442:
571:
654:
298:
147:
770:
478:
926:
G. Carleo; F. Becca; L. Sanchez-Palencia; S. Sorella & M. Fabrizio (2014). "Light-cone effect and supersonic correlations in one- and two-dimensional bosonic superfluids".
469:
282:
242:
186:
847:
580:
818:
794:
206:
437:{\displaystyle i\sum _{k^{\prime }}\langle O_{k}O_{k^{\prime }}\rangle _{t}^{c}{\dot {a}}_{k^{\prime }}=\langle O_{k}{\mathcal {H}}\rangle _{t}^{c},}
50:
670:
820:, the quantities entering the equation of motion are evaluated as statistical averages over the sampled configurations. The trajectories
797:
472:
1008:
292:. In particular one can show that the optimal parameters for the evolution satisfy at each time the equation of motion
1003:
773:
660:
566:{\displaystyle \langle AB\rangle _{t}^{c}=\langle AB\rangle _{t}-\langle A\rangle _{t}\langle B\rangle _{t}}
33:
573:
are connected averages, and the quantum expectation values are taken over the time-dependent variational
850:
285:
945:
886:
450:
21:
961:
935:
876:
664:
37:
912:
251:
211:
155:
29:
982:
953:
902:
894:
649:{\displaystyle \langle \cdots \rangle _{t}\equiv \langle \Psi (t)|\cdots |\Psi (t)\rangle }
849:
of the variational parameters are then found upon numerical integration of the associated
823:
949:
890:
907:
864:
803:
779:
191:
25:
997:
574:
289:
41:
965:
800:
is then used to sample exactly from this probability distribution and, at each time
776:
function over the multi-dimensional space spanned by the many-body configurations
957:
986:
976:
916:
898:
142:{\displaystyle \Psi (X,t)=\exp \left(\sum _{k}a_{k}(t)O_{k}(X)\right)}
765:{\displaystyle {\frac {|\Psi (X,t)|^{2}}{\int |\Psi (X,t)|^{2}\,dX}}}
245:
940:
881:
978:
Spectral and dynamical properties of strongly correlated systems
865:"Localization and glassy dynamics of many-body quantum systems"
456:
411:
386:
345:
315:
24:
approach to study the dynamics of closed, non-relativistic
863:
G. Carleo; F. Becca; M. SchirĂł & M. Fabrizio (2012).
244:
are time-independent operators that define the specific
826:
806:
782:
673:
583:
481:
453:
301:
254:
214:
194:
158:
53:
841:
812:
788:
764:
648:
565:
463:
436:
276:
236:
200:
180:
141:
18:time-dependent variational Monte Carlo (t-VMC)
8:
643:
603:
591:
584:
554:
547:
538:
531:
519:
509:
492:
482:
417:
396:
353:
322:
667:for evaluating integrals, we can interpret
188:are time-dependent variational parameters,
939:
906:
880:
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752:
746:
741:
717:
706:
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100:
90:
52:
248:. The time evolution of the parameters
208:denotes a many-body configuration and
7:
722:
682:
631:
606:
54:
36:method, in which a time-dependent
14:
981:(PhD Thesis). pp. 107–128.
40:is encoded by some variational
836:
830:
742:
737:
725:
718:
702:
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678:
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464:{\displaystyle {\mathcal {H}}}
271:
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175:
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131:
125:
112:
106:
69:
57:
44:, generally parametrized as
28:in the context of the quantum
1:
798:Metropolis–Hastings algorithm
284:can be found upon imposing a
32:. It is an extension of the
663:approach and following the
1025:
958:10.1103/PhysRevA.89.031602
152:where the complex-valued
774:probability distribution
277:{\displaystyle a_{k}(t)}
237:{\displaystyle O_{k}(X)}
181:{\displaystyle a_{k}(t)}
661:Variational Monte Carlo
34:variational Monte Carlo
843:
814:
790:
766:
650:
567:
465:
438:
278:
238:
202:
182:
143:
851:differential equation
844:
815:
791:
767:
651:
568:
466:
439:
286:variational principle
279:
239:
203:
183:
144:
842:{\displaystyle a(t)}
824:
804:
780:
671:
659:In analogy with the
581:
479:
451:
299:
252:
212:
192:
156:
51:
1009:Quantum Monte Carlo
950:2014PhRvA..89c1602C
891:2012NatSR...2E.243C
505:
430:
366:
22:quantum Monte Carlo
975:G. Carleo (2011).
839:
810:
786:
762:
665:Monte Carlo method
646:
563:
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95:
38:pure quantum state
1004:Quantum mechanics
987:20.500.11767/4289
899:10.1038/srep00243
813:{\displaystyle t}
789:{\displaystyle X}
760:
377:
305:
201:{\displaystyle X}
86:
30:many-body problem
1016:
990:
969:
943:
934:(3): 031602(R).
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475:of the system,
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26:quantum systems
12:
11:
5:
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13:
10:
9:
6:
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988:
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979:
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707:
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623:
612:
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575:wave function
558:
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295:
294:
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290:wave function
287:
268:
260:
256:
247:
228:
220:
216:
195:
172:
164:
160:
135:
128:
120:
116:
109:
101:
97:
91:
87:
82:
78:
75:
72:
66:
63:
60:
47:
46:
45:
43:
42:wave function
39:
35:
31:
27:
23:
19:
977:
931:
928:Phys. Rev. A
927:
872:
868:
658:
446:
151:
20:method is a
17:
15:
473:Hamiltonian
998:Categories
857:References
941:1310.2246
882:1109.2516
723:Ψ
715:∫
683:Ψ
644:⟩
632:Ψ
624:⋯
607:Ψ
604:⟨
601:≡
592:⟩
588:⋯
585:⟨
555:⟩
548:⟨
539:⟩
532:⟨
529:−
520:⟩
510:⟨
493:⟩
483:⟨
418:⟩
397:⟨
387:′
375:˙
354:⟩
346:′
323:⟨
316:′
307:∑
88:∑
79:
55:Ψ
966:45660254
917:22355756
869:Sci. Rep
577:, i.e.,
946:Bibcode
908:3272662
887:Bibcode
875:: 243.
471:is the
288:to the
964:
915:
905:
796:. The
447:where
246:ansatz
962:S2CID
936:arXiv
877:arXiv
772:as a
913:PMID
16:The
983:hdl
954:doi
903:PMC
895:doi
76:exp
1000::
960:.
952:.
944:.
932:89
930:.
911:.
901:.
893:.
885:.
871:.
867:.
853:.
656:.
989:.
985::
968:.
956::
948::
938::
919:.
897::
889::
879::
873:2
837:)
834:t
831:(
828:a
808:t
784:X
757:X
754:d
748:2
743:|
738:)
735:t
732:,
729:X
726:(
719:|
708:2
703:|
698:)
695:t
692:,
689:X
686:(
679:|
641:)
638:t
635:(
628:|
620:|
616:)
613:t
610:(
596:t
559:t
551:B
543:t
535:A
524:t
516:B
513:A
507:=
502:c
497:t
489:B
486:A
457:H
432:,
427:c
422:t
412:H
405:k
401:O
394:=
383:k
372:a
363:c
358:t
342:k
337:O
331:k
327:O
312:k
303:i
272:)
269:t
266:(
261:k
257:a
232:)
229:X
226:(
221:k
217:O
196:X
176:)
173:t
170:(
165:k
161:a
136:)
132:)
129:X
126:(
121:k
117:O
113:)
110:t
107:(
102:k
98:a
92:k
83:(
73:=
70:)
67:t
64:,
61:X
58:(
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