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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.

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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.

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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, François Jamet, Ivan Rungger or Cedric Weber.

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The authors declare no competing interests.

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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.

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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.

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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).

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