Quantum annealing inversion and its implementation

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1 CHINESE JOURNAL OF GEOPHYSICS Vol. 49, No. 2 Mar., 2006,,..,2006,49 (2) : We C, Zhu P M, Wang J Y. Quantum annealng nverson and ts mplementaton. Chnese J. Geophys. (n Chnese), 2006,49 (2) : ,2, 1, 1 1 ( ), , ,.,.,,,,,,.,, (2006) P , Quantum annealng nverson and ts mplementaton WEI Chao 1,2, ZHU Pe-Mn 1, WANGJa- Yng 1 1 Insttute of Geophyscs and Geomatcs, Chna Unversty of Geoscences, Wuhan, , Chna 2 Insttute of Geology and Geophyscs, Chnese Academy of Scences, Bejng , Chna Abstract Quantum annealng s a new method to solve nonlnear nverse problems based on the quantum tunnelng effect of state transtons. Compared wth smulated Annealng, the method of quantum annealng nverson has a potental advantage n some aspects such as the annealng convergent rate and avodng to be trapped n a local mnmum. So the study of quantum annealng nverson s of mportant academc sgnfcance and practcal value. In ths paper, some model tests and a practcal data nverson show that quantum annealng has hgher effcency and better accuracy. It s the frst tme to apply the quantum annealng method to contnuous geophyscal nverse problems. Keywords Quantum annealng, Smulated annealng, Geophyscal nverson 1 ( ), [1 ].,,..,, Newton Newton.,. [2 ] (Monte Carlo method), ( , , ).,,1981,,. E-mal :cug -

2 578 (Chnese J. Geophys. ) 49,,. (Smulated Annealng,SA) ( Genetc Algorthm, GA ) ( Artfcal Neural Networks,ANN) [3 ].,,,, [4 ] (Quantum Annealng,QA). Brooke et al. [5 ] LHo 0144 Y 0156 F 4,, 2 30,. [6 ],., Tsalls [4 ], ;Tadash [7 ], ( Isng), ; Huntsman [8 ] ;Santoro et al. [9 ], [10 ] ; Trugenberger [11 ] ;Martonak et al. [6 ],.,.,,.,,.,.,. 2,., D.,.,., T, r (Boltzmann) r { E( r) } = 1 Z ( T) exp - E( r) kt, (1), E( r) r, k > 0, [12 ]. Z ( T), ( Partton Functon ), ( State Summaton), [13 ] : Z ( T) = exp - s D E( s) kt. (2) (1),,, ;,,., T ().,,., [1 ]. T,, ;,,,.,.,,. Hamlton H 0 ( Hamlton H ( t) = U ( r, t) + K, [14 ] ).,,.,,,, [15 ]. H ( t),, ( Transverse Feld) [5 11] ( t) x, Hamlton H( t) = H 0 + H ( t), H( t) = H 0 + ( t) x, (3)

3 2 : 579 ( t), T ; x x ( Paul Representaton). (3) H 0, ( t) x. U = H 0, K = ( t) x, (3) H( t) = U + K. (4) T ( k = 1) [10 ] Z = Tr (e - HΠT ) = Tr (e - HΠPT ) P = Tr (e - ( K+ U)ΠPT ) P,, P,Tr (5). = (e - H ΠPT ) P ΠZ. (6),, Hamlton. K( ),. m (0), m ( = 1,2,, P), d j ( j = 1,2,, M). l, M E( m ( l) ) = ( d j - j = 1 d j ( m ( l) ) ) 2. (7),m ( l + 1), E( m ( l + 1) ), E = E ( m ( l + 1) ) -,Hamlton E( m ( l ) H = E + C ( t), (8),, x C.,,(6)., Z, : ( H) = (e - HΠPT ) P, (9) (8),,,. [16, 17 ],, : (1) m (0), 0,( t) = 0 t (, t, ( t) 0). ( = 1,2,, P), ; (2). l, m ( l) = m ( l) 1, m ( l) 2,, m ( l) P E( m ( l) ) ;, (3) [ 0,1 ] ( = 1,2,, P),l + 1 m ( l + 1) ( m 1 ) l +1 = ( m 1 ) l + (2 1-1) 1 ( m 2 ) l +1 = ( m 2 ) l + (2 2-1) 2 ( m P ) l +1 = ( m P ) l + (2 P - 1) P ; (10) (4) m ( l + 1) E ( m ( l + 1) ) E = E( m ( l + 1) ) - E ( m ( l) ), (8) Hamlton. E < 0, l + 1 m ( l + 1) m ( l) ; E > 0, (9) m ( l + 1) m ( l) ; (5) m ( = 1,2,, P),., ( t). ( t),, m., 0, ( t) MT 1 H MT ( 1a ), 1b MT Bostck SA QA 1 MT Table 1 Comparson among varous nverse methods for MT model data 1 Π m 2 Π m 3 Π m h 1 Πm h 2 Πm Bostck SA QA

4 580 (Chnese J. Geophys. ) 49,0 10 % 20 %. 2 2.,,,, ;,,,. 4 1 MT (a) MT ; (b). Fg. 1 Comparson of results of modelng MT sound curves usng three methods (a) Modelng MT sound curves ; (b) Inverse results. ( 1 ), Bostck., ( 1, 2, 3, h 1, h 2 ) ( m, 5010 m, m,35010m,50010m)., 14501, ,. [18, 19 ].,, ( - ) -, - -, 215ms ;,, ;,. 3. 3,., , 1 56, ;12mn ,, % 20 %,. 3a, 3b 3a, 45Hz Rcker Table 2 Comparson between nverse results of QA and SA ( ( ) ( ) QA SA % 20 % QA SA QA SA >

5 2 : (A) ; (B) ; (C). (a 1, a 2, a 3 ) ; (b 1, b 2, b 3 ) 10 % ; (c 1, c 2, c 3 ) 20 %. Fg. 2 Inverson results of dfferent mpedance models at dfferent nose levels (A) Model ; (B) Model ; (C) Model. (a 1, a 2, a 3 ) No nose ; (b 1, b 2, b 3 ) 10 %nose ; (c 1, c 2, c 3 ) 20 %nose. 3 Table 3 Comparson of nverse results of real acoustc mpedance between QA and SA 0 10 % 20 % QA SA > > > > QA SA > > > > QA SA > > > > a 5a, ( ),.

6 582 (Chnese J. Geophys. ) 49 4 (a) ; (b) ; (c) 10 % ; (d) 20 %. Fg. 4 Inverson results of real acoustc mpedance at dfferent nose levels for mode (a) Intal wave mpedance model ; (b) Inverse result wthout nose ; (c) Inverse result wth 10 % nose ; (d) Inverse result wth 20 % nose. 5 (a) ; (b) ; (c) 10 % ; (d) 20 %. Fg. 5 Inverson results of real acoustc mpedance at dfferent nose levels for mode (a) Intal wave mpedance model ; (b) Inverse result wthout nose ; (c) Inverse result wth 10 % nose ; (d) Inverse result wth 20 % nose.

7 2 : (a) ; (b). Fg. 3 Loggng data n Xnjang (a) Acoustc wave mpedance curve ; (b) Correspondng synthetc sesmogram. 5 (1),,,,.,,,. (2),,. (3),,., ( ) 2002 MT, Bostck ;,. (References) [ 1 ].. :,2000 Wang J Y. Inverse Theory n Geophyscs ( n Chnese). Bejng : Hgher Educaton Press, 2000 [ 2 ] Hammersley J M, Handscomb D C. Monte Carlo Methods. London : Chapman & Hall, 1964 [ 3 ]. [ ]. :,2004 Zhang L Q. Homotopy method and the applcaton to nverson of acoustc mpedance [ Ph. D. thess] (n Chnese). Wuhan : Insttute of Geophyscs and Geomatcs, Chna Unversty of Geoscences, 2004 [ 4 ] Tsalls C, Starolo D A. Generalzed smulated annealng. Physca A, 1996,233 : [ 5 ] Brooke J, Btko D, Rosenbaum T F, et al. Quantum annealng of a dsordered magnet. Scence, 1999, 284 : [ 6 ] Martonak R, Santoro G E, Tosatt E. Quantum annealng of the travelng salesman problem. http :ΠΠxxx. arxv. cornell. eduπabsπcondmatπ , 2004 [ 7 ] Tadash K, Hdetosh N. Study of optmzaton problems by quantum annealng[ Ph. D. thess ]. Tokyo : Department of Physcs, Tokyo Insttute of Technology, 1998 [ 8 ] Huntsman S. Quantum smulated annealng. http :ΠΠarxv. orgπabsπ quant-phπ , 2001 [ 9 ] Santoro G E, Martonak R, Tosatt E, et al. Theory of quantum annealng of an sng spn glass. Scence, 2002,295 : [10 ] Martonak R, Santoro G E, Tosatt E, et al. Supplemental materal to Theory of quantum annealng of an sng spn glass. Scence, 2002, 295 [11 ] Trugenberger C A. Quantum optmzaton for combnatoral searches. New Journal of Physcs, 2002, 4 : [12 ],.. :, 1999 Xng W X, Xe J X. Modern Optmzaton Calculaton Method (n Chnese). Bejng : Tsnghua Unversty Press, 1999 [13 ].. :,1984 L Z D. Statstcal Mechancs (n Chnese). Bejng : Bejng Normal Press, 1984 [14 ],.. :,1997 Lu W J, Pan Z C. Concse Quantum Mechancs ( n Chnese ). Tanjn : Tanjn Scence and Technology Translaton and Publshng Corp., 1997 [15 ].. :,1998 Zeng J Y. The Introducton of Quantum Mechancs ( n Chnese). Bejng : Pekng Unversty Press, 1998 [16 ] Metropolse N, Rosenbluth A, Rosenbluth M, et al. Equaton of state calculatons by fast computng machnes. Chern. Phys., 1953, 21 : [17 ].. :,2003 Xu G M. Inverse Theory and Its Applcaton (n Chnese). Bejng : Sesmologcal Press, 2003 [18 ],,.., 1985, 28 (1) :74 83 Huang G Y, Xu W T, Fu G H, et al. An algorthm for processng sesmc exploraton data. Chnese J. Geophys. (n Chnese), 1985, 28(1) : [19 ],,.., 2004, 47 (6) : Zhang L Q, Wang J Y, Yan T D. A homotopy method for the mpedance nverson of 1-D wave equaton. Chnese J. Geophys. (n Chnese), 2004, 47 (6) : ( )

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