Skip to main content

Thank you for visiting nature.com. 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.

  • Article
  • Published:

A polynomial time algorithm for the ground state of one-dimensional gapped local Hamiltonians

Nature Physics volume 11, pages 566–569 (2015)Cite this article

Subjects

Abstract

The density matrix renormalization group method has been extensively used to study the ground state of 1D many-body systems since its introduction two decades ago. In spite of its wide use, this heuristic method is known to fail in certain cases and no certifiably correct implementation is known, leaving researchers faced with an ever-growing toolbox of heuristics, none of which is guaranteed to succeed. Here we develop a polynomial time algorithm that provably finds the ground state of any 1D quantum system described by a gapped local Hamiltonian with constant ground-state energy. The algorithm is based on a framework that combines recently discovered structural features of gapped 1D systems with an efficient construction of a class of operators called approximate ground-state projections (AGSPs). The combination of these tools yields a method that is guaranteed to succeed in all 1D gapped systems. An AGSP-centric approach may help guide the search for algorithms for more general quantum systems, including for the central challenge of 2D systems, where even heuristic methods have had more limited success.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Figure 1: Entanglement and MPSs.
Figure 2: Boundary contraction associated with a left-half MPS on the first i qudits with an open bond.

Similar content being viewed by others

References

  1. Bednorz, J. G. & Müller, K. A. Possible high-t c superconductivity in the Ba–La–Cu–O system. Z. Phys. B 64, 189–193 (1986).

    Article  ADS  Google Scholar 

  2. Tsui, D. C., Stormer, H. L. & Gossard, A. C. Two-dimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett. 48, 1559–1562 (1982).

    Article  ADS  Google Scholar 

  3. Laughlin, R. B. Anomalous quantum Hall effect: An incompressible quantum fluid with fractionally charged excitations. Phys. Rev. Lett. 50, 1395–1398 (1983).

    Article  ADS  Google Scholar 

  4. Verstraete, F. & Cirac, J. I. Renormalization algorithms for quantum-many body systems in two and higher dimensions. Preprint at http://arXiv.org/abs/cond-mat/0407066 (2004).

  5. Vidal, G. Efficient classical simulation of slightly entangled quantum computations. Phys. Rev. Lett. 91, 147902 (2003).

    Article  ADS  Google Scholar 

  6. 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, 143–224 (2008).

    Article  ADS  Google Scholar 

  7. Eisert, J., Cramer, M. & Plenio, M. B. Colloquium. Rev. Mod. Phys. 82, 277–306 (2010).

    Article  ADS  Google Scholar 

  8. Hastings, M. B. An area law for one-dimensional quantum systems. J. Stat. Mech. 2007, P08024 (2007).

    MathSciNet  Google Scholar 

  9. Wolf, M. M., Verstraete, F., Hastings, M. B. & Cirac, J. I. Area laws in quantum systems: Mutual information and correlations. Phys. Rev. Lett. 100, 070502 (2008).

    Article  ADS  MathSciNet  Google Scholar 

  10. 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).

    Article  ADS  Google Scholar 

  11. Ge, Y. & Eisert, J. Area laws and approximations of quantum many-body states. Preprint at http://arXiv.org/abs/1411.2995 (2014).

  12. White, S. R. Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett. 69, 2863–2866 (1992).

    Article  ADS  Google Scholar 

  13. Schollwöck, U. The density-matrix renormalization group. Rev. Mod. Phys. 77, 259–315 (2005).

    Article  ADS  MathSciNet  Google Scholar 

  14. Hallberg, K. A. New trends in density matrix renormalization. Adv. Phys. 55, 477–526 (2006).

    Article  ADS  Google Scholar 

  15. White, S. Spin gaps in a frustrated Heisenberg model for CaV 4O9 . Phys. Rev. Lett. 77, 3633–3636 (1996).

    Article  ADS  Google Scholar 

  16. Hieida, Y. Application of the density matrix renormalization group method to a non-equilibrium problem. J. Phys. Soc. Jpn 67, 369–372 (1998).

    Article  ADS  Google Scholar 

  17. White, S. & Scalapino, D. Density matrix renormalization group study of the striped phase in the 2d t–J model. Phys. Rev. Lett. 80, 1272–1275 (1998).

    Article  ADS  Google Scholar 

  18. White, S. & Chernyshev, A. Neél order in square and triangular lattice Heisenberg models. Phys. Rev. Lett. 99, 127004 (2007).

    Article  ADS  Google Scholar 

  19. Eisert, J. Computational difficulty of global variations in the density matrix renormalization group. Phys. Rev. Lett. 97, 260501 (2006).

    Article  ADS  Google Scholar 

  20. White, S. R. Density matrix renormalization group algorithms with a single center site. Phys. Rev. B 72, 180403 (2005).

    Article  ADS  Google Scholar 

  21. Arad, I., Landau, Z. & Vazirani, U. Improved one-dimensional area law for frustration-free systems. Phys. Rev. B 85, 195145 (2012).

    Article  ADS  Google Scholar 

  22. Kitaev, A. Y., Shen, A. H. & Vyalyi, M. N. Classical and Quantum Computation Vol. 47 (American Mathematical Society, 2002).

    Book  Google Scholar 

  23. Oliveira, R. & Terhal, B. M. The complexity of quantum spin systems on a two-dimensional square lattice. Quantum Inf. Comput. 8, 900–924 (2008).

    MathSciNet  MATH  Google Scholar 

  24. Aharonov, D., Gottesman, D., Irani, S. & Kempe, J. The power of quantum systems on a line. Commun. Math. Phys. 287, 41–65 (2009).

    Article  ADS  MathSciNet  Google Scholar 

  25. Gottesman, D. & Irani, S. The quantum and classical complexity of translationally invariant tiling and hamiltonian problems. Theory Comput. 9, 31–116 (2013).

    Article  MathSciNet  Google Scholar 

  26. Schuch, N., Cirac, I. & Verstraete, F. Computational difficulty of finding matrix product ground states. Phys. Rev. Lett. 100, 250501 (2008).

    Article  ADS  Google Scholar 

  27. Arad, I., Kitaev, A., Landau, Z. & Vazirani, U. Proc. 4th Innov. Theor. Comput. Sci. (ITCS) (ACM, 2013).

    Google Scholar 

  28. Aharonov, D., Arad, I. & Irani, S. Efficient algorithm for approximating one-dimensional ground states. Phys. Rev. A 82, 012315 (2010).

    Article  ADS  Google Scholar 

  29. Schuch, N. & Cirac, J. I. Matrix product state and mean-field solutions for one-dimensional systems can be found efficiently. Phys. Rev. A 82, 012314 (2010).

    Article  ADS  Google Scholar 

  30. Vidal, G. Classical simulation of infinite-size quantum lattice systems in one spatial dimension. Phys. Rev. Lett. 98, 070201 (2007).

    Article  ADS  Google Scholar 

  31. Corboz, P., Rice, T. M. & Troyer, M. Competing states in the t–j model: Uniform d-wave state versus stripe state. Phys. Rev. Lett. 113, 046402 (2014).

    Article  ADS  Google Scholar 

  32. Jordan, J., Orús, R., Vidal, G., Verstraete, F. & Cirac, J. I. Classical simulation of infinite-size quantum lattice systems in two spatial dimensions. Phys. Rev. Lett. 101, 250602 (2008).

    Article  ADS  Google Scholar 

  33. Corboz, P., Orús, R., Bauer, B. & Vidal, G. Simulation of strongly correlated fermions in two spatial dimensions with fermionic projected entangled-pair states. Phys. Rev. B 81, 165104 (2010).

    Article  ADS  Google Scholar 

  34. Kraus, C. V., Schuch, N., Verstraete, F. & Cirac, J. I. Fermionic projected entangled pair states. Phys. Rev. A 81, 052338 (2010).

    Article  ADS  MathSciNet  Google Scholar 

  35. Barthel, T., Pineda, C. & Eisert, J. Contraction of fermionic operator circuits and the simulation of strongly correlated fermions. Phys. Rev. A 80, 042333 (2009).

    Article  ADS  Google Scholar 

  36. Pineda, C., Barthel, T. & Eisert, J. Unitary circuits for strongly correlated fermions. Phys. Rev. A 81, 050303 (2010).

    Article  ADS  Google Scholar 

  37. Shi, Q-Q., Li, S-H., Zhao, J-H. & Zhou, H-Q. Graded projected entangled-pair state representations and an algorithm for translationally invariant strongly correlated electronic systems on infinite-size lattices in two spatial dimensions. Preprint at http://arXiv.org/abs/0907.5520 (2009).

  38. Corboz, P. & Vidal, G. Fermionic multiscale entanglement renormalization ansatz. Phys. Rev. B 80, 165129 (2009).

    Article  ADS  Google Scholar 

  39. Pižorn, I. & Verstraete, F. Fermionic implementation of projected entangled pair states algorithm. Phys. Rev. B 81, 245110 (2010).

    Article  ADS  Google Scholar 

  40. Verstraete, F., Wolf, M. M., Perez-Garcia, D. & Cirac, J. I. Criticality, the area law, and the computational power of projected entangled pair states. Phys. Rev. Lett. 96, 220601 (2006).

    Article  ADS  MathSciNet  Google Scholar 

  41. Schuch, N., Wolf, M. M., Verstraete, F. & Cirac, J. I. Computational complexity of projected entangled pair states. Phys. Rev. Lett. 98, 140506 (2007).

    Article  ADS  MathSciNet  Google Scholar 

  42. Schuch, N., Pérez-García, D. & Cirac, I. Classifying quantum phases using matrix product states and projected entangled pair states. Phys. Rev. B 84, 165139 (2011).

    Article  ADS  Google Scholar 

  43. Perez-Garcia, D., Verstraete, F., Wolf, M. M. & Cirac, J. I. Peps as unique ground states of local hamiltonians. Quantum Inf. Comput. 8, 650–663 (2008).

    MathSciNet  MATH  Google Scholar 

  44. Cirac, J. I., Poilblanc, D., Schuch, N. & Verstraete, F. Entanglement spectrum and boundary theories with projected entangled-pair states. Phys. Rev. B 83, 245134 (2011).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The first two authors acknowledge support by ARO Grant W911NF-12-1-0541, NSF Grant CCF-0905626 and Templeton Foundation Grant 21674. The third author was supported by the National Science Foundation under Grant No. 0844626 and by the Ministry of Education, Singapore under the Tier 3 grant MOE2012-T3-1-009.

Author information

Authors and Affiliations

  1. Computer Science Division, University of California, Berkeley, Berkeley, California 94720, USA

    Zeph Landau & Umesh Vazirani

  2. Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, California 91125, USA

    Thomas Vidick

Authors
  1. Zeph Landau
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Umesh Vazirani
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Thomas Vidick
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed equally to this work.

Corresponding author

Correspondence to Thomas Vidick.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary Information

Supplementary Information (PDF 305 kb)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Landau, Z., Vazirani, U. & Vidick, T. A polynomial time algorithm for the ground state of one-dimensional gapped local Hamiltonians. Nature Phys 11, 566–569 (2015). https://doi.org/10.1038/nphys3345

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nphys3345

This article is cited by

Search

Advanced search

Quick links

Nature Briefing AI and Robotics

Sign up for the Nature Briefing: AI and Robotics newsletter — what matters in AI and robotics research, free to your inbox weekly.

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