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quantum annealing
Quantum annealing (QA) is a metaheuristic 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 the ground state of a spin glass.〔P Ray, BK Chakrabarti, A Chakrabarti "Sherrington-Kirkpatrick model in a transverse field: Absence of replica symmetry breaking due to quantum fluctuations" (Phys. Rev. B 39, 11828 (1989) )〕 It was formulated in its present form by T. Kadowaki and H. Nishimori in "Quantum annealing in the transverse Ising model"〔T. Kadowaki and H. Nishimori, "Quantum annealing in the transverse Ising model" (Phys. Rev. E 58, 5355 (1998) )〕 though a proposal in a different form had been proposed by A. B. Finilla, M. A. Gomez, C. Sebenik and J. D. Doll, in "Quantum annealing: A new method for minimizing multidimensional functions".〔A. B. Finilla, M. A. Gomez, C. Sebenik and J. D. Doll, "Quantum annealing: A new method for minimizing multidimensional functions" (Chem. Phys. Lett. 219, 343 (1994) )〕
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, i.e., adiabatic quantum computation.〔E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Ludgren and D. Preda, "A Quantum adiabatic evolution algorithm applied to random instances of an NP-Complete problem" (Science 292, 472 (2001) )〕 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.〔J. Brooke, D. Bitko, T. F. Rosenbaum and G. Aeppli, "Quantum annealing of a disordered magnet", (Science 284 779 (1999) )〕
==Comparison to simulated annealing==
Quantum annealing can be compared to simulated annealing, whose "temperature" parameter plays a similar role to QA's tunneling field strength. In simulated annealing, the temperature determines the probability of moving to a state of higher "energy" from a single current state. In quantum annealing, the strength of transverse field determines the quantum-mechanical probability to change the amplitudes of all states in parallel. Analytical 〔S. Morita and H. Nishimori, "Mathematical foundation of quantum annealing", (J.Math. Phys. 49, 125210 (2008) )〕 and numerical 〔G. E. Santoro and E. Tosatti, "Optimization using quantum mechanics: quantum annealing through adiabatic evolution" (J. Phys. A 39, R393 (2006) )〕 evidence suggests that quantum annealing outperforms simulated annealing under certain conditions (see 〔B. Heim, T. F. Rønnow, S. V. Isakov and M. Troyer, "Quantum versus classical annealing of Ising spin glasses" (Science 348, pp. 215-217 (2015) )〕 for a careful analysis).
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