# Quantum annealing

Quantum annealing (QA) is an optimization process for finding the global minimum of a given objective function over a given set of candidate solutions (candidate states), by a process using quantum fluctuations. Quantum annealing is used mainly for problems where the search space is discrete (combinatorial optimization problems) with many local minima; such as finding[1] the ground state of a spin glass or the traveling salesman problem. The term "quantum annealing" was first proposed in 1988 by B. Apolloni, N. Cesa Bianchi and D. De Falco as a quantum-inspired classical algorithm.[2][3] It was formulated in its present form by T. Kadowaki and H. Nishimori (ja) in "Quantum annealing in the transverse Ising model"[4] though an imaginary-time variant without quantum coherence had been discussed by A. B. Finnila, M. A. Gomez, C. Sebenik and J. D. Doll, in "Quantum annealing is a new method for minimizing multidimensional functions".[5]

Quantum annealing starts from a quantum-mechanical superposition of all possible states (candidate states) with equal weights. Then the system evolves following the time-dependent Schrödinger equation, a natural quantum-mechanical evolution of physical systems. The amplitudes of all candidate states keep changing, realizing a quantum parallelism, according to the time-dependent strength of the transverse field, which causes quantum tunneling between states. If the rate of change of the transverse field is slow enough, the system stays close to the ground state of the instantaneous Hamiltonian (also see adiabatic quantum computation).[6] If the rate of change of the transverse field is accelerated, the system may leave the ground state temporarily but produce a higher likelihood of concluding in the ground state of the final problem Hamiltonian, i.e., diabatic quantum computation.[7][8] The transverse field is finally switched off, and the system is expected to have reached the ground state of the classical Ising model that corresponds to the solution to the original optimization problem. An experimental demonstration of the success of quantum annealing for random magnets was reported immediately after the initial theoretical proposal.[9]

1. ^ Ray, P.; Chakrabarti, B. K.; Chakrabarti, A. (1989). "Sherrington-Kirkpatrick model in a transverse field: Absence of replica symmetry breaking due to quantum fluctuations". Physical Review B. 39 (16): 11828–11832. Bibcode:1989PhRvB..3911828R. doi:10.1103/PhysRevB.39.11828. PMID 9948016.
2. ^ Apolloni, Bruno; Cesa-Bianchi, Nicolo; De Falco, Diego (July 1988). "A numerical implementation of quantum annealing". Stochastic Processes, Physics and Geometry, Proceedings of the Ascona-Locarno Conference.
3. ^ Apolloni, Bruno; Carvalho, Maria C.; De Falco, Diego (1989). "Quantum stochastic optimization". Stoc. Proc. Appl. 33 (2): 233–244. doi:10.1016/0304-4149(89)90040-9.
4. ^ Kadowaki, T.; Nishimori, H. (1998). "Quantum annealing in the transverse Ising model". Phys. Rev. E. 58 (5): 5355. arXiv:cond-mat/9804280. Bibcode:1998PhRvE..58.5355K. doi:10.1103/PhysRevE.58.5355. S2CID 36114913. Archived from the original on 2013-08-11.
5. ^ Finnila, A.B.; Gomez, M.A.; Sebenik, C.; Stenson, C.; Doll, J.D. (1994). "Quantum annealing: A new method for minimizing multidimensional functions". Chemical Physics Letters. 219 (5–6): 343–348. arXiv:chem-ph/9404003. Bibcode:1994CPL...219..343F. doi:10.1016/0009-2614(94)00117-0. S2CID 97302385.
6. ^ Farhi, E.; Goldstone, J.; Gutmann, S.; Lapan, J.; Ludgren, A.; Preda, D. (2001). "A Quantum adiabatic evolution algorithm applied to random instances of an NP-Complete problem". Science. 292 (5516): 472–5. arXiv:quant-ph/0104129. Bibcode:2001Sci...292..472F. doi:10.1126/science.1057726. PMID 11313487. S2CID 10132718.
7. ^ Crosson, Elizabeth, Farhi, Edward, Lin, Cedric, Lin, Han-Hsuan & Shor, Peter (2014). "Different Strategies for Optimization Using the Quantum Adiabatic Algorithm". arXiv:1401.7320
8. ^ Muthukrishnan, Siddharth, Albash, Tameem & Lidar, Daniel A. (2015). "When Diabatic Trumps Adiabatic in Quantum Optimization" arXiv:1505.01249
9. ^ Brooke, J.; Bitko, D.; Rosenbaum, T. F.; Aeppli, G. (1999). "Quantum annealing of a disordered magnet". Science. 284 (5415): 779–81. arXiv:cond-mat/0105238. Bibcode:1999Sci...284..779B. doi:10.1126/science.284.5415.779. PMID 10221904. S2CID 37564720.