The Complexity of Translationally Invariant Spin Chains with Low Local Dimension

Abstract

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.

References

  1. 1.

    Hastings, M.B.: An area law for one-dimensional quantum systems. J. Stat. Mech.: Theory Exp. 08, P08024 (2007)

    MathSciNet  Google 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)

    Article  MATH  MathSciNet  Google 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)

    ADS  Article  Google 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)

    ADS  Article  Google 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)

    Article  MathSciNet  Google Scholar 

  7. 7.

    Cubitt, T., Perez-Garcia, D., Wolf, M.M.: Undecidability of the spectral gap. Nature 528(7581), 207–211 (2015)

    ADS  Article  Google 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)

    ADS  Article  MATH  MathSciNet  Google 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). https://doi.org/10.1007/978-0-387-36944-0_13

  11. 11.

    Kempe, J., Kitaev, A.Y., Regev, O.: The complexity of the local Hamiltonian problem. SIAM J. Comput. 35(5), 1070–1097 (2006)

    Article  MATH  MathSciNet  Google 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)

    MATH  MathSciNet  Google 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)

    ADS  Article  MATH  MathSciNet  Google 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)

    MathSciNet  Google Scholar 

  15. 15.

    Barthel, T., Hübener, R.: Solving condensed-matter ground-state problems by semidefinite relaxations. Phys. Rev. Lett. 108(20), 200404 (2012)

    ADS  Article  Google Scholar 

  16. 16.

    Cubitt, T., Montanaro, A.: Complexity classification of local Hamiltonian problems. SIAM J. Comput. 45(2), 268–316 (2016)

    Article  MATH  MathSciNet  Google 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)

    ADS  Article  Google Scholar 

  19. 19.

    Feynman, R.P.: Quantum mechanical computers. Found. Phys. 16(6), 507–531 (1986)

    ADS  Article  MathSciNet  Google 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). https://doi.org/10.1090/conm/536/10552

  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)

  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)

  23. 23.

    Gosset, D., Nagaj, D.: Quantum 3-SAT is QMA1-complete. SIAM J. Comput. 45(3), 1080–1128 (2016)

    Article  MATH  MathSciNet  Google 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)

    Article  Google 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)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  26. 26.

    Nagaj, D., Wocjan, P.: Hamiltonian quantum cellular automata in one dimension. Phys. Rev. A 78(3), 032311 (2008)

    ADS  Article  Google Scholar 

  27. 27.

    Nagaj, D.: Universal two-body-Hamiltonian quantum computing. Phys. Rev. A 85(3), 032330 (2012)

    ADS  Article  Google 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)

    ADS  Article  MathSciNet  Google Scholar 

  29. 29.

    Thomas, W.: When nobody else dreamed of these things—Axel Thue und die Termersetzung. Informatik-Spektrum 33(5), 504–508 (2010)

    Article  Google Scholar 

  30. 30.

    Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Boston (1979)

    MATH  Google Scholar 

  31. 31.

    Morita, K.: Reversible computing and cellular automata—a survey. Theor. Comput. Sci. 395(1), 101–131 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  32. 32.

    Bernstein, E., Vazirani, U.: Quantum complexity theory. SIAM J. Comput. 26(5), 1411–1473 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  33. 33.

    Bennett, C.H.: Logical reversibility of computation. IBM J. Res. Dev. 17(6), 525–532 (1973)

    Article  MATH  MathSciNet  Google 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)

    Google Scholar 

  36. 36.

    Vidick, T., Watrous, J.: Quantum proofs. Found. Trends Theor. Comput. Sci. 11(1–2), 1–215 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  37. 37.

    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information, 10th edn. Cambridge University Press, Cambridge (2010)

    Book  MATH  Google 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)

    Article  MATH  MathSciNet  Google 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)

    MATH  Google 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)

    Book  MATH  Google Scholar 

  44. 44.

    Fiedler, M.: Algebraic connectivity of graphs. Czechoslov. Math. J. 23(2), 298–305 (1973)

    MATH  MathSciNet  Google Scholar 

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Correspondence to Johannes Bausch.

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Communicated by David Perez-García.

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Bausch, J., Cubitt, T. & Ozols, M. The Complexity of Translationally Invariant Spin Chains with Low Local Dimension. Ann. Henri Poincaré 18, 3449–3513 (2017). https://doi.org/10.1007/s00023-017-0609-7

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