An introduction to quantum annealing

de Falco, Diego; Tamascelli, Dario

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 45 (2011) no. 1, pp. 99-116.

Quantum annealing, or quantum stochastic optimization, is a classical randomized algorithm which provides good heuristics for the solution of hard optimization problems. The algorithm, suggested by the behaviour of quantum systems, is an example of proficuous cross contamination between classical and quantum computer science. In this survey paper we illustrate how hard combinatorial problems are tackled by quantum computation and present some examples of the heuristics provided by quantum annealing. We also present preliminary results about the application of quantum dissipation (as an alternative to imaginary time evolution) to the task of driving a quantum system toward its state of lowest energy.

[1] D. Aharonov et al., Adiabatic quantum computation is equivalent to standard quantum computation. SIAM J. Comput. 37 (2007) 166. | MR | Zbl

[2] S. Albeverio, R. Hoeg-Krohn and L. Streit, Energy forms, Hamiltonians and distorted Brownian paths. J. Math. Phys. 18 (1977) 907-917. | MR | Zbl

[3] B. Altshuler, H. Krovi and J. Roland, Anderson localization casts clouds over adiabatic quantum optimization. arXiv:0912.0746v1 (2009).

[4] P. Amara, D. Hsu and J. Straub, Global minimum searches using an approximate solution of the imaginary time Schrödinger equation. J. Chem. Phys. 97 (1993) 6715-6721.

[5] A. Ambainis and O. Regev, An elementary proof of the quantum adiabatic theorem. arXiv:quant-ph/0411152 (2004).

[6] M.H.S. Amin and V. Choi. First order quantum phase transition in adiabatic quantum computation. arXiv:quant-ph/0904.1387v3 (2009).

[7] P. Anderson, Absence of diffusion in certain random lattices. Phys. Rev. 109 (1958) 1492-1505.

[8] B. Apolloni, C. Carvalho and D. De Falco, Quantum stochastic optimization. Stoc. Proc. Appl. 33 (1989) 223-244. | MR | Zbl

[9] B. Apolloni, N. Cesa-Bianchi and D. De Falco, A numerical implementation of Quantum Annealing, in Stochastic Processes, Physics and Geometry, Proceedings of the Ascona/Locarno Conference, 4-9 July 1988. Albeverio et al., Eds. World Scientific (1990), 97-111. | MR

[10] D. Battaglia, G. Santoro, L. Stella, E. Tosatti and O. Zagordi, Deterministic and stochastic quantum annealing approaches. Lecture Notes in Computer Physics 206 (2005) 171-206.

[11] E. Bernstein and U. Vazirani, Quantum complexity theory. SIAM J. Comput. 26 (1997) 1411-1473. | MR | Zbl

[12] F. Bloch, Über die Quantenmechanik der Elektronen in Kristallgittern. Z. Phys. 52 (1929) 555-600. | JFM

[13] M. Born and V. Fock, Beweis des Adiabatensatzes. Z. Phys. A 51 (1928) 165. | JFM

[14] S. Bravyi, Efficient algorithm for a quantum analogue of 2-sat. arXiV:quant-ph/0602108 (2006). | MR | Zbl

[15] H.P. Breuer and F. Petruccione, The theory of open quantum systems. Oxford University Press, New York (2002). | MR | Zbl

[16] D. De Falco and D. Tamascelli, Speed and entropy of an interacting continuous time quantum walk. J. Phys. A 39 (2006) 5873-5895. | MR | Zbl

[17] D. De Falco and D. Tamascelli, Quantum annealing and the Schrödinger-Langevin-Kostin equation. Phys. Rev. A 79 (2009) 012315.

[18] D. De Falco, E. Pertoso and D. Tamascelli, Dissipative quantum annealing, in Proceedings of the 29th Conference on Quantum Probability and Related Topics. World Scientific (2009) (in press). | MR | Zbl

[19] D. Deutsch, Quantum theory, the Church-Turing principle and the universal quantum computer. Proc. R. Soc. Lond. A 400 (1985) 97-117. | MR | Zbl

[20] S. Eleuterio and S. Vilela Mendes, Stochastic ground-state processes. Phys. Rev. B 50 (1994) 5035-5040.

[21] E. Farhi et al., Quantum computation by adiabatic evolution. arXiv:quant-ph/0001106v1 (2000).

[22] E. Farhi et al., A quantum adiabatic evolution algorithm applied to random instances of an NP-Complete problem. Science (2001) 292. | MR | Zbl

[23] R. Feynman, Simulating physics with computers. Int. J. Theor. Phys. 21 (1982) 467-488. | MR

[24] R.P. Feynman, Space-time approach to non-relativistic quantum mechanics. Rev. Mod. Phys. 20 (1948) 367-387. | MR

[25] G. Ford, M. Kac and P. Mazur, Statistical mechanics of assemblies of coupled oscillators. J. Math. Phys. 6 (1965) 504-515. | MR | Zbl

[26] T. Gregor and R. Car, Minimization of the potential energy surface of Lennard-Jones clusters by quantum optimization. Chem. Rev. Lett. 412 (2005) 125-130.

[27] J.J. Griffin and K.-K. Kan, Colliding heavy ions: Nuclei as dynamical fluids. Rev. Mod. Phys. 48 (1976) 467-477.

[28] L. Grover, A fast quantum-mechanical algorithm for database search, in Proc. 28th Annual ACM Symposium on the Theory of Computing. ACM, New York (1996). | MR | Zbl

[29] L. Grover, From Schrödinger equation to the quantum search algorithm. Am. J. Phys. 69 (2001) 769-777.

[30] T. Hogg, Adiabatic quantum computing for random satisfiability problems. Phys. Rev. A 67 (2003) 022314.

[31] D.S. Johnson, C.R. Aragon, L.A. Mcgeoch and C. Shevon, Optimization by simulated annealing: An experimental evaluation; part i, graph partitioning. Oper. Res. 37 (1989) 865-892. | Zbl

[32] G. Jona-Lasinio, F. Martinelli and E. Scoppola, New approach to the semiclassical limit of quantum mechanics. i. Multiple tunnelling in one dimension. Commun. Math. Phys. 80 (1981) 223-254. | MR | Zbl

[33] M. Kac, On distributions of certain Wiener functionals. Trans. Am. Math. Soc. (1949) 1-13. | MR | Zbl

[34] J. Kempe, A. Kitaev and O. Regev, The complexity of the local Hamiltonian problem. SIAM J. Comput. 35 (2006) 1070-1097. | MR | Zbl

[35] S. Kirkpatrik, C.D. Gelatt Jr. and M.P. Vecchi, Optimization by simulated annealing. Science 220 (1983) 671-680. | MR | Zbl

[36] M. Kostin, On the Schrödinger-Langevin equation. J. Chem. Phys. 57 (1972) 3589-3591.

[37] M. Kostin, Friction and dissipative phenomena in quantum mechanics. J. Statist. Phys. 12 (1975) 145-151.

[38] K. Kurihara, S. Tanaka and S. Miyashita, Quantum annealing for clustering. arXiv:quant-ph/09053527v2 (2009).

[39] C. Laumann et al., On product, generic and random generic quantum satisfiability. arXiv:quant-ph/0910.2058v1 (2009).

[40] C. Laumann et al., Phase transitions and random quantum satisfiability. arXiv:quant-ph/0903.1904v1 (2009).

[41] A. Messiah, Quantum Mechanics. John Wiley and Sons (1958). | Zbl

[42] S. Morita and H. Nishimori, Mathematical foundations of quantum annealing. J. Math. Phys. 49 (2008) 125210. | MR | Zbl

[43] C. Papadimitriou and K. Steiglitz, Combinatorial optimization: algorithms and complexity. Dover New York (1998). | MR | Zbl

[44] B. Reichardt, The quantum adiabatic optimization algorithm and local minima, in Proc. 36th STOC (2004) 502. | MR | Zbl

[45] G. Santoro and E. Tosatti, Optimization using quantum mechanics: quantum annealing through adiabatic evolution. J. Phys. A 39 (2006) R393-R431. | MR | Zbl

[46] G. Santoro and E. Tosatti, Optimization using quantum mechanics: quantum annealing through adiabatic evolution. J. Phys. A: Math. Theor. 41 (2008) 209801. | MR | Zbl

[47] P.W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26 (1997) 1484. | MR | Zbl

[48] L. Stella, G. Santoro and E. Tosatti, Optimization by quantum annealing: Lessons from simple cases. Phys. Rev. B 72 (2005) 014303.

[49] W. Van Dam, M. Mosca and U. Vazirani, How powerful is adiabatic quantum computation. Proc. FOCS '01 (2001). | MR

[50] J. Watrous, Succint quantum proofs for properties of finite groups, in Proc. IEEE FOCS (2000) 537-546. | MR

[51] A.P. Young, S. Knysh and V.N. Smelyanskiy. Size dependence of the minimum excitation gap in the quantum adiabatic algorithm. Phys. Rev. Lett. 101 (2008) 170503.

[52] J. Yuen-Zhou et al., Time-dependent density functional theory for open quantum systems with unitary propagation. arXiv:cond-mat.mtrl-sci/0902.4505v3 (2009).

[53] C. Zener, A theory of the electrical breakdown of solid dielectrics. Proc. R. Soc. Lond. A 145 (1934) 523-529. | Zbl

[54] M. Žnidari and M. Horvat, Exponential complexity of an adiabatic algorithm for an np-complete problem. Phys. Rev. A 73 (2006) 022329.