Finite Correlation Length Implies Efficient Preparation of Quantum Thermal States


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)

  2. 2

    Temme K., Osborne T.J., Vollbrecht K.G., Poulin D., Verstraete F.: Quantum metropolis sampling. Nature 471(7336), 87–90 (2011)

    ADS  Article  Google 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)

    ADS  Article  Google 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)

    ADS  MathSciNet  Article  Google Scholar 

  5. 5

    Guionnet, A., Zegarlinksi, B.: Lectures on logarithmic sobolev inequalities. In: Séminaire de probabilités XXXVI, pp. 1–134. Springer, Berlin (1801)

  6. 6

    Martinelli, F.: Lectures on Glauber Dynamics for Discrete Spin Systems. Lecture Notes in Mathematics, 1(7):1

  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)

  8. 8

    Fawzi O., Renner R.: Quantum conditional mutual information and approximate Markov chains. Commun. Math. Phys. 340(2), 575–611 (2015)

    ADS  MathSciNet  Article  MATH  Google 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)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  11. 11

    Ohya M., Petz D.: Quantum Entropy and Its Use. Springer, Berlin (2004)

    Google Scholar 

  12. 12

    Hastings M.B.: Quantum belief propagation: an algorithm for thermal quantum systems. Phys. Rev. B 76(20), 201102 (2007)

    ADS  Article  Google Scholar 

  13. 13

    König R., Pastawski F.: Generating topological order: no speedup by dissipation. Phys. Rev. B 90(4), 045101 (2014)

    ADS  Article  Google 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)

    ADS  Article  Google Scholar 

  15. 15

    Michalakis S., Zwolak J.P.: Stability of frustration-free Hamiltonians. Commun. Math. Phys. 322(2), 277–302 (2013)

    ADS  MathSciNet  Article  MATH  Google 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)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  17. 17

    Kitaev A.Y.: Fault-tolerant quantum computation by anyons. Ann. Phys. 303(1), 2–30 (2003)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  18. 18

    Bombin H., Martin-Delgado M.A.: Topological quantum distillation. Phys. Rev. Lett. 97(18), 180501 (2006)

    ADS  Article  Google Scholar 

  19. 19

    Kim I.H.: Perturbative analysis of topological entanglement entropy from conditional independence. Phys. Rev. B. 86(24), 245116 (2012)

    ADS  Article  Google 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)

    ADS  Article  Google 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)

    ADS  Article  Google Scholar 

  23. 23

    Bilgin E., Boixo S.: Preparing thermal states of quantum systems by dimension reduction. Phys. Rev. Lett. 105(17), 170405 (2010)

    ADS  Article  Google 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)

    ADS  MathSciNet  Article  MATH  Google 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)

    Article  Google 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)

    ADS  Article  Google Scholar 

  28. 28

    Hein M., Eisert J., Briegel H.J.: Multiparty entanglement in graph states. Phys. Rev. A 69(6), 062311 (2004)

    ADS  MathSciNet  Article  MATH  Google 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)

    ADS  MathSciNet  Article  MATH  Google 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)

    ADS  MathSciNet  Article  Google Scholar 

  32. 32

    Hastings M.B.: Topological order at nonzero temperature. Phys. Rev. Lett. 107(21), 210501 (2011)

    ADS  Article  Google Scholar 

  33. 33

    Dennis E., Kitaev A., Landahl A., Preskill J.: Topological quantum memory. J. Math. Phys. 43(9), 4452–4505 (2002)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  34. 34

    Haah J.: Local stabilizer codes in three dimensions without string logical operators. Phys. Rev. A 83(4), 042330 (2011)

    ADS  Article  Google 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)

    ADS  Article  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  38. 38

    Bravyi S., Poulin D., Terhal B.: Tradeoffs for reliable quantum information storage in 2D systems. Phys. Rev. Lett. 104, 050503 (2010)

    ADS  Article  MATH  Google 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)

  40. 40

    Swingle B., McGreevy J.: Renormalization group constructions of topological quantum liquids and beyond. Phys. Rev. B 93(4), 045127 (2016)

    ADS  Article  Google Scholar 

  41. 41

    Swingle, B., McGreevy, J.: Mixed s-sourcery: building many-body states using bubbles of nothing. arXiv preprint arXiv:1607.05753 (2016)

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Correspondence to Michael J. Kastoryano.

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Communicated by M. Wolf

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Brandão, F.G.S.L., Kastoryano, M.J. Finite Correlation Length Implies Efficient Preparation of Quantum Thermal States. Commun. Math. Phys. 365, 1–16 (2019).

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