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Local Hamiltonians and Ground States

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Quantum Information Meets Quantum Matter

Part of the book series: Quantum Science and Technology ((QST))

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Abstract

We discuss many-body systems, where the Hamiltonian involves only few-body interactions. With the tensor product structure of the many-body Hilbert space in mind, we introduce the concept of locality. It is naturally associated with the spatial geometry of the system, where the most natural interaction between degrees of freedom are those “local” ones, for instance nearest neighbor interactions. We discuss the effect of locality on the ground-state properties. We then discuss ways of determining the ground-state energy of local Hamiltonians, and their hardness. Theories have been developed in quantum information science to show that even with the existence of a quantum computer, there is no efficient way of finding the ground-state energy for a local Hamiltonian in general. However, for practical cases, special structures may lead to simpler method, such as Hartree’s mean-field theory. We also discuss a special kind of local Hamiltonians, called the frustration-free Hamiltonians, where the ground state is also ground states of all the local interaction terms. However, to determine whether a Hamiltonian is frustration-free is in general hard.

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References

  1. Affleck, I., Kennedy, T., Lieb, E. H., & Tasaki, H. (1987). Rigorous results on valence-bond ground states in antiferromagnets. Physical Review Letters, 59(7), 799–802.

    Article  ADS  Google Scholar 

  2. Aharonov, D., & Naveh, T. (2002). Quantum NP - a survey. arXiv:quant-ph/0210077.

  3. Aharonov, D., Gottesman, D., Irani, S., & Kempe, J. (2009). The power of quantum systems on a line. Communications in Mathematical Physics, 287, 41–65.

    Article  ADS  MathSciNet  Google Scholar 

  4. Bravyi, S. (2006). Efficient algorithm for a quantum analogue of 2-SAT. arXiv:quant-ph/0602108.

  5. Bravyi, S. (2011). Subsystem codes with spatially local generators. Physical Review A, 83, 012320.

    Article  ADS  Google Scholar 

  6. Bravyi, S., & Terhal, B. (2009). A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes. New Journal of Physics, 11(4), 043029.

    Article  ADS  Google Scholar 

  7. Chen, J., Chen, X., Duan, R., Ji, Z., & Zeng, B. (2011). No-go theorem for one-way quantum computing on naturally occurring two-level systems. Physical Review A, 83(5), 050301.

    Article  ADS  Google Scholar 

  8. Chen, J., Ji, Z., Kribs, D., Lütkenhaus, N., & Zeng, B. (2014). Symmetric extension of two-qubit states. Physical Review A, 90(3), 032318.

    Article  ADS  Google Scholar 

  9. Coleman, A. J. (1963). Structure of fermion density matrices. Reviews of Modern Physics, 35, 668–686.

    Article  ADS  MathSciNet  Google Scholar 

  10. Diaconis, P., & Freedman, D. (1980). Finite exchangeable sequences. The Annals of Probability, 8, 745–764.

    Article  MathSciNet  Google Scholar 

  11. Doherty, A. C., Parrilo, P. A., & Spedalieri, F. M. (2002). Distinguishing separable and entangled states. Physical Review Letters, 88(18), 187904.

    Article  ADS  Google Scholar 

  12. Eldar, L., & Regev, O. (2008). Quantum sat for a qutrit-cinquit pair is qma1-complete. In ICALP ’08 Proceedings of the 35th International Colloquium on Automata, Languages and Programming, Part I, pp. 881–892.

    Google Scholar 

  13. Garey, M. R., & Johnson, D. S. (1979). Computers and intractability. A guide to the theory of NP-completeness. San Francisco, CA: W. H. Freeman and Company.

    MATH  Google Scholar 

  14. Gosset, D., & Nagaj, D. (2013). Quantum 3-SAT is QMA1-complete. ArXiv e-prints.

    Google Scholar 

  15. Harrow, A. W. (2013). The church of the symmetric subspace. arXiv:1308.6595.

  16. Hudson, R. L., & Moody, G. R. (1976). Locally normal symmetric states and an analogue of de finetti’s theorem. Probability Theory and Related Fields, 33(4), 343–351.

    MathSciNet  MATH  Google Scholar 

  17. Ji, Z., Wei, Z., & Zeng, B. (2011). Complete characterization of the ground space structure of two-body frustration-free hamiltonians for qubits. Physical Review A, 84(4), 042338.

    Article  ADS  Google Scholar 

  18. Kempe, J., & Regev, O. (2003). 3-Local Hamiltonian is QMA-complete. Quantum Information and Computation, 3, 258–264.

    MathSciNet  MATH  Google Scholar 

  19. Kempe, J., Kitaev, A., & Regev, O. (2004). The complexity of the local hamiltonian problem. In FSTTCS 2004: Foundations of Software Technology and Theoretical Computer Science (Lecture Notes in Computer Science vol 3328), pp. 372–383. Chennai, India, Berlin: Springer. arXiv:quant-ph/0406180.

  20. Kitaev, A Yu., Shen, A. H., & Vyalyi, M. N. (2002). Classical and quantum computation (Vol. 47), Graduate studies in mathematics. Providence, RI: American Mathematical Society.

    Google Scholar 

  21. Klyachko, A. A. (2006). Quantum marginal problem and N-representability. Journal of Physics Conference Series, 36, 72–86.

    Article  ADS  Google Scholar 

  22. Kovalev, A. A., & Pryadko, L. P. (2012). Improved quantum hypergraph-product ldpc codes. In Proceedings of the 2012 IEEE International Symposium on Information Theory, pp. 348–352. arXiv:1202.0928.

  23. Kwek, L. C., Wei, Z., & Zeng, B. (2012). Measurement-based quantum computing with valence-bond. International Journal of Modern Physics B, 26, 30002.

    Article  ADS  Google Scholar 

  24. Lewin, M., Nam, P. T., & Rougerie, N. (2014). Derivation of hartree’s theory for generic mean-field bose systems. Advances in Mathematics, 254, 570–621.

    Article  MathSciNet  Google Scholar 

  25. Liu, Y.-K. (2006). Consistency of local density matrices is QMA-complete. arXiv:quant-ph/0604166.

  26. Liu, Y.-K., Christandl, M., & Verstraete, F. (2007). Quantum computational complexity of the \(n\)-representability problem: Qma complete. Physical Review Letters, 98, 110503.

    Article  ADS  Google Scholar 

  27. Myhr, G. O., & Lütkenhaus, N. (2009). Spectrum conditions for symmetric extendible states. Physical Review A, 79(6), 062307.

    Article  ADS  MathSciNet  Google Scholar 

  28. Nagaj, D. (2008). Local Hamiltonians in quantum computation. ArXiv e-prints.

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  30. Osborne, T. J. (2012). Hamiltonian complexity. Reports on Progress in Physics, 75(2), 022001.

    Article  ADS  MathSciNet  Google Scholar 

  31. Raussendorf, R., & Wei, T.-C. (2012). Quantum computation by local measurement. Annual Review of Condensed Matter Physics, 26, 239–261.

    Article  Google Scholar 

  32. Størmer, E. (1969). Symmetric states of infinite tensor products of c*-algebras. Journal of Functional Analysis, 3(1), 48–68.

    Article  MathSciNet  Google Scholar 

  33. Tillich, J.-P., & Zémor, G. (2009). Quantum ldpc codes with positive rate and minimum distance proportional to \(n^{1/2}\). In Proceedings of the 2009 IEEE International Symposium on Information Theory, pp. 799–803. arXiv:0903.0566.

  34. Wei, T.-C., Mosca, M., & Nayak, A. (2010). Interacting boson problems can be qma hard. Physical Review Letters, 104, 040501.

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. University of Guelph, Guelph, ON, Canada

    Bei Zeng

  2. Physics, California Institute of Technology, Pasadena, CA, USA

    Xie Chen

  3. Institute of Physics, Chinese Academy of Sciences, Beijing, China

    Duan-Lu Zhou

  4. Massachusetts Institute of Technology, Cambridge, MA, USA

    Xiao-Gang Wen

Authors
  1. Bei Zeng
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  2. Xie Chen
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  3. Duan-Lu Zhou
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  4. Xiao-Gang Wen
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Correspondence to Bei Zeng .

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Zeng, B., Chen, X., Zhou, DL., Wen, XG. (2019). Local Hamiltonians and Ground States. In: Quantum Information Meets Quantum Matter. Quantum Science and Technology. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-9084-9_4

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  • DOI: https://doi.org/10.1007/978-1-4939-9084-9_4

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  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4939-9082-5

  • Online ISBN: 978-1-4939-9084-9

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