Communications in Mathematical Physics

, Volume 287, Issue 1, pp 41–65 | Cite as

The Power of Quantum Systems on a Line

  • Dorit Aharonov
  • Daniel Gottesman
  • Sandy Irani
  • Julia Kempe


We study the computational strength of quantum particles (each of finite dimensionality) arranged on a line. First, we prove that it is possible to perform universal adiabatic quantum computation using a one-dimensional quantum system (with 9 states per particle). This might have practical implications for experimentalists interested in constructing an adiabatic quantum computer. Building on the same construction, but with some additional technical effort and 12 states per particle, we show that the problem of approximating the ground state energy of a system composed of a line of quantum particles is QMA-complete; QMA is a quantum analogue of NP. This is in striking contrast to the fact that the analogous classical problem, namely, one-dimensional MAX-2-SAT with nearest neighbor constraints, is in P. The proof of the QMA-completeness result requires an additional idea beyond the usual techniques in the area: Not all illegal configurations can be ruled out by local checks, so instead we rule out such illegal configurations because they would, in the future, evolve into a state which can be seen locally to be illegal. Our construction implies (assuming the quantum Church-Turing thesis and that quantum computers cannot efficiently solve QMA-complete problems) that there are one-dimensional systems which take an exponential time to relax to their ground states at any temperature, making them candidates for being one-dimensional spin glasses.


Ground State Energy Turing Machine Transition Rule Quantum Circuit Qubit State 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. ACdf88.
    Apolloni B., Carvalho C., de Falco D.: Quantum stochastic optimization. Stochastic Processes and their Applications 33(5), 233–244 (1988)MathSciNetGoogle Scholar
  2. ACdf90.
    Apolloni, B., Cesa-Bianchi, N., de Falco, D.: A numerical implementation of “quantum annealing”. In: Stochastic Processes, Physics and Geometry: Proceedings of the Ascona-Locarno Conference. River Edge, NJ: World Scientific. 1990, pp. 97–111Google Scholar
  3. AGIK07.
    Aharonov, D., Gottesman, D., Irani, S., Kempe, J.: The power of quantum systems on a line. In: FOCS. Proc. 48 th Ann. IEEE, Symp on Foundations of Computer Science, Los Alamitos, CA: IEEE Comp. Soc., 2007, pp. 373–383Google Scholar
  4. AGK07.
    Aharonov, D., Gottesman, D., Kempe, J.: The power of quantum systems on a line. [quant-ph], 2007
  5. AvDK+04.
    Aharonov, D., van Dam, W., Kempe, J., Landau, Z., Lloyd, S., Regev, O.: Adiabatic quantum computation is equivalent to standard quantum computation. In: Proc. 45th FOCS, Los Alamitos, CA: IEEE Comp. Soc., 2004, pp. 42–51Google Scholar
  6. Bar82.
    Barahona F.: On the computational complexity of Ising spin glass models. J. Phys. A: Math. Gen. 15, 3241–3253 (1982)CrossRefADSMathSciNetGoogle Scholar
  7. BY86.
    Binder K., Young A.P.: Spin glasses: Experimental facts, theoretical concepts, and open questions. Rev. Mod. Phys. 58, 801–976 (1986)CrossRefADSGoogle Scholar
  8. CFP02.
    Childs A., Farhi E., Preskill J.: Robustness of adiabatic quantum computation. Phys. Rev. A 65, 012322 (2002)CrossRefADSGoogle Scholar
  9. DRS07.
    Deift P., Ruskai M.B., Spitzer W.: Improved gap estimates for simulating quantum circuits by adiabatic evolution. Quant Infor. Proc. 6(2), 121–125 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  10. Fey85.
    Feynman R.: Quantum mechanical computers. Optics News 11, 11–21 (1985)CrossRefGoogle Scholar
  11. FGG+01.
    Farhi E., Goldstone J., Gutmann S., Lapan J., Lundgren A., Preda D.: A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem. Science 292(5516), 472–476 (2001)CrossRefADSMathSciNetGoogle Scholar
  12. FGGS00.
    Farhi, E., Goldstone, J., Gutmann, S., Sipser, M.: Quantum computation by adiabatic evolution., 2000
  13. Fis95.
    Fisher D.S.: Critical behavior of random transverse-field Ising spin chains. Phys. Rev. B 51(10), 6411–6461 (1995)CrossRefADSGoogle Scholar
  14. Has.
    Hastings, M.: Personal communicationGoogle Scholar
  15. Has07.
    Hastings, M.: An area law for one dimensional quantum systems. JSTAT. P08024 (2007)Google Scholar
  16. HT.
    Hastings, M., Terhal, B.: Personal communicationGoogle Scholar
  17. Ira07.
    Irani, S.: The complexity of quantum systems on a one-dimensional chain.[quant-ph], 2007
  18. JFS06.
    Jordan S.P., Farhi E., Shor P.W.: Error-correcting codes for adiabatic quantum computation. Phys. Rev. A 74, 052322 (2006)CrossRefADSMathSciNetGoogle Scholar
  19. JWZ07.
    Janzing, D., Wocjan, P., Zhang, S.: A single-shot measurement of the energy of product states in a translation invariant spin chain can replace any quantum computation.[quant-ph], 2007
  20. Kay08.
    Kay A.: The computational power of symmetric hamiltonians. Phys. Rev. A. 78, 012346 (2008)CrossRefADSGoogle Scholar
  21. KKR06.
    Kempe J., Kitaev A., Regev O.: The complexity of the Local Hamiltonian problem. SIAM J. Comp. 35(5), 1070–1097 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  22. KSV02.
    Kitaev, A.Y., Shen, A.H., Vyalyi, M.N.: Classical and Quantum Computation. Providence, RI: Amer. Math. Soc., 2002Google Scholar
  23. Nag08.
    Nagaj, D.: Local Hamiltonians in Quantum Computation. PhD thesis, MIT.[quant-ph], 2008
  24. NW08.
    Nagaj D., Wocjan P.: Hamiltonian quantum cellular automata in 1d. Phys. Rev. A 78, 032311 (2008)CrossRefADSGoogle Scholar
  25. Osb06.
    Osborne T.: Efficient approximation of the dynamics of one-dimensional quantum spin systems. Phys. Rev. Lett. 97, 157202 (2006)CrossRefADSMathSciNetGoogle Scholar
  26. Osb07.
    Osborne T.: Ground state of a class of noncritical one-dimensional quantum spin systems can be approximated efficiently. Phys. Rev. A 75, 042306 (2007)CrossRefADSMathSciNetGoogle Scholar
  27. OT05.
    Oliveira R., Terhal B.: The complexity of quantum spin systems on a two-dimensional square lattice. Quant. Inf. Comp. 8(10), 0900–0924 (2008)MathSciNetGoogle Scholar
  28. Sch05.
    Schollwöck U.: The density-matrix renormalization group. Rev. Mod. Phys. 77, 259–316 (2005)CrossRefADSGoogle Scholar
  29. SFW06.
    Shepherd D.J., Franz T., Werner R.F.: Universally programmable quantum cellular automaton. Phys. Rev. Lett. 97, 020502 (2006)CrossRefADSGoogle Scholar
  30. Suz76.
    Suzuki M.: Relationship between d-dimensional quantal spin systems and (d+1)-dimensional ising systems. Prog. Theor. Phys. 56(5), 1454–1469 (1976)zbMATHCrossRefADSGoogle Scholar
  31. vEB90.
    van Emde Boas, P.: Handbook of Theoretical Computer Science. volume A, Chapter 1. Cambridge, MA: MIT Press, 1990, pp. 1–66Google Scholar
  32. VV86.
    Valiant L.G., Vazirani V.V.: NP is as easy as detecting unique solutions. Theor. Comput. Sci. 47(3), 85–93 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  33. Wat95.
    Watrous, J.: On one-dimensional quantum cellular automata. In: Proc. 36th Annual IEEE Symp. on Foundations of Computer Science (FOCS), Los Alamitos, CA: IEEE Comp. Sci, 1995, pp. 528–537Google Scholar
  34. Whi92.
    White S.R.: Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett. 69, 2863 (1992)CrossRefADSGoogle Scholar
  35. Whi93.
    White S.R.: Density-matrix algorithms for quantum renormalization groups. Phys. Rev. B 48, 10345 (1993)CrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Dorit Aharonov
    • 1
  • Daniel Gottesman
    • 2
  • Sandy Irani
    • 3
  • Julia Kempe
    • 4
  1. 1.School of Computer Science and EngineeringHebrew UniversityJerusalemIsrael
  2. 2.Perimeter InstituteWaterlooCanada
  3. 3.Computer Science DepartmentUniversity of CaliforniaIrvineUSA
  4. 4.School of Computer ScienceTel Aviv UniversityTel AvivIsrael

Personalised recommendations