Complexity of Quantum Impurity Problems


We give a quasi-polynomial time classical algorithm for estimating the ground state energy and for computing low energy states of quantum impurity models. Such models describe a bath of free fermions coupled to a small interacting subsystem called an impurity. The full system consists of n fermionic modes and has a Hamiltonian \({H=H_0+H_{imp}}\), where H 0 is quadratic in creation–annihilation operators and H imp is an arbitrary Hamiltonian acting on a subset of O(1) modes. We show that the ground energy of H can be approximated with an additive error \({2^{-b}}\) in time \({n^3 \exp{[O(b^3)]}}\). Our algorithm also finds a low energy state that achieves this approximation. The low energy state is represented as a superposition of \({\exp{[O(b^3)]}}\) fermionic Gaussian states. To arrive at this result we prove several theorems concerning exact ground states of impurity models. In particular, we show that eigenvalues of the ground state covariance matrix decay exponentially with the exponent depending very mildly on the spectral gap of H 0. A key ingredient of our proof is Zolotarev’s rational approximation to the \({\sqrt{x}}\) function. We anticipate that our algorithms may be used in hybrid quantum-classical simulations of strongly correlated materials based on dynamical mean field theory. We implemented a simplified practical version of our algorithm and benchmarked it using the single impurity Anderson model.

This is a preview of subscription content, log in to check access.


  1. 1

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

    ADS  MathSciNet  Article  Google Scholar 

  2. 2

    Kondo J.: Resistance minimum in dilute magnetic alloys. Prog. Theor. Phys. 32(1), 37–49 (1964)

    ADS  Article  Google Scholar 

  3. 3

    Wilson K.G.: The renormalization group: critical phenomena and the Kondo problem. Rev. Mod. Phys. 47(4), 773 (1975)

    ADS  MathSciNet  Article  Google Scholar 

  4. 4

    Kouwenhoven L., Glazman L.: Revival of the Kondo effect. Phys. World 14(1), 33 (2001)

    Article  Google Scholar 

  5. 5

    De Haas W.J., Van Den Berg G.J.: The electrical resistance of gold and silver at low temperatures. Physica 3(6), 440–449 (1936)

    ADS  Article  Google Scholar 

  6. 6

    Georges A.: Strongly correlated electron materials: dynamical mean field theory and electronic structure. AIP Conf. Prod. 715(1), 3–74 (2004)

    ADS  Article  Google Scholar 

  7. 7

    Kotliar G., Savrasov S., Haule K., Oudovenko V., Parcollet O., Marianetti C.: Electronic structure calculations with dynamical mean-field theory. Rev. Mod. Phys. 78(3), 865 (2006)

    ADS  Article  Google Scholar 

  8. 8

    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 

  9. 9

    Kreula J.M., García-Álvarez L., Lamata L., Clark S.R., Solano E., Jaksch D.: Few-qubit quantum-classical simulation of strongly correlated lattice fermions. EPJ Quantum Technol. 3, 11 (2016)

    Article  Google Scholar 

  10. 10

    Gharibian S., Huang Y., Landau Z., Shin S.W.: Quantum Hamiltonian complexity. Found. Trends Theor. Comput. Sci. 10(3), 159–282 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11

    Kitaev A.Yu., Shen A.H., Vyalyi M.N.: Classical and Quantum Computation. American Mathematical Society, Boston (2002)

    Google Scholar 

  12. 12

    Schuch N., Verstraete F.: Computational complexity of interacting electrons and fundamental limitations of density functional theory. Nat. Phys. 5(10), 732–735 (2009)

    Article  Google Scholar 

  13. 13

    Bini D., Pan V.Y.: Computing matrix eigenvalues and polynomial zeros where the output is real. SIAM J. Comput. 27(4), 1099–1115 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14

    Wiegmann P.B., Tsvelick A.M.: Exact solution of the Anderson model: I. J. Phys. C: Solid State Phys. 16(12), 2281 (1983)

    ADS  Article  Google Scholar 

  15. 15

    Kawakami N., Okiji A.: Exact expression of the ground-state energy for the symmetric Anderson model. Phys. Lett. A 86(9), 483–486 (1981)

    ADS  MathSciNet  Article  Google Scholar 

  16. 16

    Zolotarev E.: Application of elliptic functions to questions of functions deviating least and most from zero. Zap. Imp. Akad. Nauk. St. Petersburg 30(5), 1–59 (1877)

    Google Scholar 

  17. 17

    Brod D.J., Childs A.M.: The computational power of matchgates and the XY interaction on arbitrary graphs. Quantum Inf. Comput. 14(11-12), 901–916 (2014)

    MathSciNet  Google Scholar 

  18. 18

    Kraus C., Cirac J.I.: Generalized Hartree–Fock theory for interacting fermions in lattices: numerical methods. New J. Phys. 12(11), 113004 (2010)

    ADS  Article  Google Scholar 

  19. 19

    Landau Z., Vazirani U., Vidick T.: A polynomial time algorithm for the ground state of one-dimensional gapped local Hamiltonians. Nat. Phys. 11(7), 566–569 (2015)

    Article  Google Scholar 

  20. 20

    Aharonov, D., Naveh, T.: Quantum NP—A Survey. arXiv:quant-ph/0210077, (2002)

  21. 21

    Terhal B.M., DiVincenzo D.P.: Classical simulation of noninteracting-fermion quantum circuits. Phys. Rev. A 65(3), 032325 (2002)

    ADS  Article  Google Scholar 

  22. 22

    Bravyi S.: Lagrangian representation for fermionic linear optics. Quantum Inf. Comput. 5(3), 216–238 (2005)

    MathSciNet  MATH  Google Scholar 

  23. 23

    Wimmer M.: Efficient numerical computation of the Pfaffian for dense and banded skew-symmetric matrices. ACM Trans. Math. Softw. 38, 4 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24

    Rubow J., Wolff U.: A factorization algorithm to compute Pfaffians. Comput. Phys. Commun. 182(12), 2530–2532 (2011)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  25. 25

    Schliemann J., Cirac J.I., Kuś M., Lewenstein M., Loss D.: Quantum correlations in two-fermion systems. Phys. Rev. A 64(2), 022303 (2001)

    ADS  Article  Google Scholar 

  26. 26

    Kitaev A.Yu.: Unpaired Majorana fermions in quantum wires. Physics-Uspekhi 44(10S), 131 (2001)

    ADS  Article  Google Scholar 

  27. 27

    Löwdin P.-O.: Quantum theory of many-particle systems I. Phys. Rev. 97(6), 1474 (1955)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  28. 28

    Kennedy A.: Approximation theory for matrices. Nucl. Phys. B Proc. Suppl. 128, 107–116 (2004)

    ADS  Article  Google Scholar 

  29. 29

    Chiu T.-W., Hsieh T.-H., Huang C.-H., Huang T.-R.: Note on the Zolotarev optimal rational approximation for the overlap Dirac operator. Phys. Rev. D 66(11), 114502 (2002)

    ADS  Article  Google Scholar 

  30. 30

    Gončar A.A.: Zolotarev problems connected with rational functions. Sbornik: Mathematics 7(4), 623–635 (1969)

    ADS  Article  Google Scholar 

  31. 31

    Nakatsukasa, Y., Freund, R.W.: Computing fundamental matrix decompositions accurately via the matrix sign function in two iterations: The power of Zolotarev’s functions., (2015)

  32. 32

    Anderson G.D., Vamanamurthy M.K., Vuorinen M.: Functional inequalities for complete elliptic integrals and their ratios. SIAM J. Math. Anal. 21(2), 536–549 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  33. 33

    Newman D.J.: Rational approximation to \({|x|}\). Mich. Math. J. 11(1), 11–14 (1964)

    Article  MATH  Google Scholar 

  34. 34

    Schultz T.D., Mattis D.C., Lieb E.H.: Two-dimensional Ising model as a soluble problem of many fermions. Rev. Mod. Phys. 36(3), 856 (1964)

    ADS  MathSciNet  Article  Google Scholar 

  35. 35

    Pfeuty P.: The one-dimensional Ising model with a transverse field. Ann. Phys. 57(1), 79–90 (1970)

    ADS  Article  Google Scholar 

  36. 36

    Mazziotti D.A.: Variational two-electron reduced density matrix theory for many-electron atoms and molecules: implementation of the spin-and symmetry-adapted T2 condition through first-order semidefinite programming. Phys. Rev. A 72(3), 032510 (2005)

    ADS  Article  Google Scholar 

  37. 37

    Francesco P., Mathieu P., Sénéchal D.: Conformal Field Theory. Springer, Berlin (2012)

    Google Scholar 

  38. 38

    Wecker D., Hastings M.B., Wiebe N., Clark B.K., Nayak C., Troyer M.: Solving strongly correlated electron models on a quantum computer. Phys. Rev. A 92(6), 062318 (2015)

    ADS  Article  Google Scholar 

  39. 39

    Golub, G.H., Van Loan, C.F.: Matrix Computations. Johns Hopkins University Press, Baltimore (1996)

  40. 40

    Nielsen M.A., Chuang I.L.: Quantum Computation and Quantum Information. Cambridge Series on Information and the Natural Sciences. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  41. 41

    Aharonov, D., Ta-Shma, A.: Adiabatic quantum state generation and statistical zero knowledge. In: Proceedings of the Thirty-fifth Annual ACM Symposium on Theory of Computing, STOC ’03, pages 20–29, New York, NY, USA, ACM. (2003)

Download references

Author information



Corresponding author

Correspondence to David Gosset.

Additional information

Communicated by M. M. Wolf

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bravyi, S., Gosset, D. Complexity of Quantum Impurity Problems. Commun. Math. Phys. 356, 451–500 (2017).

Download citation