Communications in Mathematical Physics

, Volume 365, Issue 1, pp 1–16 | Cite as

Finite Correlation Length Implies Efficient Preparation of Quantum Thermal States

  • Fernando G. S. L. Brandão
  • Michael J. KastoryanoEmail author


Preparing quantum thermal states on a quantum computer is in general a difficult task. We provide a procedure to prepare a thermal state on a quantum computer with a logarithmic depth circuit of local quantum channels assuming that the thermal state correlations satisfy the following two properties: (i) the correlations between two regions are exponentially decaying in the distance between the regions, and (ii) the thermal state is an approximate Markov state for shielded regions. We require both properties to hold for the thermal state of the Hamiltonian on any induced subgraph of the original lattice. Assumption (ii) is satisfied for all commuting Gibbs states, while assumption (i) is satisfied for every model above a critical temperature. Both assumptions are satisfied in one spatial dimension. Moreover, both assumptions are expected to hold above the thermal phase transition for models without any topological order at finite temperature. As a building block, we show that exponential decay of correlation (for thermal states of Hamiltonians on all induced subgraphs) is sufficient to efficiently estimate the expectation value of a local observable. Our proof uses quantum belief propagation, a recent strengthening of strong sub-additivity, and naturally breaks down for states with topological order.


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  1. 1.
    Kastoryano, M.J., Brandao, F.G.S.L.: Quantum Gibbs Samplers: the commuting case. Commun. Math. Phys., 1–43 (2014)Google Scholar
  2. 2.
    Temme K., Osborne T.J., Vollbrecht K.G., Poulin D., Verstraete F.: Quantum metropolis sampling. Nature 471(7336), 87–90 (2011)ADSCrossRefGoogle Scholar
  3. 3.
    Terhal B.M., DiVincenzo D.P.: Problem of equilibration and the computation of correlation functions on a quantum computer. Phys. Rev. A 61(2), 022301 (2000)ADSCrossRefGoogle Scholar
  4. 4.
    Poulin D., Wocjan P.: Sampling from the thermal quantum Gibbs state and evaluating partition functions with a quantum computer. Phys. Rev. Lett. 103(22), 220502 (2009)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Guionnet, A., Zegarlinksi, B.: Lectures on logarithmic sobolev inequalities. In: Séminaire de probabilités XXXVI, pp. 1–134. Springer, Berlin (1801)Google Scholar
  6. 6.
    Martinelli, F.: Lectures on Glauber Dynamics for Discrete Spin Systems. Lecture Notes in Mathematics, 1(7):1Google Scholar
  7. 7.
    Diaconis, P., Saloff-Coste, L.: What do we know about the metropolis algorithm? In Proceedings of the Twenty-Seventh Annual ACM Symposium on Theory of Computing, pp. 112–129. ACM (1995)Google Scholar
  8. 8.
    Fawzi O., Renner R.: Quantum conditional mutual information and approximate Markov chains. Commun. Math. Phys. 340(2), 575–611 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Junge, M., Renner, R., Sutter, D., Wilde, M.M., Winter, A.: Universal recovery from a decrease of quantum relative entropy. arXiv preprint arXiv:1509.07127 (2015)
  10. 10.
    Audenaert K.M.R.: A sharp continuity estimate for the von Neumann entropy. J. Phys. A Math. Theor. 40, 8127 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ohya M., Petz D.: Quantum Entropy and Its Use. Springer, Berlin (2004)zbMATHGoogle Scholar
  12. 12.
    Hastings M.B.: Quantum belief propagation: an algorithm for thermal quantum systems. Phys. Rev. B 76(20), 201102 (2007)ADSCrossRefGoogle Scholar
  13. 13.
    König R., Pastawski F.: Generating topological order: no speedup by dissipation. Phys. Rev. B 90(4), 045101 (2014)ADSCrossRefGoogle Scholar
  14. 14.
    Ge Y., Molnár A., Cirac J.I.: Rapid adiabatic preparation of injective projected entangled pair states and Gibbs states. Phys. Rev. Lett. 116(8), 080503 (2016)ADSCrossRefGoogle Scholar
  15. 15.
    Michalakis S., Zwolak J.P.: Stability of frustration-free Hamiltonians. Commun. Math. Phys. 322(2), 277–302 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Martinelli F., Olivieri E., Schonmann R.: For 2-D lattice spin systems weak mixing implies strong mixing. Commun. Math. Phys. 165, 33 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kitaev A.Y.: Fault-tolerant quantum computation by anyons. Ann. Phys. 303(1), 2–30 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Bombin H., Martin-Delgado M.A.: Topological quantum distillation. Phys. Rev. Lett. 97(18), 180501 (2006)ADSCrossRefGoogle Scholar
  19. 19.
    Kim I.H.: Perturbative analysis of topological entanglement entropy from conditional independence. Phys. Rev. B. 86(24), 245116 (2012)ADSCrossRefGoogle Scholar
  20. 20.
    Kato, K., Brandao, F.G.S.L.: Quantum approximate Markov chains are thermal. arXiv preprint arXiv:1609.06636v1 (2016)
  21. 21.
    Poulin D., Bilgin E.: Belief propagation algorithm for computing correlation functions in finite-temperature quantum many-body systems on loopy graphs. Phys. Rev. A 77(5), 052318 (2008)ADSCrossRefGoogle Scholar
  22. 22.
    Bilgin E., Poulin D.: Coarse-grained belief propagation for simulation of interacting quantum systems at all temperatures. Phys. Rev. B 81(5), 054106 (2010)ADSCrossRefGoogle Scholar
  23. 23.
    Bilgin E., Boixo S.: Preparing thermal states of quantum systems by dimension reduction. Phys. Rev. Lett. 105(17), 170405 (2010)ADSCrossRefGoogle Scholar
  24. 24.
    Wolf M.M., Verstraete F., Hastings M.B., Cirac J.I.: Area laws in quantum systems: mutual information and correlations. Phys. Rev. Lett. 100(7), 070502 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Verstraete F., Wolf M.M., Cirac J.I.: Quantum computation and quantum-state engineering driven by dissipation. Nat. Phys. 5(9), 633–636 (2009)CrossRefGoogle Scholar
  26. 26.
    Gottesman, D.: Stabilizer codes and quantum error correction. arXiv preprint arXiv:quant-ph/9705052 (1997)
  27. 27.
    Levin M.A., Wen X.-G.: String-net condensation: a physical mechanism for topological phases. Phys. Rev. B 71(4), 045110 (2005)ADSCrossRefGoogle Scholar
  28. 28.
    Hein M., Eisert J., Briegel H.J.: Multiparty entanglement in graph states. Phys. Rev. A 69(6), 062311 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Schwarz, M., Buerschaper, O., Eisert, J.: Approximating local observables on projected entangled pair states. arXiv preprint arXiv:1606.06301 (2016)
  30. 30.
    Araki H.: Gibbs states of a one dimensional quantum lattice. Commun. Math. Phys. 14(2), 120–157 (1969)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Brown B.J., Loss D., Pachos J.K., Self C.N., Wootton J.R.: Quantum memories at finite temperature. Rev. Mod. Phys. 88(4), 045005 (2016)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Hastings M.B.: Topological order at nonzero temperature. Phys. Rev. Lett. 107(21), 210501 (2011)ADSCrossRefGoogle Scholar
  33. 33.
    Dennis E., Kitaev A., Landahl A., Preskill J.: Topological quantum memory. J. Math. Phys. 43(9), 4452–4505 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Haah J.: Local stabilizer codes in three dimensions without string logical operators. Phys. Rev. A 83(4), 042330 (2011)ADSCrossRefGoogle Scholar
  35. 35.
    Michnicki, K.: 3-d quantum stabilizer codes with a power law energy barrier. arXiv preprint arXiv:1208.3496 (2012)
  36. 36.
    Brell C.G.: A proposal for self-correcting stabilizer quantum memories in 3 dimensions (or slightly less). New J. Phys. 18(1), 013050 (2016)ADSCrossRefGoogle Scholar
  37. 37.
    Alicki R., Horodecki M., Horodecki P., Horodecki R..: On thermal stability of topological qubit in Kitaev’s 4D model. Open Syst. Inf. Dyn. 17(01), 1–20 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Bravyi S., Poulin D., Terhal B.: Tradeoffs for reliable quantum information storage in 2D systems. Phys. Rev. Lett. 104, 050503 (2010)ADSCrossRefzbMATHGoogle Scholar
  39. 39.
    Kitaev, A.Y.: On the classification of short-range entangled states. Simons Center for Geometry and Physics, Topological Phases of Matter (2016)Google Scholar
  40. 40.
    Swingle B., McGreevy J.: Renormalization group constructions of topological quantum liquids and beyond. Phys. Rev. B 93(4), 045127 (2016)ADSCrossRefGoogle Scholar
  41. 41.
    Swingle, B., McGreevy, J.: Mixed s-sourcery: building many-body states using bubbles of nothing. arXiv preprint arXiv:1607.05753 (2016)

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Fernando G. S. L. Brandão
    • 1
  • Michael J. Kastoryano
    • 2
    Email author
  1. 1.IQIMCalifornia Institute of TechnologyPasadenaUSA
  2. 2.NBIA, Niels Bohr InstituteUniversity of CopenhagenCopenhagenDenmark

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