Suppose we have an n-qubit system, and we are given a collection of local density matrices ρ 1,...,ρ m , where each ρ i describes a subset C i of the qubits. We say that the ρ i are “consistent” if there exists some global state σ (on all n qubits) that matches each of the ρ i on the subsets C i . This generalizes the classical notion of the consistency of marginal probability distributions.

We show that deciding the consistency of local density matrices is QMA-complete (where QMA is the quantum analogue of NP). This gives an interesting example of a hard problem in QMA. Our proof is somewhat unusual: we give a Turing reduction from Local Hamiltonian, using a convex optimization algorithm by Bertsimas and Vempala, which is based on random sampling. Unlike in the classical case, simple mapping reductions do not seem to work here.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aharonov, D.: Private communication (2004)Google Scholar
  2. 2.
    Kitaev, A.Y., Shen, A.H., Vyalyi, M.N.: Classical and Quantum Computation. AMS (2002)Google Scholar
  3. 3.
    Aharonov, D., Naveh, T.: Quantum NP - A Survey, Arxiv: quant-ph/0210077Google Scholar
  4. 4.
    Kempe, J., Regev, O.: 3-Local Hamiltonian is QMA-complete. Quantum Info. and Comput. 3(3), 258–264 (2079) Arxiv: quant-ph/0302079MathSciNetGoogle Scholar
  5. 5.
    Kempe, J., Kitaev, A., Regev, O.: The Complexity of the Local Hamiltonian Problem. In: Lodaya, K., Mahajan, M. (eds.) FSTTCS 2004. LNCS, vol. 3328, pp. 372–383. Springer, Heidelberg (2004) Arxiv: quant-ph/0406180CrossRefGoogle Scholar
  6. 6.
    Oliveira, R., Terhal, B.M.: The complexity of quantum spin systems on a two-dimensional square lattice, Arxiv: quant-ph/0504050Google Scholar
  7. 7.
    Aharonov, D., van Dam, W., Kempe, J., Landau, Z., Lloyd, S., Regev, O.: Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation. In: FOCS 2004, pp. 42–51 (2004), Arxiv: quant-ph/0405098Google Scholar
  8. 8.
    Janzing, D., Wocjan, P., Beth, T.: Identity check is QMA-complete, Arxiv: quant-ph/0305050Google Scholar
  9. 9.
    Bertsimas, D., Vempala, S.: Solving Convex Programs by Random Walks. Journal of the ACM 51(4), 540–556 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Kalai, A., Vempala, S.: Convex Optimization by Simulated Annealing (preprint, 2004)Google Scholar
  11. 11.
    Vempala, S.: Geometric Random Walks: A Survey. MSRI volume on Combinatorial and Computational Geometry (2005)Google Scholar
  12. 12.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, Heidelberg (1988)zbMATHGoogle Scholar
  13. 13.
    Bravyi, S., Vyalyi, M.: Commutative version of the local Hamiltonian problem and common eigenspace problem. Quantum Info. and Comput. 5(3), 187–215 (2005) Arxiv: quant-ph/0308021zbMATHMathSciNetGoogle Scholar
  14. 14.
    Bravyi, S.: Efficient algorithm for a quantum analogue of 2-SAT, Arxiv: quant-ph/0602108Google Scholar
  15. 15.
    Aharonov, D., Regev, O.: A Lattice Problem in Quantum NP. In: FOCS 2003, pp. 210–219 (2003), Arxiv: quant-ph/0307220Google Scholar
  16. 16.
    Lovász, L., Simonovits, M.: Random Walks in a Convex Body and an Improved Volume Algorithm. Random Structures and Algorithms 4(4) (1993)Google Scholar
  17. 17.
    Bhatia, R.: Matrix Analysis. Springer, Heidelberg (1997)Google Scholar
  18. 18.
    Liu, Y.-K.: (in preparation)Google Scholar
  19. 19.
    Watrous, J.: Zero-knowledge against quantum attacks, Arxiv: quant-ph/0511020Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Yi-Kai Liu
    • 1
  1. 1.Computer Science and EngineeringUniversity of CaliforniaSan Diego

Personalised recommendations