rohringer, G., Valli, A. & Toschi, A. Local electronic correlation at the two-particle level. Phys. Rev. B 86, 125114 (2012).

    Google Scholar 

  48. 48.

    Rohringer, G. et al. Diagrammatic routes to nonlocal correlations beyond dynamical mean field theory. Rev. Mod. Phys. 90, 025003 (2018).

    MathSciNet  Google Scholar 

  49. 49.

    Pashov, D. et al. Questaal: a package of electronic structure methods based on the linear muffin-tin orbital technique. Comput. Phys. Commun. 249, 107065 (2020).

    MathSciNet  Google Scholar 

  50. 50.

    Seth, P., Krivenko, I., Ferrero, M. & Parcollet, O. TRIQS/CTHYB: a continuous-time quantum Monte Carlo hybridisation expansion solver for quantum impurity problems. Comput. Phys. Commun. 200, 274–284 (2016).

    Google Scholar 

  51. 51.

    Wiersema, R. et al. Exploring entanÿglement and optimization within the Hamiltonian variational ansatz. PRX Quantum 1, 020319 (2020).

    Google Scholar 

  52. 52.

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

    Google Scholar 

  53. 53.

    Nakanishi, K. M., Mitarai, K. & Fujii, K. Subspace-search variational quantum eigensolver for excited states. Phys. Rev. Res. 1, 033062 (2019).

    Google Scholar 

  54. 54.

    Higgott, O., Wang, D. & Brierley, S. Variational quantum computation of excited states. Quantum 3, 156 (2019).

    Google Scholar 

  55. 55.

    Jones, T., Endo, S., McArdle, S., Yuan, X. & Benjamin, S. C. Variational quantum algorithms for discovering Hamiltonian spectra. Phys. Rev. A 99, 062304 (2019).

    Google Scholar 

  56. 56.

    Lupo, C., Tse, T., Jamet, F., Rungger, I. & Weber, C. Maximally localized dynamical embedding (MLDE) (Code Ocean, 2021);

Download references


C.W. acknowledges insightful and stimulating discussions with A. Mitchell. C.W. is supported by grant no. EP/R02992X/1 from the UK Engineering and Physical Sciences Research Council (EPSRC). C.L. is supported by the EPSRC Centre for Doctoral Training in Cross-Disciplinary Approaches to Non-Equilibrium Systems (CANES, EP/L015854/1). I.R. and F.J. acknowledge the support of the UK Government Department for Business, Energy and Industrial Strategy through the UK National Quantum Technologies Programme (InnovateUK Industrial Strategy Challenge Fund (ISCF) QUANTIFI project). F.J. was also partly supported by the Simons Many-Electron Collaboration and by the EPSRC Centre for Doctoral Training in Cross-Disciplinary Approaches to Non-Equilibrium Systems (CANES, EP/L015854/1). This work was performed using resources provided by the ARCHER UK National Supercomputing Service and the Cambridge Service for Data Driven Discovery (CSD3) operated by the University of Cambridge Research Computing Service (, provided by Dell EMC and Intel using Tier-2 funding from the Engineering and Physical Sciences Research Council (capital grant no. EP/P020259/1), and DiRAC funding from the Science and Technology Facilities Council (

Author information




C.W. designed the research. I.R. and F.J. developed the quantum computing algorithm. All authors contributed to the code design, results production and analysis. All authors reviewed the manuscript.

Corresponding authors

Correspondence to Carla Lupo or François Jamet or Ivan Rungger or Cedric Weber.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Reviewer recognition statement

Peer review information Nature Computational Science thanks Orazio Scarlatella and the other, anonymous, reviewers for their contribution to the peer review of this work. Handling editor: Jie Pan, in collaboration with the Nature Computational Science team.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data

Extended Data Fig. 1 Error scaling.

In panel (a) we show the two cases of interest: on the left we consider a system with two uncorrelated bath sites which embed a correlated impurity (star symbol). The bath sites energy levels are ϵi = − ϵj = 0.5, the hopping amplitudes are Vi and the impurity correlation energy is Uimp = 4 and ϵimp = − U/2. On the right we consider a system with a correlated bath site with ϵi = − 0.5, Ub = 1 and correlated impurity (Uimp = 4 and ϵimp = − U/2) coupled to the bath sites via \({V}_{j}={V}_{i}\sqrt{2}\). Despite being two different systems, the bath Green’s function Gd(ω) is equivalent. (b) Difference between the impurity self energies obtained solving the systems described in panel a), as function of the leakage field. The hybridisation of both systems is identical when the non-local self energy is small.

Extended Data Fig. 2 Model hybridization.

Real part and imaginary part of the model hybridization used for the MLDE benchmark.

Extended Data Fig. 3 Strength of correlations.

Imaginary part of the self-energy obtained by ED for Nb = 2 (dashed lines) compared to the exact solution (continuous line).

Extended Data Fig. 4 Benchmark between ED and MLDE in real frequency.

We compare the results obtained with ED (9 bath sites) and MLDE (3 bath sites). In panel a) we report the real par of the impurity Green’s function obtained solving the AIM for fixed U = 4. In panel b) we report the local Green’s function obtained solving the self-consistent DMFT loop in the case of the square lattice, for fixed value U = 12.

Extended Data Fig. 5 Model hybridization of the TOV molecule on Gold surface.

Real part and imaginary part of the hybridization used for the application of MLDE to study the case of the deposition of the TOV molecule on gold surface. The impurity energy is set to ϵf = − 0.13 (eV) and the temperature is T=5K. (Units are in eV).

Extended Data Fig. 6 TOV molecule on Gold surface.

a)Density of states as function of real energies for different temperatures and b) data points of the self-energy at T=20K (circles) fitted at low frequencies with a quadratic function.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lupo, C., Jamet, F., Tse, W.H.T. et al. Maximally localized dynamical quantum embedding for solving many-body correlated systems. Nat Comput Sci 1, 410–420 (2021).

Download citation


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing