The quantum adiabatic algorithm applied to random optimization problems: The quantum spin glass perspective
Tài liệu tham khảo
Garey, 1979
Papadimitriou, 1994
Papadimitriou, 1998
Feynman, 1982, Simulating physics with computers, Internat. J. Theoret. Phys., 21, 467, 10.1007/BF02650179
Deutsch, 1985, Quantum theory, the Church-Turing principle and the universal quantum computer, Proc. R. Soc. Lond. A, 400, 97, 10.1098/rspa.1985.0070
Nielsen, 2000
Mermin, 2007, 10.1017/CBO9780511813870
2007
Deutsch, 1992, Rapid solutions of problems by quantum computation, Proc. R. Soc. Lond. A, 439, 553, 10.1098/rspa.1992.0167
D.R. Simon, On the power of quantum computation, in: Foundations of Computer Science, 1994 Proceedings., 35th Annual Symposium, 1994, p. 116.
P. Shor, Algorithms for quantum computation: discrete logarithms and factoring, in: Foundations of Computer Science, 1994 Proceedings., 35th Annual Symposium, 1994, p. 124.
Grover, 1997, Quantum mechanics helps in searching for a needle in a haystack, Phys. Rev. Lett., 79, 325, 10.1103/PhysRevLett.79.325
Bernstein, 1997, Quantum complexity theory, SIAM J. Comput., 26, 1411, 10.1137/S0097539796300921
Watrous, 2000, Succinct quantum proofs for properties of finite groups
Kitaev, 2002, 10.1090/gsm/047
Apolloni, 1989, Quantum stochastic optimization, Stochastics Process Appl., 33, 233, 10.1016/0304-4149(89)90040-9
Finnila, 1994, Quantum annealing: a new method for minimizing multidimensional functions, Chem. Phys. Lett., 219, 343, 10.1016/0009-2614(94)00117-0
Kadowaki, 1998, Quantum annealing in the transverse Ising model, Phys. Rev. E, 58, 5355, 10.1103/PhysRevE.58.5355
Brooke, 1999, Quantum annealing of a disordered magnet, Science, 284, 779, 10.1126/science.284.5415.779
Farhi, 2001, A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem, Science, 292, 472, 10.1126/science.1057726
Santoro, 2006, Optimization using quantum mechanics: quantum annealing through adiabatic evolution, J. Phys. A: Math. Gen., 39, R393, 10.1088/0305-4470/39/36/R01
2005
Das, 2008, Colloquium: quantum annealing and analog quantum computation, Rev. Modern Phys., 80, 1061, 10.1103/RevModPhys.80.1061
Morita, 2008, Mathematical foundation of quantum annealing, J. Math. Phys., 49, 125210, 10.1063/1.2995837
Messiah, 1962
W. van Dam, M. Mosca, U. Vazirani, How powerful is adiabatic quantum computation? in: Proc. 42nd FOCS, 2001, p. 279.
W. van Dam, U. Vazirani, Limits on quantum adiabatic optimization (unpublished).
Farhi, 2008, How to make the quantum adiabatic algorithm fail, Int. J. Quantum Comput., 6, 503, 10.1142/S021974990800358X
Znidaric, 2006, Exponential complexity of an adiabatic algorithm for an NP-complete problem, Phys. Rev. A, 73, 022329, 10.1103/PhysRevA.73.022329
Janson, 2000
Mitchell, 1992, Hard and easy distributions for SAT problems, 459
Mézard, 1987
Sachdev, 2001
Goldschmidt, 1990, Solvable model of the quantum spin glass in a transverse field, Phys. Rev. B, 41, 4858, 10.1103/PhysRevB.41.4858
Nieuwenhuizen, 1998, Quantum phase transition in spin glasses with multi-spin interactions, Physica A, 250, 8, 10.1016/S0378-4371(97)00546-3
Biroli, 2001, Quantum Thouless-Anderson-Palmer equations for glassy systems, Phys. Rev. B, 64, 014206, 10.1103/PhysRevB.64.014206
Cugliandolo, 2001, Imaginary-time replica formalism study of a quantum spherical p-spin-glass model, Phys. Rev. B, 64, 014403, 10.1103/PhysRevB.64.014403
Jörg, 2008, Simple glass models and their quantum annealing, Phys. Rev. Lett., 101, 147204, 10.1103/PhysRevLett.101.147204
Amin, 2009, First-order quantum phase transition in adiabatic quantum computation, Phys. Rev. A, 80, 062326, 10.1103/PhysRevA.80.062326
Altshuler, 2010, Anderson localization makes adiabatic quantum optimization fail, Proc. Natl. Acad. Sci., 107, 12446, 10.1073/pnas.1002116107
Farhi, 2011, Quantum adiabatic algorithms, small gaps, and different paths, Quantum Inf. Comput., 11, 181
Choi, 2011, Different adiabatic quantum optimization algorithms for the NP-complete exact cover and 3-SAT problems, Quantum Inf. Comput., 11, 638
Dickson, 2012, Algorithmic approach to adiabatic quantum optimization, Phys. Rev. A, 85, 032303, 10.1103/PhysRevA.85.032303
Vazirani, 2001
Dziarmaga, 2010, Dynamics of a quantum phase transition and relaxation to a steady state, Adv. Phys., 59, 1063, 10.1080/00018732.2010.514702
Hastad, 2001, Some optimal inapproximability results, J. ACM, 48, 798, 10.1145/502090.502098
Foini, 2010, Solvable model of quantum random optimization problems, Phys. Rev. Lett., 105, 167204, 10.1103/PhysRevLett.105.167204
Jörg, 2010, First-order transitions and the performance of quantum algorithms in random optimization problems, Phys. Rev. Lett., 104, 207206, 10.1103/PhysRevLett.104.207206
V. Bapst, G. Semerjian, F. Zamponi, The effect of quantum fluctuations on the coloring of random graphs (in preparation).
V.N. Smelyanskiy, E.G. Rieffel, S.I. Knysh, C.P. Williams, M.W. Johnson, M.C. Thom, W.G. Macready, K.L. Pudenz, A near-term quantum computing approach for hard computational problems in space exploration, 2012. arXiv:1204.2821.
S. Tanaka, R. Tamura, Quantum annealing: from viewpoints of statistical physics, condensed matter physics, and computational physics, 2012. arXiv:1204.2907.
M. Ohzeki, Spin glass: A bridge between quantum computation and statistical mechanics, 2012. arXiv:1204.2865.
Richardson, 2008
DiVincenzo, 2000, The physical implementation of quantum computation, Fortschr. Phys., 48, 771, 10.1002/1521-3978(200009)48:9/11<771::AID-PROP771>3.0.CO;2-E
C.R. Laumann, R. Moessner, A. Scardicchio, S.L. Sondhi, Statistical mechanics of classical and quantum computational complexity, 2010. arXiv:1009.1635.
DiVincenzo, 1995, Two-bit gates are universal for quantum computation, Phys. Rev. A, 51, 1015, 10.1103/PhysRevA.51.1015
Deutsch, 1995, Universality in quantum computation, Proc. Math. Phys. Sci., 449, 669, 10.1098/rspa.1995.0065
Lloyd, 1995, Almost any quantum logic gate is universal, Phys. Rev. Lett., 75, 346, 10.1103/PhysRevLett.75.346
Barenco, 1995, Elementary gates for quantum computation, Phys. Rev. A, 52, 3457, 10.1103/PhysRevA.52.3457
Cleve, 1998, Quantum algorithms revisited, Proc. R. Soc. Lond. A, 454, 339, 10.1098/rspa.1998.0164
Rivest, 1978, A method for obtaining digital signatures and public-key cryptosystems, Commun. ACM, 21, 120, 10.1145/359340.359342
Vandersypen, 2001, Experimental realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance, Nature, 414, 883, 10.1038/414883a
Lu, 2007, Demonstration of a compiled version of Shor’s quantum factoring algorithm using photonic qubits, Phys. Rev. Lett., 99, 250504, 10.1103/PhysRevLett.99.250504
Lanyon, 2007, Experimental demonstration of a compiled version of Shor’s algorithm with quantum entanglement, Phys. Rev. Lett., 99, 250505, 10.1103/PhysRevLett.99.250505
E. Lucero, R. Barends, Y. Chen, J. Kelly, M. Mariantoni, A. Megrant, P. O’Malley, D. Sank, A. Vainsencher, J. Wenner, T. White, Y. Yin, A.N. Cleland, J.M. Martinis, Computing prime factors with a Josephson phase qubit quantum processor, 2012. arXiv:1202.5707.
Agrawal, 2004, PRIMES is in P, Ann. of Math. (2), 160, 781, 10.4007/annals.2004.160.781
Bennett, 1997, Strengths and weaknesses of quantum computing, SIAM J. Comput., 26, 1510, 10.1137/S0097539796300933
Dewes, 2012, Quantum speeding-up of computation demonstrated in a superconducting two-qubit processor, Phys. Rev. B, 85, 140503, 10.1103/PhysRevB.85.140503
Harrow, 2009, Quantum algorithm for linear systems of equations, Phys. Rev. Lett., 103, 150502, 10.1103/PhysRevLett.103.150502
Ambainis, 2012, Variable time amplitude amplification and quantum algorithms for linear algebra problems, 636
Kempe, 2003, 3-local hamiltonian is QMA-complete, Quantum Inf. Comput., 3, 258
Kempe, 2006, The complexity of the local hamiltonian problem, SIAM J. Comput., 35, 1070, 10.1137/S0097539704445226
Aharonov, 2009, The power of quantum systems on a line, Comm. Math. Phys., 287, 41, 10.1007/s00220-008-0710-3
S. Bravyi, Efficient algorithm for a quantum analogue of 2-SAT, 2006. arXiv:quant-ph/0602108.
S. Bravyi, C. Moore, A. Russell, Bounds on the quantum satisfiability threshold, 2009. arXiv:0907.1297.
A. Ambainis, J. Kempe, O. Sattath, A quantum Lovasz local lemma, in: Proc. 42nd STOC, 2010, p. 151.
Laumann, 2010, Phase transitions and random quantum satisfiability, Quantum. Inf. Comput., 10, 1
Laumann, 2010, Product, generic, and random generic quantum satisfiability, Phys. Rev. A, 81, 062345, 10.1103/PhysRevA.81.062345
Nayak, 2008, Non-abelian anyons and topological quantum computation, Rev. Modern Phys., 80, 1083, 10.1103/RevModPhys.80.1083
Raussendorf, 2001, A one-way quantum computer, Phys. Rev. Lett., 86, 5188, 10.1103/PhysRevLett.86.5188
Farhi, 1998, Quantum computation and decision trees, Phys. Rev. A, 58, 915, 10.1103/PhysRevA.58.915
Kempe, 2003, Quantum random walks: an introductory overview, Contemp. Phys., 44, 307, 10.1080/00107151031000110776
Ambainis, 2003, Quantum walks and their algorithmic applications, Int. J. Quantum Inf., 1, 507, 10.1142/S0219749903000383
Childs, 2009, Universal computation by quantum walk, Phys. Rev. Lett., 102, 180501, 10.1103/PhysRevLett.102.180501
Reitzner, 2011, Quantum walks, Acta Phys. Slovaca, 61, 603, 10.2478/v10155-011-0006-6
Kirkpatrick, 1983, Optimization by simulated annealing, Science, 220, 671, 10.1126/science.220.4598.671
Steffen, 2003, Experimental implementation of an adiabatic quantum optimization algorithm, Phys. Rev. Lett., 90, 067903, 10.1103/PhysRevLett.90.067903
Z. Bian, F. Chudak, W.G. Macready, L. Clark, F. Gaitan, Experimental determination of Ramsey numbers with quantum annealing, 2012. arXiv:1201.1842.
Born, 1928, Beweis des Adiabatensatzes, Zeit. Phys. A, 51, 165, 10.1007/BF01343193
Kato, 1950, On the adiabatic theorem of quantum mechanics, J. Phys. Soc. Japan, 5, 435, 10.1143/JPSJ.5.435
Cheung, 2011, Improved error bounds for the adiabatic approximation, J. Phys. A, 44, 415302, 10.1088/1751-8113/44/41/415302
A. Elgart, G. Hagedorn, A note on the switching adiabatic theorem, 2012. arXiv:1204.2318.
E. Farhi, J. Goldstone, S. Gutmann, M. Sipser, Quantum computation by adiabatic evolution, 2000. arXiv:quant-ph/0001106.
Rigolin, 2012, Adiabatic theorem for quantum systems with spectral degeneracy, Phys. Rev. A, 85, 062111, 10.1103/PhysRevA.85.062111
Wilczek, 1984, Appearance of gauge structure in simple dynamical systems, Phys. Rev. Lett., 52, 2111, 10.1103/PhysRevLett.52.2111
Valiant, 1986, NP is as easy as detecting unique solutions, Theoret. Comput. Sci., 47, 85, 10.1016/0304-3975(86)90135-0
Landau, 1932, Zur Theorie der Energieubertragung. II, Phys. Soviet Union, 2, 46
Zener, 1932, Non-adiabatic crossing of energy levels, Proc. R. Soc. Lond., 137, 696, 10.1098/rspa.1932.0165
Vitanov, 1996, Landau–Zener model: effects of finite coupling duration, Phys. Rev. A, 53, 4288, 10.1103/PhysRevA.53.4288
Vitanov, 1999, Transition times in the Landau-Zener model, Phys. Rev. A, 59, 988, 10.1103/PhysRevA.59.988
Volkov, 2007, Analytical results for state-to-state transition probabilities in the multistate Landau-Zener model by nonstationary perturbation theory, Phys. Rev. A, 75, 022105, 10.1103/PhysRevA.75.022105
Santoro, 2002, Theory of quantum annealing of an Ising spin glass, Science, 295, 2427, 10.1126/science.1068774
Bapst, 2012, On quantum mean-field models and their quantum annealing, J. Stat. Mech., 2012, P06007, 10.1088/1742-5468/2012/06/P06007
Aharonov, 2007, Adiabatic quantum computation is equivalent to standard quantum computation, SIAM J. Comput., 37, 166, 10.1137/S0097539705447323
Rezakhani, 2010, Intrinsic geometry of quantum adiabatic evolution and quantum phase transitions, Phys. Rev. A, 82, 012321, 10.1103/PhysRevA.82.012321
Roland, 2002, Quantum search by local adiabatic evolution, Phys. Rev. A, 65, 042308, 10.1103/PhysRevA.65.042308
Caneva, 2009, Optimal control at the quantum speed limit, Phys. Rev. Lett., 103, 240501, 10.1103/PhysRevLett.103.240501
Caneva, 2011, Speeding up critical system dynamics through optimized evolution, Phys. Rev. A, 84, 012312, 10.1103/PhysRevA.84.012312
J. Nehrkorn, S. Montangero, A. Ekert, A. Smerzi, R. Fazio, T. Calarco, Staying adiabatic with unknown energy gap, 2011. arXiv:1105.1707.
Seki, 2012, Quantum annealing with antiferromagnetic fluctuations, Phys. Rev. E, 85, 051112, 10.1103/PhysRevE.85.051112
Seoane, 2012, Many-body transverse interactions in the quantum annealing of the p-spin ferromagnet, J. Phys. A, 45, 435301, 10.1088/1751-8113/45/43/435301
Ribeiro, 2006, Adiabatic computation: a toy model, Phys. Rev. A, 74, 042333, 10.1103/PhysRevA.74.042333
Arora, 1998, Proof verification and the hardness of approximation problems, J. ACM, 45, 501, 10.1145/278298.278306
D. Aharonov, I. Arad, Z. Landau, U. Vazirani, The detectibility lemma and quantum gap amplification, in: Proc. 41st Annual ACM Symposium on Theory of Computing, vol. 287, 2009, pp. 417–426.
M. Hastings, Trivial low energy states for commuting hamiltonians, and the quantum PCP conjecture, 2012. arXiv:1201.3387.
S. Gharibian, J. Kempe, Approximation algorithms for QMA-complete problems, in: Proc. 26th CCC’11, 2011, p. 178.
S. Gharibian, J. Kempe, Hardness of approximation for quantum problems, 2012. arXiv:1209.1055.
Cheeseman, 1991, Where the really hard problems are, 331
Friedgut, 1999, Sharp thresholds of graph properties, and the k-sat problem, J. Amer. Math. Soc., 12, 1017, 10.1090/S0894-0347-99-00305-7
Franco, 2001, Results related to threshold phenomena research in satisfiability: lower bounds, Theoret. Comput. Sci., 265, 147, 10.1016/S0304-3975(01)00158-X
Achlioptas, 2001, Lower bounds for random 3-SAT via differential equations., Theoret. Comput. Sci., 265, 159, 10.1016/S0304-3975(01)00159-1
Dubois, 2001, Upper bounds on the satisfiability threshold, Theoret. Comput. Sci., 265, 187, 10.1016/S0304-3975(01)00161-X
Achlioptas, 2004, The threshold for random k-SAT is 2k log 2−O(k), J. Amer. Math. Soc., 17, 947, 10.1090/S0894-0347-04-00464-3
Monasson, 1997, Statistical mechanics of the random k-satisfiability model, Phys. Rev. E, 56, 1357, 10.1103/PhysRevE.56.1357
Mézard, 2002, Random k-satisfiability problem: from an analytic solution to an efficient algorithm, Phys. Rev. E, 66, 056126, 10.1103/PhysRevE.66.056126
Mézard, 2002, Analytic and algorithmic solution of random satisfiability problems, Science, 297, 812, 10.1126/science.1073287
Mertens, 2006, Threshold values of random k-SAT from the cavity method, Random Structures Algorithms, 28, 340, 10.1002/rsa.20090
Krzakala, 2004, Threshold values, stability analysis, and high-q asymptotics for the coloring problem on random graphs, Phys. Rev. E, 70, 046705, 10.1103/PhysRevE.70.046705
Biroli, 2000, A variational description of the ground state structure in random satisfiability problems, Eur. Phys. J. B, 14, 551, 10.1007/s100510051065
Krzakala, 2007, Gibbs states and the set of solutions of random constraint satisfaction problems, Proc. Natl. Acad. Sci., 104, 10318, 10.1073/pnas.0703685104
Franz, 2003, Replica bounds for optimization problems and diluted spin systems, J. Stat. Phys., 111, 535, 10.1023/A:1022885828956
Panchenko, 2004, Bounds for diluted mean-fields spin glass models, Probab. Theory Related Fields, 130, 319, 10.1007/s00440-004-0342-2
Daudé, 2008, Pairs of sat assignments and clustering in random boolean formulae, Theoret. Comput. Sci., 393, 260, 10.1016/j.tcs.2008.01.005
D. Achlioptas, F. Ricci-Tersenghi, On the solution-space geometry of random constraint satisfaction problems, in: Proc. of the 38th Annual ACM Symposium on Theory of Computing.
A. Coja-Oghlan, On belief propagation guided decimation for random k-sat, in: Proc. 22nd SODA, 2011, p. 957.
Monasson, 2007, Introduction to phase transitions in random optimization problems
Mézard, 2009
Moore, 2011, 10.1093/acprof:oso/9780199233212.001.0001
Anderson, 1958, Absence of diffusion in certain random lattices, Phys. Rev., 109, 1492, 10.1103/PhysRev.109.1492
Derrida, 1981, Random-energy model: an exactly solvable model of disordered systems, Phys. Rev. B, 24, 2613, 10.1103/PhysRevB.24.2613
Edwards, 1975, Theory of spin-glasses, J. Phys. F, 5, 965, 10.1088/0305-4608/5/5/017
Sherrington, 1975, Solvable model of a spin-glass, Phys. Rev. Lett., 35, 1792, 10.1103/PhysRevLett.35.1792
Parisi, 1980, A sequence of approximated solutions to the S-K model for spin glasses, J. of Phys. A, 13, L115, 10.1088/0305-4470/13/4/009
Fischer, 1991
Talagrand, 2006, The Parisi formula, Ann. of Math., 163, 221, 10.4007/annals.2006.163.221
Guerra, 2002, The thermodynamic limit in mean field spin glass models, Comm. Math. Phys., 230, 71, 10.1007/s00220-002-0699-y
Gross, 1984, The simplest spin glass, Nuclear Phys. B, 240, 431, 10.1016/0550-3213(84)90237-2
Viana, 1985, Phase diagrams for dilute spin glasses, J. Phys. C, 18, 3037, 10.1088/0022-3719/18/15/013
Mézard, 1985, Replicas and optimization, J. Phys., 46, L771
Fu, 1986, Application of statistical mechanics to NP-complete problems in combinatorial optimization, J. Phys. A, 19, 1605, 10.1088/0305-4470/19/9/033
Monasson, 1998, Optimization problems and replica symmetry breaking in finite connectivity spin glasses, J. Phys. A, 31, 513, 10.1088/0305-4470/31/2/012
Mézard, 2001, The Bethe lattice spin glass revisited, Eur. Phys. J. B, 20, 217, 10.1007/PL00011099
Mézard, 2003, The cavity method at zero temperature, J. Stat. Phys., 111, 1, 10.1023/A:1022221005097
Talagrand, 2003
Castellani, 2005, Spin-glass theory for pedestrians, J. Stat. Mech., 2005, P05012, 10.1088/1742-5468/2005/05/P05012
Kirkpatrick, 1987, Stable and metastable states in mean-field Potts and structural glasses, Phys. Rev. B, 36, 8552, 10.1103/PhysRevB.36.8552
Kirkpatrick, 1988, Mean-field soft-spin Potts glass model: statics and dynamics, Phys. Rev. B, 37, 5342, 10.1103/PhysRevB.37.5342
Kirkpatrick, 1987, Connections between some kinetic and equilibrium theories of the glass transition, Phys. Rev. A, 35, 3072, 10.1103/PhysRevA.35.3072
Kirkpatrick, 1989, Scaling concepts for the dynamics of viscous liquids near an ideal glassy state, Phys. Rev. A, 40, 1045, 10.1103/PhysRevA.40.1045
Lubchenko, 2007, Theory of structural glasses and supercooled liquids, Ann. Rev. Phys. Chem., 58, 235, 10.1146/annurev.physchem.58.032806.104653
Cavagna, 2009, Supercooled liquids for pedestrians, Phys. Rep., 476, 51, 10.1016/j.physrep.2009.03.003
Biroli, 2012, The Random First-Order Transition Theory of Glasses: A Critical Assessment
Berthier, 2011, Theoretical perspective on the glass transition and amorphous materials, Rev. Modern Phys., 83, 587, 10.1103/RevModPhys.83.587
Götze, 2009
Kauzmann, 1948, The nature of the glassy state and the behavior of liquids at low temperatures, Chem. Rev., 43, 219, 10.1021/cr60135a002
Gardner, 1985, Spin glasses with p-spin interactions, Nuclear Phys. B, 257, 747, 10.1016/0550-3213(85)90374-8
Mézard, 2003, Two solutions to diluted p-spin models and XORSAT problems., J. Stat. Phys., 111, 505, 10.1023/A:1022886412117
Cocco, 2003, Rigorous decimation-based construction of ground pure states for spin-glass models on random lattices, Phys. Rev. Lett., 90, 047205, 10.1103/PhysRevLett.90.047205
Montanari, 2006, On the dynamics of the glass transition on Bethe lattices, J. Stat. Phys., 124, 103, 10.1007/s10955-006-9103-1
Ibrahimi, 2012, The set of solutions of random XORSAT formulae, 760
D. Achlioptas, M. Molloy, The solution space geometry of random linear equations, 2011. arXiv:1107.5550.
Mora, 2008, Random subcubes as a toy model for constraint satisfaction problems, J. Stat. Phys., 131, 1121, 10.1007/s10955-008-9543-x
Zdeborová, 2007, Phase transitions in the coloring of random graphs, Phys. Rev. E, 76, 031131, 10.1103/PhysRevE.76.031131
Semerjian, 2008, On the freezing of variables in random constraint satisfaction problems, J. Stat.Phys., 130, 251, 10.1007/s10955-007-9417-7
Achlioptas, 2008, Algorithmic barriers from phase transitions, 793
Alava, 2008, Circumspect descent prevails in solving random constraint satisfaction problems, Proc. Natl. Acad. Sci., 105, 15253, 10.1073/pnas.0712263105
Krzakala, 2007, Landscape analysis of constraint satisfaction problems, Phys. Rev. E, 76, 021122, 10.1103/PhysRevE.76.021122
Krzakala, 2008, Potts glass on random graphs, Europhys. Lett., 81, 57005, 10.1209/0295-5075/81/57005
Montanari, 2006, Rigorous inequalities between length and time scales in glassy systems, J. Stat. Phys., 125, 23, 10.1007/s10955-006-9175-y
Gross, 1985, Mean-field theory of the Potts glass, Phys. Rev. Lett., 55, 304, 10.1103/PhysRevLett.55.304
Geman, 1984, Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, IEEE Trans. Pattern Anal. Mach. Intell., 6, 721, 10.1109/TPAMI.1984.4767596
Monthus, 1996, Models of traps and glass phenomenology, J. Phys. A, 29, 3847, 10.1088/0305-4470/29/14/012
Arous, 2008, Universality of the REM for dynamics of mean-field spin glasses, Comm. Math. Phys., 282, 663, 10.1007/s00220-008-0565-7
Bray, 1987, Chaotic nature of the spin-glass phase, Phys. Rev. Lett., 58, 57, 10.1103/PhysRevLett.58.57
Fisher, 1988, Equilibrium behavior of the spin-glass ordered phase, Phys. Rev. B, 38, 386, 10.1103/PhysRevB.38.386
Krzakala, 2002, Chaotic temperature dependence in a model of spin glasses, Eur. Phys. J. B, 28, 199, 10.1140/epjb/e2002-00221-y
Krzakala, 2010, Following Gibbs states adiabatically the energy landscape of mean-field glassy systems, Europhys. Lett., 90, 66002, 10.1209/0295-5075/90/66002
Zdeborová, 2010, Generalization of the cavity method for adiabatic evolution of Gibbs states, Phys. Rev. B, 81, 224205, 10.1103/PhysRevB.81.224205
van Mourik, 2002, Random graph coloring: statistical physics approach, Phys. Rev. E, 66, 056120, 10.1103/PhysRevE.66.056120
Ardelius, 2006, Behavior of heuristics on large and hard satisfiability problems, Phys. Rev. E, 74, 037702, 10.1103/PhysRevE.74.037702
Zdeborová, 2008, Locked constraint satisfaction problems, Phys. Rev. Lett., 101, 078702, 10.1103/PhysRevLett.101.078702
Zdeborová, 2009, Statistical physics of hard optimization problems, Acta Phys. Slovaca, 59, 169, 10.2478/v10155-010-0096-6
Braunstein, 2002, Complexity transitions in global algorithms for sparse linear systems over finite fields, J. Phys. A, 35, 7559, 10.1088/0305-4470/35/35/301
Haanpää, 2006, Hard satisfiable clause sets for benchmarking equivalence reasoning techniques, J. Sat. Boolean Model. Comput., 2, 27
Ricci-Tersenghi, 2010, Being glassy without being hard to solve, Science, 330, 1639, 10.1126/science.1189804
Zdeborová, 2011, Quiet planting in the locked constraint satisfaction problems, SIAM J. Discrete Math., 25, 750, 10.1137/090750755
Young, 2010, First-order phase transition in the quantum adiabatic algorithm, Phys. Rev. Lett., 104, 020502, 10.1103/PhysRevLett.104.020502
Foini, 2011, Quantum Biroli–Mézard model: glass transition and superfluidity in a quantum lattice glass model, Phys. Rev. B, 83, 094513, 10.1103/PhysRevB.83.094513
Botet, 1983, Large-size critical behavior of infinitely coordinated systems, Phys. Rev. B, 28, 3955, 10.1103/PhysRevB.28.3955
Dusuel, 2004, Finite-size scaling exponents of the Lipkin-Meshkov-Glick model, Phys. Rev. Lett., 93, 237204, 10.1103/PhysRevLett.93.237204
Dusuel, 2005, Continuous unitary transformations and finite-size scaling exponents in the Lipkin–Meshkov–Glick model, Phys. Rev. B, 71, 224420, 10.1103/PhysRevB.71.224420
Das, 2006, Infinite-range Ising ferromagnet in a time-dependent transverse magnetic field: quench and ac dynamics near the quantum critical point, Phys. Rev. B, 74, 144423, 10.1103/PhysRevB.74.144423
Fisher, 1995, Critical behavior of random transverse-field Ising spin chains, Phys. Rev. B, 51, 6411, 10.1103/PhysRevB.51.6411
Fisher, 1998, Distributions of gaps and end-to-end correlations in random transverse-field Ising spin chains, Phys. Rev. B, 58, 9131, 10.1103/PhysRevB.58.9131
Young, 1996, Numerical study of the random transverse-field Ising spin chain, Phys. Rev. B, 53, 8486, 10.1103/PhysRevB.53.8486
Caneva, 2007, Adiabatic quantum dynamics of a random Ising chain across its quantum critical point, Phys. Rev. B, 76, 144427, 10.1103/PhysRevB.76.144427
Bray, 1980, Replica theory of quantum spin glasses, J. Phys. C, 13, L655, 10.1088/0022-3719/13/24/005
A. Andreanov, M. Müller, Collective excitations and marginal stability of quantum Ising spin glasses, 2012. arXiv:1204.4156.
Jörg, 2010, Energy gaps in quantum first-order mean-field-like transitions: the problems that quantum annealing cannot solve, EPL, 89, 40004, 10.1209/0295-5075/89/40004
Filippone, 2011, Quantum phase transitions in fully connected spin models: an entanglement perspective, Phys. Rev. A, 83, 022327, 10.1103/PhysRevA.83.022327
C. Laumann, R. Moessner, A. Scardicchio, S. Sondhi, The quantum adiabatic algorithm and scaling of gaps at first order quantum phase transition, 2012. arXiv:1202.3646.
Dobrosavljevic, 1990, 1/p expansion for a p-spin interaction spin-glass model in a transverse field, J. Phys. A, 23, L767, 10.1088/0305-4470/23/15/013
Jörg, 2010, Quantum annealing of hard problems, Progr. Theoret. Phys. Suppl., 184, 290, 10.1143/PTPS.184.290
Buccheri, 2011, Structure of typical states of a disordered Richardson model and many-body localization, Phys. Rev. B, 84, 094203, 10.1103/PhysRevB.84.094203
Dickson, 2011, Elimination of perturbative crossings in adiabatic quantum optimization, New J. Phys., 13, 073011, 10.1088/1367-2630/13/7/073011
S. Knysh, V. Smelyanskiy, On the relevance of avoided crossings away from quantum critical point to the complexity of quantum adiabatic algorithm, 2010. arXiv:1005.3011.
V. Choi, Adiabatic quantum algorithms for the NP-complete maximum-weight independent set, exact cover and 3-SAT problems, 2010. arXiv:1004.2226.
V. Choi, Avoid first order quantum phase transition by changing problem hamiltonians, 2010. arXiv:1010.1220.
Dickson, 2011, Does adiabatic quantum optimization fail for np-complete problems?, Phys. Rev. Lett., 106, 050502, 10.1103/PhysRevLett.106.050502
Thompson, 1976, The behavior of eigenvalues and singular values under perturbations of restricted rank, Linear Algebra Appl., 13, 69, 10.1016/0024-3795(76)90044-6
Montanari, 2008, Clusters of solutions and replica symmetry breaking in random k-satisfiability, J. Stat. Mech., 2008, P04004, 10.1088/1742-5468/2008/04/P04004
Dembo, 2010, Ising models on locally tree-like graphs, Ann. Appl. Probab., 20, 565, 10.1214/09-AAP627
Laumann, 2008, Cavity method for quantum spin glasses on the Bethe lattice, Phys. Rev. B, 78, 134424, 10.1103/PhysRevB.78.134424
Krzakala, 2008, Path-integral representation for quantum spin models: application to the quantum cavity method and monte carlo simulations, Phys. Rev. B, 78, 134428, 10.1103/PhysRevB.78.134428
Leifer, 2008, Quantum graphical models and belief propagation, Ann. Physics, 323, 1899, 10.1016/j.aop.2007.10.001
Poulin, 2008, Belief propagation algorithm for computing correlation functions in finite-temperature quantum many-body systems on loopy graphs, Phys. Rev. A, 77, 052318, 10.1103/PhysRevA.77.052318
Bilgin, 2010, Coarse-grained belief propagation for simulation of interacting quantum systems at all temperatures, Phys. Rev. B, 81, 054106, 10.1103/PhysRevB.81.054106
Poulin, 2011, Markov entropy decomposition: a variational dual for quantum belief propagation, Phys. Rev. Lett., 106, 80403, 10.1103/PhysRevLett.106.080403
Ioffe, 2010, Disorder-driven quantum phase transitions in superconductors and magnets, Phys. Rev. Lett., 105, 037001, 10.1103/PhysRevLett.105.037001
Dimitrova, 2011, The cavity method for quantum disordered systems: from transverse random field ferromagnets to directed polymers in random media, J. Stat. Mech., 2011, P01020, 10.1088/1742-5468/2011/01/P01020
Ramezanpour, 2012, Cavity approach to variational quantum mechanics, Phys. Rev. B, 85, 125131, 10.1103/PhysRevB.85.125131
Semerjian, 2009, Exact solution of the Bose–Hubbard model on the Bethe lattice, Phys. Rev. B, 80, 014524, 10.1103/PhysRevB.80.014524
Kschischang, 2001, Factor graphs and the sum-product algorithm, IEEE Trans. Inform. Theory, 47, 498, 10.1109/18.910572
Molloy, 1995, A critical point for random graphs with a given degree sequence, Random Structures Algorithms, 6, 161, 10.1002/rsa.3240060204
Monasson, 1995, Structural glass transition and the entropy of the metastable states, Phys. Rev. Lett., 75, 2847, 10.1103/PhysRevLett.75.2847
Cugliandolo, 1993, Analytical solution of the off-equilibrium dynamics of a long-range spin-glass model, Phys. Rev. Lett., 71, 173, 10.1103/PhysRevLett.71.173
Abou-Chacra, 1973, A selfconsistent theory of localization, J. Phys. C, 6, 1734, 10.1088/0022-3719/6/10/009
Arulampalam, 2002, A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking, IEEE Trans. Signal Process., 50, 174, 10.1109/78.978374
Andrea Montanari, 2004, Instability of one-step replica-symmetry-broken phase in satisfiability problems, J. Phys. A, 37, 2073, 10.1088/0305-4470/37/6/008
Ginibre, 1968, Reduced density matrices of the anisotropic Heisenberg model, Comm. Math. Phys., 10, 140, 10.1007/BF01654238
Gallavotti, 1968, Analyticity properties of the anisotropic Heisenberg model, Comm. Math. Phys, 10, 311, 10.1007/BF03399504
Farhi, 1992, The functional integral constructed directly from the Hamiltonian, Ann. Physics, 213, 182, 10.1016/0003-4916(92)90288-W
Aizenman, 1994, Geometric aspects of quantum spin states, Comm. Math. Phys., 164, 17, 10.1007/BF02108805
Chayes, 2008, The phase diagram of the quantum Curie–Weiss model, J. Stat. Phys., 133, 131, 10.1007/s10955-008-9608-x
Ioffe, 2009, Stochastic geometry of classical and quantum Ising models, vol. 1970, 87
Martinelli, 2012, Glauber dynamics for the quantum Ising model in a transverse field on a regular tree, J. Stat. Phys., 146, 1059, 10.1007/s10955-012-0436-7
Hastings, 2007, Quantum belief propagation: an algorithm for thermal quantum systems, Phys. Rev. B, 76, 201102, 10.1103/PhysRevB.76.201102
Feigel’man, 2010, Superconductor–insulator transition and energy localization, Phys. Rev. B, 82, 184534, 10.1103/PhysRevB.82.184534
M. Mueller, Giant positive magnetoresistance and localization in bosonic insulators, arXiv:1109.0245.
Georges, 1996, Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions, Rev. Modern Phys., 68, 13, 10.1103/RevModPhys.68.13
A. Globerson, T. Jaakkola, Approximate inference using conditional entropy decompositions, in: Proc. of the 11th International Conference on Artificial Intelligence and Statistics, 2007.
Jastrow, 1955, Many-body problem with strong forces, Phys. Rev., 98, 1479, 10.1103/PhysRev.98.1479
Ramezanpour, 2012, The sign problem in the Bethe approximation, Phys. Rev. B, 86, 155147, 10.1103/PhysRevB.86.155147
White, 1992, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett., 69, 2863, 10.1103/PhysRevLett.69.2863
Vidal, 2003, Efficient classical simulation of slightly entangled quantum computations, Phys. Rev. Lett., 91, 147902, 10.1103/PhysRevLett.91.147902
Nagaj, 2008, Quantum transverse-field Ising model on an infinite tree from matrix product states, Phys. Rev. B, 77, 214431, 10.1103/PhysRevB.77.214431
Shi, 2006, Classical simulation of quantum many-body systems with a tree tensor network, Phys. Rev. A, 74, 022320, 10.1103/PhysRevA.74.022320
Nagy, 2012, Simulating quantum systems on the Bethe lattice by translationally invariant infinite-tree tensor network, Ann. Physics, 327, 542, 10.1016/j.aop.2011.11.012
W. Li, J. von Delft, T. Xiang, Efficient simulation of infinite tree tensor network states on the Bethe lattice, 2012. arXiv:1209.2387.
Vidal, 2007, Classical simulation of infinite-size quantum lattice systems in one spatial dimension, Phys. Rev. Lett., 98, 70201, 10.1103/PhysRevLett.98.070201
Lepetit, 2000, Density-matrix renormalization study of the Hubbard model on a Bethe lattice, Eur. Phys. J. B, 13, 421, 10.1007/s100510050053
Hatano, 2005, Finding exponential product formulas of higher orders, Optimization, 22
Huyghebaert, 1990, Product formula methods for time-dependent schrodinger problems, J. Phys. A, 23, 5777, 10.1088/0305-4470/23/24/019
Poulin, 2011, Quantum simulation of time-dependent hamiltonians and the convenient illusion of hilbert space, Phys. Rev. Lett., 106, 170501, 10.1103/PhysRevLett.106.170501
Cazalilla, 2002, Time-dependent density-matrix renormalization group: a systematic method for the study of quantum many-body out-of-equilibrium systems, Phys. Rev. Lett., 88, 256403, 10.1103/PhysRevLett.88.256403
White, 2004, Real-time evolution using the density matrix renormalization group, Phys. Rev. Lett., 93, 76401, 10.1103/PhysRevLett.93.076401
Daley, 2004, Time-dependent density-matrix renormalization-group using adaptive effective hilbert spaces, J. Stat. Mech., 2004, P04005, 10.1088/1742-5468/2004/04/P04005
Ceperley, 1995, Path integrals in the theory of condensed helium, Rev. Mod. Phys., 67, 279, 10.1103/RevModPhys.67.279
Krauth, 1991, Superfluid-insulator transition in disordered boson systems, Phys. Rev. Lett., 67, 2307, 10.1103/PhysRevLett.67.2307
Rieger, 1994, Zero-temperature quantum phase transition of a two-dimensional Ising spin glass, Phys. Rev. Lett., 72, 4141, 10.1103/PhysRevLett.72.4141
Batrouni, 1996, World line simulations of the bosonic Hubbard model in the ground state, Comput. Phys. Comm., 97, 63, 10.1016/0010-4655(96)00022-7
Beard, 1996, Simulations of discrete quantum systems in continuous euclidean time, Phys. Rev. Lett., 77, 5130, 10.1103/PhysRevLett.77.5130
Prokof’ev, 1998, “Worm” algorithm in quantum Monte Carlo simulations, Phys. Lett. A, 238, 253, 10.1016/S0375-9601(97)00957-2
Rieger, 1999, Application of a continuous time cluster algorithm to the two-dimensional random quantum Ising ferromagnet, Eur. Phys. J. B, 9, 233, 10.1007/s100510050761
Farhi, 2011, A quantum Monte Carlo method at fixed energy, Comput. Phys. Comm., 182, 1663, 10.1016/j.cpc.2011.04.021
Sandvik, 1999, Stochastic series expansion method with operator-loop update, Phys. Rev. B, 59, R14157, 10.1103/PhysRevB.59.R14157
Hen, 2011, Exponential complexity of the quantum adiabatic algorithm for certain satisfiability problems, Phys. Rev. E, 84, 061152, 10.1103/PhysRevE.84.061152
Young, 2008, Size dependence of the minimum excitation gap in the quantum adiabatic algorithm, Phys. Rev. Lett., 101, 170503, 10.1103/PhysRevLett.101.170503
Hen, 2012, Excitation gap from optimized correlation functions in quantum Monte Carlo simulations, Phys. Rev. E, 85, 036705, 10.1103/PhysRevE.85.036705
Markland, 2010, The quantum liquid-glass transition, Nat. Phys., 7, 134, 10.1038/nphys1865
Guidetti, 2011, Complexity of several constraint-satisfaction problems using the heuristic classical algorithm walksat, Phys. Rev. E, 84, 011102, 10.1103/PhysRevE.84.011102
E. Farhi, D. Gosset, I. Hen, A. Sandvik, P. Shor, A. Young, F. Zamponi, The performance of the quantum adiabatic algorithm on random instances of two optimization problems on regular hypergraphs, 2012. arXiv:1208.3757.
Franz, 2001, Exact solutions for diluted spin glasses and optimization problems, Phys. Rev. Lett., 87, 127209, 10.1103/PhysRevLett.87.127209
Franz, 2001, A ferromagnet with a glass transition, Europhys. Lett., 55, 465, 10.1209/epl/i2001-00438-4
http://code.google.com/p/relsat.
http://www.laria.u-picardie.fr/~cli/maxsatz2009.c.
D. Gosset, Ph.D. Thesis, Case Studies in Quantum Adiabatic Optimization, 2011.
B. Olmos, I. Lesanovsky, J. Garrahan, Facilitated spin models of dissipative quantum glasses, 2012. arXiv:1203.6585.
Carleo, 2009, Bose–Einstein condensation in quantum glasses, Phys. Rev. Lett., 103, 215302, 10.1103/PhysRevLett.103.215302
Childs, 2001, Robustness of adiabatic quantum computation, Phys. Rev. A, 65, 012322, 10.1103/PhysRevA.65.012322
Roland, 2005, Noise resistance of adiabatic quantum computation using random matrix theory, Phys. Rev. A, 71, 032330, 10.1103/PhysRevA.71.032330
Sarandy, 2005, Adiabatic quantum computation in open systems, Phys. Rev. Lett., 95, 250503, 10.1103/PhysRevLett.95.250503