Annales Henri Poincaré

, Volume 18, Issue 11, pp 3449–3513 | Cite as

The Complexity of Translationally Invariant Spin Chains with Low Local Dimension

  • Johannes BauschEmail author
  • Toby Cubitt
  • Maris Ozols
Open Access


We prove that estimating the ground state energy of a translationally invariant, nearest-neighbour Hamiltonian on a 1D spin chain is \(\textsf {QMA}_{{\textsf {EXP}}}\)-complete, even for systems of low local dimension (\(\approx 40\)). This is an improvement over the best previously known result by several orders of magnitude, and it shows that spin-glass-like frustration can occur in translationally invariant quantum systems with a local dimension comparable to the smallest-known non-translationally invariant systems with similar behaviour. While previous constructions of such systems rely on standard models of quantum computation, we construct a new model that is particularly well-suited for encoding quantum computation into the ground state of a translationally invariant system. This allows us to shift the proof burden from optimizing the Hamiltonian encoding a standard computational model, to proving universality of a simple model. Previous techniques for encoding quantum computation into the ground state of a local Hamiltonian allow only a linear sequence of gates, hence only a linear (or nearly linear) path in the graph of all computational states. We extend these techniques by allowing significantly more general paths, including branching and cycles, thus enabling a highly efficient encoding of our computational model. However, this requires more sophisticated techniques for analysing the spectrum of the resulting Hamiltonian. To address this, we introduce a framework of graphs with unitary edge labels. After relating our Hamiltonian to the Laplacian of such a unitary labelled graph, we analyse its spectrum by combining matrix analysis and spectral graph theory techniques.


  1. 1.
    Hastings, M.B.: An area law for one-dimensional quantum systems. J. Stat. Mech.: Theory Exp. 08, P08024 (2007)MathSciNetGoogle Scholar
  2. 2.
    Irani, S.: The complexity of quantum systems on a one-dimensional chain (2007). arXiv:0705.4067
  3. 3.
    Gottesman, D., Irani, S.: The quantum and classical complexity of translationally invariant tiling and Hamiltonian problems. Theory Comput. 9(2), 31–116 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bravyi, S., Caha, L., Movassagh, R., Nagaj, D., Shor, P.W.: Criticality without frustration for quantum spin-1 chains. Phys. Rev. Lett. 109(20), 207202 (2012)ADSCrossRefGoogle Scholar
  5. 5.
    Chen, J., Chen, X., Duan, R., Ji, Z., Zeng, B.: No-go theorem for one-way quantum computing on naturally occurring two-level systems. Phys. Rev. A 83(5), 050301 (2011)ADSCrossRefGoogle Scholar
  6. 6.
    Movassagh, R., Shor, P.W.: Supercritical entanglement in local systems: Counterexample to the area law for quantum matter. PNAS 113(47), 13278–13282 (2016)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Cubitt, T., Perez-Garcia, D., Wolf, M.M.: Undecidability of the spectral gap. Nature 528(7581), 207–211 (2015)ADSCrossRefGoogle Scholar
  8. 8.
    Bravyi, S., Gosset, D.: Gapped and gapless phases of frustration-free spin-1/2 chains. J. Math. Phys. 56(6), 061902 (2015)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Bausch, J., Cubitt, T., Lucia, A., Perez-Garcia, D., Wolf, M.M.: Size-driven quantum phase transitions (Dec. 2016). arXiv:1512.05687
  10. 10.
    Kitaev, A.Y., Shen, A., Vyalyi, M.N.: Classical and quantum computing. In: Quantum Information, pp. 203–217. Springer, New York (2002).
  11. 11.
    Kempe, J., Kitaev, A.Y., Regev, O.: The complexity of the local Hamiltonian problem. SIAM J. Comput. 35(5), 1070–1097 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Oliveira, R.I., Terhal, B.M.: The complexity of quantum spin systems on a two-dimensional square lattice. Quantum Inf. Comput. 8(10), 0900–0924 (2008)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Aharonov, D., Gottesman, D., Irani, S., Kempe, J.: The power of quantum systems on a line. Commun. Math. Phys. 287(1), 41–65 (2009)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Hallgren, S., Nagaj, D., Narayanaswami, S.: The local Hamiltonian problem on a line with eight states is QMA-complete. Quantum Inf. Comput. 13(9&10), 0721–0750 (2013)MathSciNetGoogle Scholar
  15. 15.
    Barthel, T., Hübener, R.: Solving condensed-matter ground-state problems by semidefinite relaxations. Phys. Rev. Lett. 108(20), 200404 (2012)ADSCrossRefGoogle Scholar
  16. 16.
    Cubitt, T., Montanaro, A.: Complexity classification of local Hamiltonian problems. SIAM J. Comput. 45(2), 268–316 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Piddock, S., Montanaro, A.: The complexity of antiferromagnetic interactions and 2D lattices (June 2015). arXiv:1506.04014
  18. 18.
    Wei, T.-C., Liang, J.C.: Hamiltonian quantum computer in one dimension. Phys. Rev. A 92(6), 062334 (2015)ADSCrossRefGoogle Scholar
  19. 19.
    Feynman, R.P.: Quantum mechanical computers. Found. Phys. 16(6), 507–531 (1986)ADSCrossRefMathSciNetGoogle Scholar
  20. 20.
    Bravyi, S.: Efficient algorithm for a quantum analogue of 2-SAT. In: Mahdavi, K., et al. (eds.) Cross Disciplinary Advances in Quantum Computing, vol. 536, pp. 33–48. AMS (2011).
  21. 21.
    Arad, I., Santha, M., Sundaram, A., Zhang, S.: Linear time algorithm for quantum 2SAT. In: Chatzigiannakis, I., et al. (eds) 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016) Leibniz International Proceedings in Informatics (LIPIcs), vol. 55, pp. 15:1-15:14. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl (2016)Google Scholar
  22. 22.
    de Beaudrap, N., Gharibian, S.: A linear time algorithm for quantum 2-SAT. In: Raz, R. (ed.) 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), vol. 50, pp. 27:1–27:21. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, Germany (2016)Google Scholar
  23. 23.
    Gosset, D., Nagaj, D.: Quantum 3-SAT is QMA1-complete. SIAM J. Comput. 45(3), 1080–1128 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    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)CrossRefGoogle Scholar
  25. 25.
    Breuckmann, N.P., Terhal, B.M.: Space-time circuit-to-Hamiltonian construction and its applications. J. Phys. A: Math. Theor. 47(19), 195304 (2014)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Nagaj, D., Wocjan, P.: Hamiltonian quantum cellular automata in one dimension. Phys. Rev. A 78(3), 032311 (2008)ADSCrossRefGoogle Scholar
  27. 27.
    Nagaj, D.: Universal two-body-Hamiltonian quantum computing. Phys. Rev. A 85(3), 032330 (2012)ADSCrossRefGoogle Scholar
  28. 28.
    Gosset, D., Terhal, B.M., Vershynina, A.: Universal adiabatic quantum computation via the space-time circuit-to-Hamiltonian construction. Phys. Rev. Lett. 114(14), 140501 (2015)ADSCrossRefMathSciNetGoogle Scholar
  29. 29.
    Thomas, W.: When nobody else dreamed of these things—Axel Thue und die Termersetzung. Informatik-Spektrum 33(5), 504–508 (2010)CrossRefGoogle Scholar
  30. 30.
    Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Boston (1979)zbMATHGoogle Scholar
  31. 31.
    Morita, K.: Reversible computing and cellular automata—a survey. Theor. Comput. Sci. 395(1), 101–131 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Bernstein, E., Vazirani, U.: Quantum complexity theory. SIAM J. Comput. 26(5), 1411–1473 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Bennett, C.H.: Logical reversibility of computation. IBM J. Res. Dev. 17(6), 525–532 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Perumalla, K.S.: Introduction to Reversible Computing (Chapman & Hall/CRC Computational Science Series). CRC Press, Boca Raton (2013)Google Scholar
  35. 35.
    Watrous, J.: Quantum computational complexity. In: Meyers, R.A. (ed.) Computational Complexity: Theory, Techniques, and Applications, pp. 2361–2387. Springer, New York (2012)CrossRefGoogle Scholar
  36. 36.
    Vidick, T., Watrous, J.: Quantum proofs. Found. Trends Theor. Comput. Sci. 11(1–2), 1–215 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information, 10th edn. Cambridge University Press, Cambridge (2010)CrossRefzbMATHGoogle Scholar
  38. 38.
    Gharibian, S., Huang, Y., Landau, Z., Woo Shin, S.: Quantum Hamiltonian complexity. Found. Trends Theor. Comput. Sci. 10(3), 159–282 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Kitaev, A.Y., Shen, A., Vyalyi, M.N.: Classical and Quantum Computing (Graduate Studies in Mathematics), vol. 47. American Mathematical Society, Providence (2002)zbMATHGoogle Scholar
  40. 40.
    Bausch, J., Crosson, E.: Increasing the quantum UNSAT penalty of the circuit-to-Hamiltonian construction, Sept 2016. arXiv:1609.08571
  41. 41.
    Trudeau, R.J.: Introduction to Graph Theory (Dover Books on Mathematics). Dover Publications, New York (2013)Google Scholar
  42. 42.
    Diestel, R.: Graph Theory (Graduate Texts in Mathematics), vol. 173, 5th edn. Springer, New York (2016)Google Scholar
  43. 43.
    Godsil, C., Royle, G.: Algebraic Graph Theory (Graduate Texts in Mathematics), vol. 207. Springer, New York (2001)CrossRefzbMATHGoogle Scholar
  44. 44.
    Fiedler, M.: Algebraic connectivity of graphs. Czechoslov. Math. J. 23(2), 298–305 (1973)zbMATHMathSciNetGoogle Scholar

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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

  1. 1.DAMTP, Centre for Mathematical SciencesUniversity of CambridgeCambridgeUK
  2. 2.Department of Computer ScienceUniversity College LondonLondonUK

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