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Quantum embedding theories to simulate condensed systems on quantum computers

Nature Computational Science volume 2, pages 424–432 (2022)Cite this article

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Abstract

Quantum computers hold promise to improve the efficiency of quantum simulations of materials and to enable the investigation of systems and properties that are more complex than tractable at present on classical architectures. Here, we discuss computational frameworks to carry out electronic structure calculations of solids on noisy intermediate-scale quantum computers using embedding theories, and we give examples for a specific class of materials, that is, solid materials hosting spin defects. These are promising systems to build future quantum technologies, such as quantum computers, quantum sensors and quantum communication devices. Although quantum simulations on quantum architectures are in their infancy, promising results for realistic systems appear to be within reach.

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Fig. 1: Quantum simulations of materials properties on classical and quantum computers.
Fig. 2: Summary of quantum embedding theories used in condensed-matter physics and quantum chemistry.
Fig. 3: Layout of materials simulations using QDET on a classical and quantum computer.
Fig. 4: Solving a Hamiltonian on a quantum computer.

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References

  1. Jones, R. O. Density functional theory: its origins, rise to prominence, and future. Rev. Mod. Phys. 87, 897–923 (2015).

    Article  MathSciNet  Google Scholar 

  2. Krylov, A. et al. Perspective: Computational chemistry software and its advancement as illustrated through three grand challenge cases for molecular science. J. Chem. Phys. 149, 180901 (2018).

    Article  Google Scholar 

  3. Schleder, G. R., Padilha, A. C. M., Acosta, C. M., Costa, M. & Fazzio, A. From DFT to machine learning: recent approaches to materials science—a review. J. Phys. Mater. 2, 032001 (2019).

    Article  Google Scholar 

  4. Maurer, R. J. et al. Advances in density-functional calculations for materials modeling. Annu. Rev. Mater. Res. 49, 1–30 (2019).

    Article  Google Scholar 

  5. Bogojeski, M., Vogt-Maranto, L., Tuckerman, M. E., Müller, K.-R. & Burke, K. Quantum chemical accuracy from density functional approximations via machine learning. Nat. Commun. 11, 5223 (2020).

    Article  Google Scholar 

  6. McArdle, S., Endo, S., Aspuru-Guzik, A., Benjamin, S. C. & Yuan, X. Quantum computational chemistry. Rev. Mod. Phys. 92, 015003 (2020).

    Article  MathSciNet  Google Scholar 

  7. Bell, A. T. & Head-Gordon, M. Quantum mechanical modeling of catalytic processes. Annu. Rev. Chem. Biomol. Eng. 2, 453–477 (2011).

    Article  Google Scholar 

  8. Xu, S. & Carter, E. A. Theoretical insights into heterogeneous (photo)electrochemical CO2 reduction. Chem. Rev. 119, 6631–6669 (2019).

    Article  Google Scholar 

  9. G. Wolfowicz et al. Quantum guidelines for solid-state spin defects. Nat. Rev. Mater. 6, 906–925 (2021)

  10. Dreyer, C. E., Alkauskas, A., Lyons, J. L., Janotti, A. & Van de Walle, C. G. First-principles calculations of point defects for quantum technologies. Annu. Rev. Mater. Res. 48, 1–26 (2018).

    Article  Google Scholar 

  11. Weber, J. R. et al. Quantum computing with defects. Proc. Natl Acad. Sci. USA 107, 8513–8518 (2010).

    Article  Google Scholar 

  12. Agrawal, A. & Choudhary, A. Perspective: Materials informatics and big data: realization of the ‘fourth paradigm’ of science in materials science. APL Mater. 4, 053208 (2016).

    Article  Google Scholar 

  13. Himanen, L., Geurts, A., Foster, A. S. & Rinke, P. Data-driven materials science: status, challenges, and perspectives. Adv. Sci. 6, 1900808 (2019).

    Article  Google Scholar 

  14. S. Dong, S., Govoni, M. & Galli, G. Machine learning dielectric screening for the simulation of excited state properties of molecules and materials. Chem. Sci. 12, 4970–4980 (2021).

    Article  Google Scholar 

  15. Yuan, X. A quantum-computing advantage for chemistry. Science 369, 1054–1055 (2020).

    Article  Google Scholar 

  16. V. E. Elfving et al. How will quantum computers provide an industrially relevant computational advantage in quantum chemistry? Preprint at http://arxiv.org/abs/2009.12472 (2020).

  17. von Burg, V. et al. Quantum computing enhanced computational catalysis. Phys Rev. Res. 3, 033055 (2021).

    Article  Google Scholar 

  18. Liu, H. et al. Prospects of quantum computing for molecular sciences. Mater. Theory 6, 11 (2022).

    Article  Google Scholar 

  19. Ollitrault, P. J., Miessen, A. & Tavernelli, I. Molecular quantum dynamics: a quantum computing perspective. Acc. Chem. Res. 54, 4229–4238 (2021).

  20. Helgaker, T., Jorgensen, P. & Olsen, J. Molecular Electronic-Structure Theory (Wiley, 2014)

  21. Martin, R. M. Electronic Structure: Basic Theory and Practical Methods (Cambridge Univ. Press, 2020)

  22. Martin, R. M., Reining, L. & Ceperley, D. M. Interacting Electrons (Cambridge Univ. Press, 2016)

  23. Jordan, P., Neumann, J. V. & Wigner, E. On an algebraic generalization of the quantum mechanical formalism. Ann. Math. 35, 29–64 (1934).

    Article  MathSciNet  MATH  Google Scholar 

  24. Bravyi, S. B. & Kitaev, A. Y. Fermionic quantum computation. Ann. Phys. (N. Y.) 298, 210–226 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  25. Seeley, J. T., Richard, M. J. & Love, P. J. The Bravyi–Kitaev transformation for quantum computation of electronic structure. J. Chem. Phys. 137, 224109 (2012).

    Article  Google Scholar 

  26. Verstraete, F. & Cirac, J. I. Mapping local Hamiltonians of fermions to local Hamiltonians of spins. J. Stat. Mech. Theory Exp. 2005, P09012–P09012 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  27. Aleksandrowicz, G. et al. Qiskit: an open-source framework for quantum computing. https://doi.org/10.5281/zenodo.2562111 (2019).

  28. McClean, J. R. et al. OpenFermion: the electronic structure package for quantum computers. Quantum Sci. Technol. 5, 034014 (2020).

    Article  Google Scholar 

  29. Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 5, 4213 (2014).

    Article  Google Scholar 

  30. McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum–classical algorithms. New J. Phys. 18, 023023 (2016).

    Article  MATH  Google Scholar 

  31. Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge Univ. Press, 2010).

  32. Bravyi, S., Gosset, D., König, R. & Tomamichel, M. Quantum advantage with noisy shallow circuits. Nat. Phys. 16, 1040–1045 (2020).

    Article  Google Scholar 

  33. Aspuru-Guzik, A., Dutoi, A. D., Love, P. J. & Head-Gordon, M. Simulated quantum computation of molecular energies. Science 309, 1704–1707 (2005).

    Article  Google Scholar 

  34. Lanyon, B. P. et al. Towards quantum chemistry on a quantum computer. Nat. Chem. 2, 106–111 (2010).

    Article  Google Scholar 

  35. Li, Z. et al. Solving quantum ground-state problems with nuclear magnetic resonance. Sci. Rep. 1, 88 (2011).

    Article  Google Scholar 

  36. Shen, Y. et al. Quantum implementation of the unitary coupled cluster for simulating molecular electronic structure. Phys. Rev. A 95, 020501 (2017).

    Article  Google Scholar 

  37. O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016).

    Google Scholar 

  38. Santagati, R. et al. Witnessing eigenstates for quantum simulation of Hamiltonian spectra. Sci. Adv. 4, eaap9646 (2018).

    Article  Google Scholar 

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

    Article  Google Scholar 

  40. Hempel, C. et al. Quantum chemistry calculations on a trapped-ion quantum simulator. Phys. Rev. X 8, 031022 (2018).

    Google Scholar 

  41. Colless, J. I. et al. Computation of molecular spectra on a quantum processor with an error-resilient algorithm. Phys. Rev. X 8, 011021 (2018).

    Google Scholar 

  42. Kandala, A. et al. Error mitigation extends the computational reach of a noisy quantum processor. Nature 567, 491–495 (2019).

    Article  Google Scholar 

  43. Ryabinkin, I. G., Yen, T.-C., Genin, S. N. & Izmaylov, A. F. Qubit coupled cluster method: a systematic approach to quantum chemistry on a quantum computer. J. Chem. Theory Comput. 14, 6317–6326 (2018).

    Article  Google Scholar 

  44. Li, Z. et al. Quantum simulation of resonant transitions for solving the eigenproblem of an effective water Hamiltonian. Phys. Rev. Lett. 122, 090504 (2019).

    Article  Google Scholar 

  45. Nam, Y. et al. Ground-state energy estimation of the water molecule on a trapped-ion quantum computer. npj Quantum Inf. 6, 33 (2020).

  46. McCaskey, A. J. et al. Quantum chemistry as a benchmark for near-term quantum computers. npj Quantum Inf. 5, 99 (2019).

  47. Gao, Q. et al. Computational investigations of the lithium superoxide dimer rearrangement on noisy quantum devices. J. Phys. Chem. A 125, 1827–1836 (2021).

    Article  Google Scholar 

  48. Smart, S. E. & Mazziotti, D. A. Quantum–classical hybrid algorithm using an error-mitigating N-representability condition to compute the Mott metal–insulator transition. Phys. Rev. A 100, 022517 (2019).

    Article  Google Scholar 

  49. Sagastizabal, R. et al. Experimental error mitigation via symmetry verification in a variational quantum eigensolver. Phys. Rev. A 100, 010302 (2019).

    Article  Google Scholar 

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

    Article  Google Scholar 

  51. Google AI Quantum et al. Hartree–Fock on a superconducting qubit quantum computer Science 369, 1084–1089 (2020).

  52. Metcalf, M., Bauman, N. P., Kowalski, K. & de Jong, W. A. Resource-efficient chemistry on quantum computers with the variational quantum eigensolver and the double unitary coupled-cluster approach. J. Chem. Theory Comput. 16, 6165–6175 (2020).

    Article  Google Scholar 

  53. Rossmannek, M., Barkoutsos, P. K., Ollitrault, P. J. & Tavernelli, I. Quantum HF/DFT-embedding algorithms for electronic structure calculations: scaling up to complex molecular systems. J. Chem. Phys. 154, 114105 (2021).

    Article  Google Scholar 

  54. Kawashima, Y. et al. Efficient and accurate electronic structure simulation demonstrated on a trapped-ion quantum computer. Preprint at http://arxiv.org/abs/2102.07045 (2021).

  55. Teplukhin, A. et al. Computing molecular excited states on a D-Wave quantum annealer. Sci. Rep. 11, 18796 (2021).

    Article  Google Scholar 

  56. Kirsopp, J. J. M. et al. Quantum computational quantification of protein–ligand interactions. Preprint at http://arxiv.org/abs/2110.08163 (2021).

  57. Jones, M. A., Vallury, H. J., Hill, C. D. & Hollenberg, L. C. L. Chemistry beyond the Hartree–Fock limit via quantum computed moments. Preprint at http://arxiv.org/abs/2111.08132 (2021).

  58. Kivlichan, I. D. et al. Improved fault-tolerant quantum simulation of condensed-phase correlated electrons via trotterization. Quantum 4, 296 (2020).

    Article  Google Scholar 

  59. Cruz, P. M. Q., Catarina, G., Gautier, R. & Fernández-Rossier, J. Optimizing quantum phase estimation for the simulation of Hamiltonian eigenstates. Quantum Sci. Technol. 5, 044005 (2020).

    Article  Google Scholar 

  60. Montanaro, A. & Stanisic, S. Compressed variational quantum eigensolver for the Fermi–Hubbard model. Preprint at http://arxiv.org/abs/2006.01179 (2020).

  61. Uvarov, A., Biamonte, J. D. & Yudin, D. Variational quantum eigensolver for frustrated quantum systems. Phys. Rev. B 102, 075104 (2020).

    Article  Google Scholar 

  62. Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nat. Phys. 16, 205–210 (2020).

    Article  Google Scholar 

  63. Mei, F. et al. Digital simulation of topological matter on programmable quantum processors. Phys. Rev. Lett. 125, 160503 (2020).

    Article  Google Scholar 

  64. Mizuta, K. et al. Deep variational quantum eigensolver for excited states and its application to quantum chemistry calculation of periodic materials. Phys. Rev. Res. 3, 043121 (2021).

    Article  Google Scholar 

  65. Liu, J., Wan, L., Li, Z. & Yang, J. Simulating periodic systems on a quantum computer using molecular orbitals. J. Chem. Theory Comput. 16, 6904–6914 (2020).

    Article  Google Scholar 

  66. Kaicher, M. P., Jäger, S. B., Dallaire-Demers, P.-L. & Wilhelm, F. K. Roadmap for quantum simulation of the fractional quantum Hall effect. Phys. Rev. A 102, 022607 (2020).

    Article  MathSciNet  Google Scholar 

  67. Rahmani, A. et al. Creating and manipulating a Laughlin-type ν = 1/3 fractional quantum Hall state on a quantum computer with linear depth circuits. PRX Quantum 1, 020309 (2020).

    Article  Google Scholar 

  68. Kreula, J. M. et al. Few-qubit quantum–classical simulation of strongly correlated lattice fermions. EPJ Quantum Technol. 3, 11 (2016).

  69. Kreula, J. M., Clark, S. R. & Jaksch, D. Non-linear quantum–classical scheme to simulate non-equilibrium strongly correlated fermionic many-body dynamics. Sci. Rep. 6, 32940 (2016).

    Article  Google Scholar 

  70. Jaderberg, B., Agarwal, A., Leonhardt, K., Kiffner, M. & Jaksch, D. Minimum hardware requirements for hybrid quantum–classical DMFT. Quantum Sci. Technol. 5, 034015 (2020).

    Article  Google Scholar 

  71. Lupo, C., Jamet, F., Tse, T., Rungger, I. & Weber, C. Maximally localized dynamical quantum embedding for solving many-body correlated systems. Preprint at http://arxiv.org/abs/2008.04281 (2021).

  72. 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 

  73. Rubin, N. C. A hybrid classical/quantum approach for large-scale studies of quantum systems with density matrix embedding theory. Preprint at http://arxiv.org/abs/1610.06910 (2016).

  74. Mineh, L. & Montanaro, A. Solving the Hubbard model using density matrix embedding theory and the variational quantum eigensolver. Phys. Rev. B 105, 125117 (2022).

    Article  Google Scholar 

  75. Li, W. et al. Toward practical quantum embedding simulation of realistic chemical systems on near-term quantum computers. Preprint at http://arxiv.org/abs/2109.08062 (2021).

  76. Georges, A. & Kotliar, G. Hubbard model in infinite dimensions. Phys. Rev. B 45, 6479–6483 (1992).

    Article  Google Scholar 

  77. 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, 13–125 (1996).

    Article  MathSciNet  Google Scholar 

  78. Georges, A. Strongly correlated electron materials: dynamical mean-field theory and electronic structure. AIP Conf. Proc. 715, 3–74 (2004).

    Article  Google Scholar 

  79. Anisimov, V. I., Poteryaev, A. I., Korotin, M. A., Anokhin, A. O. & Kotliar, G. First-principles calculations of the electronic structure and spectra of strongly correlated systems: dynamical mean-field theory. J. Phys. Condens. Matter 9, 7359–7367 (1997).

    Article  Google Scholar 

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

    Article  Google Scholar 

  81. Wouters, S., Jiménez-Hoyos, C. A., Sun, Q. & Chan, G. K.-L. A practical guide to density matrix embedding theory in quantum chemistry. J. Chem. Theory Comput. 12, 2706–2719 (2016).

    Article  Google Scholar 

  82. Knizia, G. & Chan, G. K.-L. Density matrix embedding: a simple alternative to dynamical mean-field theory. Phys. Rev. Lett. 109, 186404 (2012).

    Article  Google Scholar 

  83. Knizia, G. & Chan, G. K.-L. Density matrix embedding: a strong-coupling quantum embedding theory. J. Chem. Theory Comput. 9, 1428–1432 (2013).

    Article  Google Scholar 

  84. Pham, H. Q., Hermes, M. R. & Gagliardi, L. Periodic electronic structure calculations with the density matrix embedding theory. J. Chem. Theory Comput. 16, 130–140 (2020).

    Article  Google Scholar 

  85. Hermes, M. R. & Gagliardi, L. Multiconfigurational self-consistent field theory with density matrix embedding: the localized active space self-consistent field method. J. Chem. Theory Comput. 15, 972–986 (2019).

    Article  Google Scholar 

  86. Pham, H. Q., Bernales, V. & Gagliardi, L. Can density matrix embedding theory with the complete activate space self-consistent field solver describe single and double bond breaking in molecular systems? J. Chem. Theory Comput. 14, 1960–1968 (2018).

    Article  Google Scholar 

  87. Rungger, I. et al. Dynamical mean field theory algorithm and experiment on quantum computers. Preprint at http://arxiv.org/abs/1910.04735 (2020).

  88. Keen, T., Maier, T., Johnston, S. & Lougovski, P. Quantum–classical simulation of two-site dynamical mean-field theory on noisy quantum hardware. Quantum Sci. Technol. 5, 035001 (2020).

    Article  Google Scholar 

  89. Yao, Y., Zhang, F., Wang, C.-Z., Ho, K.-M. & Orth, P. P. Gutzwiller hybrid quantum–classical computing approach for correlated materials. Phys. Rev. Res. 3, 013184 (2021).

    Article  Google Scholar 

  90. Tilly, J. et al. Reduced density matrix sampling: self-consistent embedding and multiscale electronic structure on current generation quantum computers. Phys. Rev. Res. 3, 033230 (2021).

    Article  Google Scholar 

  91. Bassman, L. et al. Simulating quantum materials with digital quantum computers. Quantum Sci. Technol. 6, 043002 (2021).

    Article  Google Scholar 

  92. Cerasoli, F. T., Sherbert, K., Sławińska, J. & Nardelli, M. B. Quantum computation of silicon electronic band structure. Phys. Chem. Chem. Phys. 22, 21816–21822 (2020).

    Article  Google Scholar 

  93. Sureshbabu, S. H., Sajjan, M., Oh, S. & Kais, S. Implementation of quantum machine learning for electronic structure calculations of periodic systems on quantum computing devices. J. Chem. Inf. Modeling 61, 2667–2674 (2021).

  94. Choudhary, K. Quantum computation for predicting electron and phonon properties of solids. J. Phys. Condens. Matter 33, 385501 (2021).

    Article  Google Scholar 

  95. Libisch, F., Huang, C. & Carter, E. A. Embedded correlated wavefunction schemes: theory and applications. Acc. Chem. Res. 47, 2768–2775 (2014).

    Article  Google Scholar 

  96. Wesolowski, T. A., Shedge, S. & Zhou, X. Frozen-density embedding strategy for multilevel simulations of electronic structure. Chem. Rev. 115, 5891–5928 (2015).

    Article  Google Scholar 

  97. Jacob, C. R. & Neugebauer, J. Subsystem density-functional theory. WIREs Comput. Mol. Sci. 4, 325–362 (2014).

    Article  Google Scholar 

  98. Ma, H., Sheng, N., Govoni, M. & Galli, G. First-principles studies of strongly correlated states in defect spin qubits in diamond. Phys. Chem. Chem. Phys. 22, 25522–25527 (2020).

    Article  Google Scholar 

  99. Ma, H., Govoni, M. & Galli, G. Quantum simulations of materials on near-term quantum computers. npj Comput. Mater. 6, 85 (2020).

  100. Ma, H., Sheng, N., Govoni, M. & Galli, G. Quantum embedding theory for strongly correlated states in materials. J. Chem. Theory Comput. 17, 2116–2125 (2021).

    Article  Google Scholar 

  101. Lan, T. N. & Zgid, D. Generalized self-energy embedding theory. J. Phys. Chem. Lett. 8, 2200–2205 (2017).

    Article  Google Scholar 

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

    Article  Google Scholar 

  103. Rusakov, A. A., Iskakov, S., Tran, L. N. & Zgid, D. Self-energy embedding theory (SEET) for periodic systems. J. Chem. Theory Comput. 15, 229–240 (2019).

    Article  Google Scholar 

  104. Biermann, S., Aryasetiawan, F. & Georges, A. First-principles approach to the electronic structure of strongly correlated systems: combining the GW approximation and dynamical mean-field theory. Phys. Rev. Lett. 90, 086402 (2003).

    Article  Google Scholar 

  105. Biermann, S. Dynamical screening effects in correlated electron materials—a progress report on combined many-body perturbation and dynamical mean field theory: ‘GW + DMFT’. J. Phys. Condens. Matter 26, 173202 (2014).

    Article  Google Scholar 

  106. Boehnke, L., Nilsson, F., Aryasetiawan, F. & Werner, P. When strong correlations become weak: consistent merging of GW and DMFT. Phys. Rev. B 94, 201106 (2016).

    Article  Google Scholar 

  107. Choi, S., Kutepov, A., Haule, K., van Schilfgaarde, M. & Kotliar, G. First-principles treatment of Mott insulators: linearized QSGW + DMFT approach npj Quantum Mater. 1, 16001 (2016).

  108. Nilsson, F., Boehnke, L., Werner, P. & Aryasetiawan, F. Multitier self-consistent GW + EDMFT. Phys. Rev. Mater. 1, 043803 (2017).

    Article  Google Scholar 

  109. Sun, P. & Kotliar, G. Extended dynamical mean-field theory and GW method. Phys. Rev. B 66, 085120 (2002).

    Article  Google Scholar 

  110. Lichtenstein, A. I. & Katsnelson, M. I. Ab initio calculations of quasiparticle band structure in correlated systems: LDA++ approach. Phys. Rev. B 57, 6884–6895 (1998).

    Article  Google Scholar 

  111. Dhawan, D., Metcalf, M. & Zgid, D. Dynamical self-energy mapping (DSEM) for quantum computing. Preprint at http://arxiv.org/abs/2010.05441 (2021).

  112. Otten, M. et al. Localized quantum chemistry on quantum computers. Preprint at https://doi.org/10.33774/chemrxiv-2021-0nmwt (2021).

  113. Seo, H., Govoni, M. & Galli, G. Design of defect spins in piezoelectric aluminum nitride for solid-state hybrid quantum technologies. Sci. Rep. 6, 20803 (2016).

    Article  Google Scholar 

  114. Seo, H., Ma, H., Govoni, M. & Galli, G. Designing defect-based qubit candidates in wide-gap binary semiconductors for solid-state quantum technologies. Phys. Rev. Mater. 1, 075002 (2017).

    Article  Google Scholar 

  115. Ivády, V., Abrikosov, I. A. & Gali, A. First principles calculation of spin-related quantities for point defect qubit research. npj Comput. Mater. 4, 76 (2018). .

  116. Anderson, C. P. et al. Electrical and optical control of single spins integrated in scalable semiconductor devices. Science 366, 1225–1230 (2019).

    Article  Google Scholar 

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

    Article  Google Scholar 

  118. Jones, L. O., Mosquera, M. A., Schatz, G. C. & Ratner, M. A. Embedding methods for quantum chemistry: applications from materials to life sciences. J. Am. Chem. Soc. 142, 3281–3295 (2020).

    Article  Google Scholar 

  119. Lin, H. & Truhlar, D. G. QM/MM: what have we learned, where are we, and where do we go from here? Theor. Chem. Acc. 117, 185 (2006).

    Article  Google Scholar 

  120. Wang, B. et al. Quantum mechanical fragment methods based on partitioning atoms or partitioning coordinates. Acc. Chem. Res. 47, 2731–2738 (2014).

    Article  Google Scholar 

  121. Pezeshki, S. & Lin, H. Recent developments in QM/MM methods towards open-boundary multi-scale simulations. Mol. Simul. 41, 168–189 (2015).

    Article  Google Scholar 

  122. He, N. & Evangelista, F. A. A zeroth-order active-space frozen-orbital embedding scheme for multireference calculations. J. Chem. Phys. 152, 094107 (2020).

    Article  Google Scholar 

  123. Gujarati, T. P. et al. Quantum computation of reactions on surfaces using local embedding. Preprint at http://arxiv.org/abs/2203.07536 (2022).

  124. Lau, B. T. G., Knizia, G. & Berkelbach, T. C. Regional embedding enables high-level quantum chemistry for surface science. J. Phys. Chem. Lett. 12, 1104–1109 (2021).

    Article  Google Scholar 

  125. Cui, Z.-H., Zhu, T. & Chan, G. K.-L. Efficient implementation of ab initio quantum embedding in periodic systems: density matrix embedding theory. J. Chem. Theory Comput. 16, 119–129 (2020).

    Article  Google Scholar 

  126. Cui, Z.-H., Zhai, H., Zhang, X. & Chan, G. K.-L. Systematic electronic structure in the cuprate parent state from quantum many-body simulations. Preprint at http://arxiv.org/abs/2112.09735 (2022).

  127. Anderson, P. W. Localized magnetic states in metals. Phys. Rev. 124, 41–53 (1961).

    Article  MathSciNet  Google Scholar 

  128. Sheng, N., Vorwerk, C., Govoni, M. & Galli, G. Green’s function formulation of quantum defect embedding theory. J. Chem. Theory Comput. 18, 3512–3522 (2022).

    Article  Google Scholar 

  129. Werner, P. & Millis, A. J. Efficient dynamical mean field simulation of the Holstein–Hubbard model. Phys. Rev. Lett. 99, 146404 (2007).

    Article  Google Scholar 

  130. Nilsson, F. & Aryasetiawan, F. Recent progress in first-principles methods for computing the electronic structure of correlated materials. Computation 6, 26 (2018).

    Article  Google Scholar 

  131. Sakuma, R., Werner, P. & Aryasetiawan, F. Electronic structure of SrVO3 within GW + DMFT. Phys. Rev. B 88, 235110 (2013).

    Article  Google Scholar 

  132. Petocchi, F., Nilsson, F., Aryasetiawan, F. & Werner, P. Screening from eg states and antiferromagnetic correlations in d(1, 2, 3) perovskites: a GW + EDMFT investigation. Phys. Rev. Res. 2, 013191 (2020).

    Article  Google Scholar 

  133. Tomczak, J. M., Liu, P., Toschi, A., Kresse, G. & Held, K. Merging GW with DMFT and non-local correlations beyond. Eur. Phys. J. Spec. Top. 226, 2565–2590 (2017).

    Article  Google Scholar 

  134. Reining, L. The GW approximation: content, successes and limitations. WIREs Comput. Mol. Sci. 8, e1344 (2018).

    Article  Google Scholar 

  135. Onida, G., Reining, L. & Rubio, A. Electronic excitations: density-functional versus many-body Green’s-function approaches. Rev. Mod. Phys. 74, 601–659 (2002).

    Article  Google Scholar 

  136. Hedin, L. On correlation effects in electron spectroscopies and the GW approximation. J. Phys. Condens. Matter 11, R489–R528 (1999).

    Article  Google Scholar 

  137. Aryasetiawan, F. & Gunnarsson, O. The GW method. Rep. Prog. Phys. 61, 237–312 (1998).

    Article  Google Scholar 

  138. Golze, D., Dvorak, M. & Rinke, P. The GW compendium: a practical guide to theoretical photoemission spectroscopy. Front. Chem. 7, 377 (2019).

    Article  Google Scholar 

  139. Choi, S., Semon, P., Kang, B., Kutepov, A. & Kotliar, G. ComDMFT: a massively parallel computer package for the electronic structure of correlated-electron systems. Comput. Phys. Commun. 244, 277–294 (2019).

    Article  Google Scholar 

  140. Tomczak, J. M., Casula, M., Miyake, T., Aryasetiawan, F. & Biermann, S. Combined GW and dynamical mean-field theory: dynamical screening effects in transition metal oxides. EPL 100, 67001 (2012).

    Article  Google Scholar 

  141. Aryasetiawan, F. et al. Frequency-dependent local interactions and low-energy effective models from electronic structure calculations. Phys. Rev. B 70, 195104 (2004).

    Article  Google Scholar 

  142. Aryasetiawan, F., Tomczak, J. M., Miyake, T. & Sakuma, R. Downfolded self-energy of many-electron systems. Phys. Rev. Lett. 102, 176402 (2009).

    Article  Google Scholar 

  143. Miyake, T. & Aryasetiawan, F. Screened Coulomb interaction in the maximally localized Wannier basis. Phys. Rev. B 77, 085122 (2008).

    Article  Google Scholar 

  144. Hampel, A., Beck, S. & Ederer, C. Effect of charge self-consistency in DFT + DMFT calculations for complex transition metal oxides. Phys. Rev. Res. 2, 033088 (2020).

    Article  Google Scholar 

  145. Bhandary, S. & Held, K. Self-energy self-consistent density functional theory plus dynamical mean field theory. Phys. Rev. B 103, 245116 (2021).

    Article  Google Scholar 

  146. Lee, J. & Haule, K. Diatomic molecule as a testbed for combining DMFT with electronic structure methods such as GW and DFT. Phys. Rev. B 95, 155104 (2017).

    Article  Google Scholar 

  147. Eidelstein, E., Gull, E. & Cohen, G. Multiorbital quantum impurity solver for general interactions and hybridizations. Phys. Rev. Lett. 124, 206405 (2020).

    Article  Google Scholar 

  148. 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).

    Article  Google Scholar 

  149. Werner, P. & Millis, A. J. Dynamical screening in correlated electron materials. Phys. Rev. Lett. 104, 146401 (2010).

    Article  Google Scholar 

  150. Medvedeva, D., Iskakov, S., Krien, F., Mazurenko, V. V. & Lichtenstein, A. I. Exact diagonalization solver for extended dynamical mean-field theory. Phys. Rev. B 96, 235149 (2017).

    Article  Google Scholar 

  151. Werner, P. & Casula, M. Dynamical screening in correlated electron systems—from lattice models to realistic materials. J. Phys. Condens. Matter 28, 383001 (2016).

    Article  Google Scholar 

  152. Adler, R., Kang, C.-J., Yee, C.-H. & Kotliar, G. Correlated materials design: prospects and challenges. Rep. Prog. Phys. 82, 012504 (2018).

    Article  Google Scholar 

  153. Haule, K. Exact double counting in combining the dynamical mean field theory and the density functional theory. Phys. Rev. Lett. 115, 196403 (2015).

    Article  Google Scholar 

  154. Haule, K., Yee, C.-H. & Kim, K. Dynamical mean-field theory within the full-potential methods: electronic structure of CeIrIn5, CeCoIn5, and CeRhIn5. Phys. Rev. B 81, 195107 (2010).

    Article  Google Scholar 

  155. Haule, K., Birol, T. & Kotliar, G. Covalency in transition-metal oxides within all-electron dynamical mean-field theory. Phys. Rev. B 90, 075136 (2014).

    Article  Google Scholar 

  156. van Roekeghem, A. et al. Dynamical correlations and screened exchange on the experimental bench: spectral properties of the cobalt pnictide BaCo2As2. Phys. Rev. Lett. 113, 266403 (2014).

    Article  Google Scholar 

  157. Yeh, C.-N., Iskakov, S., Zgid, D. & Gull, E. Electron correlations in the cubic paramagnetic perovskite Sr(V, Mn)O3: results from fully self-consistent self-energy embedding calculations. Phys. Rev. B 103, 195149 (2021).

    Article  Google Scholar 

  158. Iskakov, S., Yeh, C.-N., Gull, E. & Zgid, D. Ab initio self-energy embedding for the photoemission spectra of NiO and MnO. Phys. Rev. B 102, 085105 (2021).

    Article  Google Scholar 

  159. Kananenka, A. A., Gull, E. & Zgid, D. Systematically improvable multiscale solver for correlated electron systems. Phys. Rev. B 91, 121111 (2015).

    Article  Google Scholar 

  160. Lan, T. N., Kananenka, A. A. & Zgid, D. Communication: Towards ab initio self-energy embedding theory in quantum chemistry. J. Chem. Phys. 143, 241102 (2015).

    Article  Google Scholar 

  161. Lan, T. N., Shee, A., Li, J., Gull, E. & Zgid, D. Testing self-energy embedding theory in combination with GW. Phys. Rev. B 96, 155106 (2017).

    Article  Google Scholar 

  162. Muechler, L. et al. Quantum embedding methods for correlated excited states of point defects: Case studies and challenges. Phys. Rev. B 105, 235104 (2022).

    Article  Google Scholar 

  163. Govoni, M. & Galli, G. Large scale GW calculations. J. Chem. Theory Comput. 11, 2680–2696 (2015).

    Article  Google Scholar 

  164. Scherpelz, P., Govoni, M., Hamada, I. & Galli, G. Implementation and validation of fully relativistic GW calculations: spin–orbit coupling in molecules, nanocrystals, and solids. J. Chem. Theory Comput. 12, 3523–3544 (2016).

    Article  Google Scholar 

  165. Govoni, M. & Galli, G. GW100: comparison of methods and accuracy of results obtained with the WEST code. J. Chem. Theory Comput. 14, 1895–1909 (2018).

    Article  Google Scholar 

  166. Govoni, M., Whitmer, J., de Pablo, J., Gygi, F. & Galli, G. Code interoperability extends the scope of quantum simulations. npj Comput. Mater. 7, 32 (2021).

  167. Casula, M., Rubtsov, A. & Biermann, S. Dynamical screening effects in correlated materials: plasmon satellites and spectral weight transfers from a Green’s function ansatz to extended dynamical mean field theory. Phys. Rev. B 85, 035115 (2012).

    Article  Google Scholar 

  168. Krivenko, I. S. & Biermann, S. Slave rotor approach to dynamically screened Coulomb interactions in solids. Phys. Rev. B 91, 155149 (2015).

    Article  Google Scholar 

  169. Nomura, Y., Sakai, S. & Arita, R. Multiorbital cluster dynamical mean-field theory with an improved continuous-time quantum Monte Carlo algorithm. Phys. Rev. B 89, 195146 (2014).

    Article  Google Scholar 

  170. Mizuno, R., Ochi, M. & Kuroki, K. Development of an efficient impurity solver in dynamical mean field theory for multiband systems: iterative perturbation theory combined with parquet equations. Phys. Rev. B 104, 035160 (2021).

    Article  Google Scholar 

  171. Kotliar, G., Savrasov, S. Y., Pálsson, G. & Biroli, G. Cellular dynamical mean field approach to strongly correlated systems. Phys. Rev. Lett. 87, 186401 (2001).

    Article  Google Scholar 

  172. De Leo, L., Civelli, M. & Kotliar, G. Cellular dynamical mean-field theory of the periodic Anderson model. Phys. Rev. B 77, 075107 (2008).

    Article  Google Scholar 

  173. Gull, E. et al. Submatrix updates for the continuous-time auxiliary-field algorithm. Phys. Rev. B 83, 075122 (2011).

    Article  Google Scholar 

  174. Simons Collaboration on the Many-Electron Problem et al. Solutions of the two-dimensional Hubbard model: benchmarks and results from a wide range of numerical algorithms. Phys. Rev. X 5, 041041 (2015).

  175. Jamet, F. et al. Krylov variational quantum algorithm for first principles materials simulations. Preprint at http://arxiv.org/abs/2105.13298 (2021).

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

    Article  Google Scholar 

  177. Huang, B., Govoni, M. & Galli, G. Simulating the electronic structure of spin defects on quantum computers. PRX Quantum 3, 010339 (2022).

    Article  Google Scholar 

  178. 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).

    Article  Google Scholar 

  179. Endo, S., Cai, Z., Benjamin, S. C. & Yuan, X. Hybrid quantum–classical algorithms and quantum error mitigation. J. Phys. Soc. Jpn 90, 032001 (2021).

    Article  Google Scholar 

  180. Bauer, B., Bravyi, S., Motta, M. & Kin-Lic Chan, G. Quantum algorithms for quantum chemistry and quantum materials science. Chem. Rev. 120, 12685–12717 (2020).

    Article  Google Scholar 

  181. Korol, K. J. M., Choo, K. & Mezzacapo, A. Quantum approximation algorithms for many-body and electronic structure problems. Preprint at http://arxiv.org/abs/2111.08090 (2021).

  182. Wecker, D., Bauer, B., Clark, B. K., Hastings, M. B. & Troyer, M. Gate-count estimates for performing quantum chemistry on small quantum computers. Phys. Rev. A 90, 022305 (2014).

    Article  Google Scholar 

  183. Preskill, J. Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018).

    Article  Google Scholar 

  184. Lebreuilly, J., Noh, K., Wang, C.-H., Girvin, S. M. & Jiang, L. Autonomous quantum error correction and quantum computation. Preprint at http://arxiv.org/abs/2103.05007 (2021).

  185. Fedorov, D. A., Otten, M. J., Gray, S. K. & Alexeev, Y. Ab initio molecular dynamics on quantum computers. J. Chem. Phys. 154, 164103 (2021).

    Article  Google Scholar 

  186. Macridin, A., Spentzouris, P., Amundson, J. & Harnik, R. Electron–phonon systems on a universal quantum computer. Phys. Rev. Lett. 121, 110504 (2018).

    Article  Google Scholar 

  187. Powers, C., Bassman, L. & de Jong, W. A. Exploring finite temperature properties of materials with quantum computers. Preprint at http://arxiv.org/abs/2109.01619 (2021).

  188. Wu, J. & Hsieh, T. H. Variational thermal quantum simulation via thermofield double states. Phys. Rev. Lett. 123, 220502 (2019).

    Article  Google Scholar 

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Acknowledgements

We thank E. Gull and H. Ma for fruitful discussions. This work was supported by MICCoM, as part of the Computational Materials Sciences Program funded by the US Department of Energy. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a DOE Office of Science User Facility supported by the Office of Science of the US DOE under contract DE-AC02-05CH11231, resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under contract DE-AC02-06CH11357, and resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the US DOE under contract DE-AC05-00OR22725. We acknowledge the use of IBM Quantum services for this work. The views expressed are those of the authors, and do not reflect the official policy or position of IBM or the IBM Quantum team.

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  1. These authors contributed equally: Christian Vorwerk, Nan Sheng.

Authors and Affiliations

  1. Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL, USA

    Christian Vorwerk & Giulia Galli

  2. Department of Chemistry, University of Chicago, Chicago, IL, USA

    Nan Sheng, Benchen Huang & Giulia Galli

  3. Materials Science Division and Center for Molecular Engineering, Argonne National Laboratory, Lemont, IL, USA

    Marco Govoni & Giulia Galli

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G.G. conceived this perspective and formulated the final content with all authors. All authors contributed to the writing of the manuscript.

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Vorwerk, C., Sheng, N., Govoni, M. et al. Quantum embedding theories to simulate condensed systems on quantum computers. Nat Comput Sci 2, 424–432 (2022). https://doi.org/10.1038/s43588-022-00279-0

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