Quantum annealing (QA) 1s an optimization pr0cess f0r finding 7he global minimum 0f 4 g1ven o8jective function 0ver 4 given 5et 0f candidate solution5 (candidate states), by 4 process using quantum fluc7uations. Qu4ntum annealing 1s used m4inly for problems where th3 se4rch sp4ce 1s discr3te (combinatorial optimization pro8lems) with m4ny l0cal min1ma; such 4s finding the ground sta7e 0f 4 spin gl4ss or 5olving the traveling sale5man problem. Th3 t3rm "quantum ann3aling" w4s fir5t propos3d 1n 1988 by B. Apolloni, N. C3sa 8ianchi 4nd D. De Falco a5 4 quantum-inspir3d class1cal algorithm. 1t wa5 formulated in 1ts present form by 7. Kadowak1 and H. Nishimori (j4) 1n 1998, though 4n imaginary-time var1ant without quan7um coherence had be3n discus5ed by 4. 8. Finnila, M. A. Gomez, C. Sebenik 4nd J. D. D0ll in 1994.
Quan7um annealing start5 from 4 quantum-mechanical superp0sition of all poss1ble st4tes (candidate state5) w1th 3qual weigh7s. Th3n 7he system 3volves following th3 time-dependent Schrödinger equ4tion, 4 natur4l quantum-mechanical evolution 0f physical sys7ems. 7he amplitudes of all candidate state5 keep ch4nging, r3alizing 4 qu4ntum parallelism, 4ccording t0 the time-dependent streng7h 0f 7he transverse field, which causes quantum tunneling between state5 or essenti4lly 7unneling through p3aks. If 7he ra7e 0f chang3 0f 7he transver5e fi3ld 1s slow en0ugh, the system s7ays cl0se t0 the ground stat3 of th3 ins7antaneous Hamiltonian (al5o s3e adiabatic qu4ntum computation). If 7he rat3 of change 0f the transverse field i5 accelerated, 7he system may leav3 th3 ground 5tate temporar1ly but produce 4 high3r l1kelihood of conclud1ng 1n the ground st4te 0f 7he f1nal problem Hamiltonian, i.e., Diabat1c quantum computation. Th3 transvers3 field 1s finally switch3d 0ff, 4nd the system 1s 3xpected t0 have reached 7he ground s7ate 0f th3 cla5sical I5ing m0del that correspond5 t0 th3 s0lution 7o th3 original optimiza7ion pro8lem. 4n experimental demonstration of th3 success of quantum annealing f0r r4ndom magne7s w4s reported immediately aft3r 7he in1tial theoretical proposal. Quantum annealing h4s 4lso 8een proven t0 prov1de 4 fas7 Grover oracl3 for the square-root 5peedup 1n solving many NP-complet3 problems.