Ground State Connectivity of Local Hamiltonians

  • Sevag Gharibian
  • Jamie SikoraEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9134)


The study of ground state energies of local Hamiltonians has played a fundamental role in quantum complexity theory. In this paper, we take a new direction by introducing the physically motivated notion of “ground state connectivity” of local Hamiltonians, which captures problems in areas ranging from quantum stabilizer codes to quantum memories. We show that determining how “connected” the ground space of a local Hamiltonian is can range from QCMA-complete to PSPACE-complete, as well as NEXP-complete for an appropriately defined “succinct” version of the problem. As a result, we obtain a natural QCMA-complete problem, a goal which has generally proven difficult since the conception of QCMA over a decade ago. Our proofs rely on a new technical tool, the Traversal Lemma, which analyzes the Hilbert space a local unitary evolution must traverse under certain conditions. We show that this lemma is essentially tight with respect to the length of the unitary evolution in question.


Ground State Energy Quantum Circuit Full Version Local Unitaries Satisfying Assignment 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Kitaev, A., Shen, A., Vyalyi, M.: Classical and Quantum Computation. American Mathematical Society (2002)Google Scholar
  2. 2.
    Cook, S.: The complexity of theorem proving procedures. In: Proceedings of the 3rd ACM Symposium on Theory of Computing (STOC 1972), pp. 151–158 (1972)Google Scholar
  3. 3.
    Levin, L.: Universal search problems. Problems of Information Transmission 9(3), 265–266 (1973)Google Scholar
  4. 4.
    Oliveira, R., Terhal, B.M.: The complexity of quantum spin systems on a two-dimensional square lattice. Quantum Information & Computation 8(10), 0900–0924 (2008)MathSciNetGoogle Scholar
  5. 5.
    Bravyi, S., Vyalyi, M.: Commutative version of the local Hamiltonian problem and common eigenspace problem. Quantum Information & Computation 5(3), 187–215 (2005)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Gharibian, S., Huang, Y., Landau, Z., Shin, S.W.: Quantum Hamiltonian complexity (2014). e-Print quant-ph/1401.3916v1Google Scholar
  7. 7.
    Cubitt, T., Montanaro, A.: Complexity classification of local hamiltonian problems (2013). e-Print quant-ph/1311.3161Google Scholar
  8. 8.
    Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings of the 10th Symposium on Theory of computing, pp. 216–226 (1978)Google Scholar
  9. 9.
    Kitaev, A.: Unpaired majorana fermions in quantum wires. Physics-Uspekhi 44, 131 (2001)CrossRefGoogle Scholar
  10. 10.
    Kitaev, A.: Fault-tolerant quantum computation by anyons. Annals of Physics 303(1), 2–30 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Kitaev, A., Laumann, C.: Topological phases and quantum computation (2009). e-Print quant-ph/0904.2771Google Scholar
  12. 12.
    Aharonov, D., Naveh, T.: Quantum NP - A survey (2002). e-Print quant-ph/0210077v1
  13. 13.
    Gharibian, S., Sikora, J.: Ground state connectivity of local hamiltonians (2014). e-Print quant-ph/1409.3182Google Scholar
  14. 14.
    Winter, A.: Coding theorem and strong converse for quantum channels, 45(7), 2481–2485 (1999)Google Scholar
  15. 15.
    Gopalan, P., Kolaitis, P.G., Maneva, E.N., Papadimitriou, C.: The connectivity of boolean satisfiability: computational and structural dichotomies. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 346–357. Springer, Heidelberg (2006) CrossRefGoogle Scholar
  16. 16.
    Mouawad, A., Nishimura, N., Pathak, V., Raman, V.: Shortest reconfiguration paths in the solution space of Boolean formulas (2014). e-Print cs.CC/1404.3801v2Google Scholar
  17. 17.
    Cereceda, L., van den Heuvel, J., Johnson, M.: Connectedness of the graph of vertex-colorings. Discrete Mathematics 308(56), 913–919 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Bonsma, P., Cereceda, L.: Finding paths between graph colourings: PSPACE-completeness and superpolynomial distances. Theoretical Computer Science 410(50), 5215–5226 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Cereceda, L., van den Heuvel, J., Johnson, M.: Finding paths between 3-colorings. Journal of Graph Theory 67(1), 69–82 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Ito, T., Kamiński, M., Demaine, E.D.: Reconfiguration of list edge-colorings in a graph. Discrete Applied Mathematics 160(15), 2199–2207 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Wocjan, P., Janzing, D., Beth, T.: Two QCMA-complete problems. Quantum Information & Computation 3(6), 635–643 (2003)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Wocjan, P., Yard, J.: The Jones polynomial: quantum algorithms and applications in quantum complexity theory. Quantum Information & Computation 8(1), 147–180 (2008)zbMATHMathSciNetGoogle Scholar
  23. 23.
    Janzing, D., Wocjan, P.: BQP-complete problems concerning mixing properties of classical random walks on sparse graphs (2006). e-Print quant-ph/0610235v2
  24. 24.
    Gharibian, S., Kempe, J.: Hardness of approximation for quantum problems. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012, Part I. LNCS, vol. 7391, pp. 387–398. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  25. 25.
    Brown, B., Flammia, S., Schuch, N.: Computational difficulty of computing the density of states. Physical Review Letters 104, 040501 (2011)CrossRefGoogle Scholar
  26. 26.
    Ambainis, A.: On physical problems that are slightly more difficult than QMA. In: Proceedings of 29th IEEE Conference on Computational Complexity (CCC 2014), pp. 32–43 (2014)Google Scholar
  27. 27.
    Kempe, J., Regev, O.: 3-local Hamiltonian is QMA-complete. Quantum Information & Computation 3(3), 258–264 (2003)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Computer ScienceVirginia Commonwealth UniversityRichmondUSA
  2. 2.Centre for Quantum Technologies and MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit, UMI 3654National University of SingaporeSingaporeSingapore

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