Communications in Mathematical Physics

, Volume 356, Issue 1, pp 65–105 | Cite as

Rigorous RG Algorithms and Area Laws for Low Energy Eigenstates in 1D

  • Itai Arad
  • Zeph Landau
  • Umesh Vazirani
  • Thomas VidickEmail author
Open Access


One of the central challenges in the study of quantum many-body systems is the complexity of simulating them on a classical computer. A recent advance (Landau et al. in Nat Phys, 2015) gave a polynomial time algorithm to compute a succinct classical description for unique ground states of gapped 1D quantum systems. Despite this progress many questions remained unsolved, including whether there exist efficient algorithms when the ground space is degenerate (and of polynomial dimension in the system size), or for the polynomially many lowest energy states, or even whether such states admit succinct classical descriptions or area laws. In this paper we give a new algorithm, based on a rigorously justified RG type transformation, for finding low energy states for 1D Hamiltonians acting on a chain of n particles. In the process we resolve some of the aforementioned open questions, including giving a polynomial time algorithm for poly(n) degenerate ground spaces and an n O(log n) algorithm for the poly(n) lowest energy states (under a mild density condition). For these classes of systems the existence of a succinct classical description and area laws were not rigorously proved before this work. The algorithms are natural and efficient, and for the case of finding unique ground states for frustration-free Hamiltonians the running time is \({\tilde{O}(nM(n))}\) , where M(n) is the time required to multiply two n × n matrices.


  1. 1.
    Aharonov D., Arad I., Vazirani U., Landau Z.: The detectability lemma and its applications to quantum hamiltonian complexity. New J. Phys. 13(11), 113043 (2011)ADSCrossRefGoogle Scholar
  2. 2.
    Anshu A., Arad I., Vidick T.: Simple proof of the detectability lemma and spectral gap amplification. Phys. Rev. B 93(20), 205142 (2016)ADSCrossRefGoogle Scholar
  3. 3.
    Arad, I., Kitaev, A., Landau, Z., Vazirani, U.: An area law and sub-exponential algorithm for 1D systems. In: Proceedings of the 4th Innovations in Theoretical Computer Science (ITCS) (2013)Google Scholar
  4. 4.
    Arad, I., Kuwahara, T., Landau, Z.: Connecting global and local energy distributions in quantum spin models on a lattice. Technical report, J. Stat. Mech. 3, 033301 (2016)Google Scholar
  5. 5.
    Arad I., Landau Z., Vazirani U.: Improved one-dimensional area law for frustration-free systems. Phys. Rev. B 85, 195145 (2012)ADSCrossRefGoogle Scholar
  6. 6.
    Bravyi S., Gosset D.: Gapped and gapless phases of frustration-free spin-1 2 chains. J. Math. Phys. 56(6), 061902 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bravyi S., Hastings M.B., Michalakis S.: Topological quantum order: stability under local perturbations. J. Math. Phys. 51(9), 093512 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bridgeman J.C., Chubb C.T.: Hand-waving and interpretive dance: An introductory course on tensor networks. J. Phys. A Math. Theor. 50, 223001 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chubb, C.T., Flammia, S.T.: Computing the degenerate ground space of gapped spin chains in polynomial time. Chic. J. Theor. Comput. 9, 1–35 (2016)Google Scholar
  10. 10.
    Dasgupta S., Gupta A.: An elementary proof of a theorem of Johnson and Lindenstrauss. Random Struct. Algorithms 22(1), 60–65 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    de Beaudrap N., Osborne T.J., Eisert J.: Ground states of unfrustrated spin hamiltonians satisfy an area law. New J. Phys. 12(9), 095007 (2010)ADSCrossRefGoogle Scholar
  12. 12.
    Eisert J., Cramer M., Plenio M.B.: Colloquium: area laws for the entanglement entropy. Rev. Mod. Phys. 82(1), 277–306 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Evenbly G., Vidal G.: Tensor network renormalization yields the multiscale entanglement renormalization ansatz. Phys. Rev. Lett. 115(20), 200401 (2015)ADSCrossRefGoogle Scholar
  14. 14.
    Feynman R.P.: Simulating physics with computers. Int. J. Theor. Phys. 21(6), 467–488 (1982)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hastings M.B.: Solving gapped hamiltonians locally. Phys. Rev. B 73(8), 085115 (2006)ADSCrossRefGoogle Scholar
  16. 16.
    Hastings M.B.: An area law for one-dimensional quantum systems. J. Stat. Mech. Theory Exp. 2007(08), P08024 (2007)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Huang, Y.: A polynomial-time algorithm for the ground state of one-dimensional gapped Hamiltonians (2014). arXiv:1406.6355
  18. 18.
    Huang, Y.: A simple efficient algorithm in frustration-free one-dimensional gapped systems (2015). arXiv:1510.01303
  19. 19.
    Keating J., Linden N., Wells H.: Spectra and eigenstates of spin chain hamiltonians. Commun. Math. Phys. 338(1), 81–102 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kliesch, M., Gogolin, C., Kastoryano, J., Riera, M.A., Eisert, J.: Locality of temperature. Phys. Rev. X 4, 031019 (2014)Google Scholar
  21. 21.
    Kuwahara, T., Arad, I., Amico, L., Vedral, V.: Local reversibility and entanglement structure of many-body ground states. Quantum Sci. Technol. 2, 015005 (2017)Google Scholar
  22. 22.
    Landau, Z., Vazirani, U., Vidick, T.: A polynomial time algorithm for the ground state of one-dimensional gapped local Hamiltonians. Nat. Phys. (2015). doi: 10.1038/nphys3345
  23. 23.
    Maldacena J.: Eternal black holes in anti-de sitter. J. High Energy Phys. 2003(04), 021 (2003)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Masanes L.: Area law for the entropy of low-energy states. Phys. Rev. A 80(5), 052104 (2009)ADSCrossRefGoogle Scholar
  25. 25.
    Molnar A., Schuch N., Verstraete F., Cirac J.I.: Approximating gibbs states of local hamiltonians efficiently with projected entangled pair states. Phys. Rev. B 91, 045138 (2015)ADSCrossRefGoogle Scholar
  26. 26.
    Pastawski, F., Yoshida, B., Harlow, D., Preskill, J.: Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence. JHEP 06, 149 (2015)Google Scholar
  27. 27.
    Roberts, B., Vidick, T., Motrunich, O.I.: Rigorous renormalization group method for ground space and low-energy states of local Hamiltonians. arXiv:1703.01994 (2017)
  28. 28.
    Schollwöck U.: The density-matrix renormalization group in the age of matrix product states. Ann. Phys. 326(1), 96–192 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Vershynin, R.: Introduction to the non-asymptotic analysis of random matrices. In: Eldar, Y. et al. (eds.) Compressed Sensing, Theory and Applications. Cambridge University Press. arXiv:1011.3027 (2012)
  30. 30.
    Verstraete F., Murg V., Cirac J.I.: Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems. Adv. Phys. 57(2), 143–224 (2008)ADSCrossRefGoogle Scholar
  31. 31.
    Vidal G.: Class of quantum many-body states that can be efficiently simulated. Phys. Rev. Lett. 101, 110501 (2008)ADSCrossRefGoogle Scholar
  32. 32.
    Vidal G.: Entanglement renormalization: an introduction. arXiv:0912.1651 (2009)
  33. 33.
    White S.R.: Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett. 69, 2863–2866 (1992)ADSCrossRefGoogle Scholar
  34. 34.
    White S.R.: Density-matrix algorithms for quantum renormalization groups. Phys. Rev. B 48, 10345–10356 (1993)ADSCrossRefGoogle Scholar
  35. 35.
    Wilson K.G.: The renormalization group: Critical phenomena and the kondo problem. Rev. Mod. Phys. 47(4), 773 (1975)ADSMathSciNetCrossRefGoogle Scholar

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© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Itai Arad
    • 1
  • Zeph Landau
    • 2
  • Umesh Vazirani
    • 2
  • Thomas Vidick
    • 3
    Email author
  1. 1.Centre for Quantum Technologies (CQT)National University of SingaporeSingaporeSingapore
  2. 2.Electrical Engineering and Computer SciencesUniversity of CaliforniaBerkeleyUSA
  3. 3.Department of Computing and Mathematical SciencesCalifornia Institute of TechnologyPasadenaUSA

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