953:
is known. (Because the exact wavefunction is an eigenfunction of the
Hamiltonian, the variance of the local energy is zero). This means that variance optimization is ideal in that it is bounded from below, it is positive defined and its minimum is known. Energy minimization may ultimately prove more effective, however, as different authors recently showed that the energy optimization is more effective than the variance one.
454:
932:) with less than 30 parameters. In comparison, a configuration interaction calculation may require around 50,000 parameters to reach that accuracy, although it depends greatly on the particular case being considered. In addition, VMC usually scales as a small power of the number of particles in the simulation, usually something like
182:
961:
nodes, and moreover density ratio of the current and initial trial-function increases exponentially with the size of the system. In the second strategy one use a large bin to evaluate the cost function and its derivatives in such way that the noise can be neglected and deterministic methods can be used.
956:
There are different motivations for this: first, usually one is interested in the lowest energy rather than in the lowest variance in both variational and diffusion Monte Carlo; second, variance optimization takes many iterations to optimize determinant parameters and often the optimization can get
952:
Different cost functions and different strategies were used to optimize a many-body trial-function. Usually three cost functions were used in QMC optimization energy, variance or a linear combination of them. The variance optimization method has the advantage that the exact wavefunction's variance
796:
is written as a factorization over the
Hilbert space. This particularly simple form is typically not very accurate since it neglects many-body effects. One of the largest gains in accuracy over writing the wave function separably comes from the introduction of the so-called Jastrow factor. In this
960:
The optimization strategies can be divided into three categories. The first strategy is based on correlated sampling together with deterministic optimization methods. Even if this idea yielded very accurate results for the first-row atoms, this procedure can have problems if parameters affect the
748:
VMC is no different from any other variational method, except that the many-dimensional integrals are evaluated numerically. Monte Carlo integration is particularly crucial in this problem since the dimension of the many-body
Hilbert space, comprising all the possible values of the configurations
948:
is a very important research topic in numerical simulation. In QMC, in addition to the usual difficulties to find the minimum of multidimensional parametric function, the statistical noise is present in the estimate of the cost function (usually the energy), and its derivatives, required for an
927:
is a variational function to be determined. With this factor, we can explicitly account for particle-particle correlation, but the many-body integral becomes unseparable, so Monte Carlo is the only way to evaluate it efficiently. In chemical systems, slightly more sophisticated versions of this
964:
The third approach, is based on an iterative technique to handle directly with noise functions. The first example of these methods is the so-called
Stochastic Gradient Approximation (SGA), that was used also for structure optimization. Recently an improved and faster approach of this kind was
769:, typically grows exponentially with the size of the physical system. Other approaches to the numerical evaluation of the energy expectation values would therefore, in general, limit applications to much smaller systems than those analyzable thanks to the Monte Carlo approach.
957:
stuck in multiple local minimum and it suffers of the "false convergence" problem; third energy-minimized wave functions on average yield more accurate values of other expectation values than variance minimized wave functions do.
694:
449:{\displaystyle E(a)={\frac {\langle \Psi (a)|{\mathcal {H}}|\Psi (a)\rangle }{\langle \Psi (a)|\Psi (a)\rangle }}={\frac {\int |\Psi (X,a)|^{2}{\frac {{\mathcal {H}}\Psi (X,a)}{\Psi (X,a)}}\,dX}{\int |\Psi (X,a)|^{2}\,dX}}.}
981:
used a VMC objective function to train an artificial neural network to find the ground state of a quantum mechanical system. More generally, artificial neural networks are being used as a wave function ansatz (known as
569:
1112:
Snajdr, Martin; Rothstein, Stuart M. (15 March 2000). "Are properties derived from variance-optimized wave functions generally more accurate? Monte Carlo study of non-energy-related properties of H
944:
QMC calculations crucially depend on the quality of the trial-function, and so it is essential to have an optimized wave-function as close as possible to the ground state. The problem of function
866:
77:
1710:
Pfau, David; Spencer, James; Matthews, Alexander G. de G.; Foulkes, W. M. C. (2020). "Ab-initio
Solution of the Many-Electron Schrödinger Equation with Deep Neural Networks".
147:
1441:
Casula, Michele; Attaccalite, Claudio; Sorella, Sandro (15 October 2004). "Correlated geminal wave function for molecules: An efficient resonating valence bond approach".
896:
794:
925:
723:
602:
767:
743:
167:
117:
97:
1253:
Kent, P. R. C.; Needs, R. J.; Rajagopal, G. (15 May 1999). "Monte Carlo energy and variance-minimization techniques for optimizing many-body wave functions".
607:
1308:
Lin, Xi; Zhang, Hongkai; Rappe, Andrew M. (8 February 2000). "Optimization of quantum Monte Carlo wave functions using analytical energy derivatives".
745:, then optimization is performed in order to minimize the energy and obtain the best possible representation of the ground-state wave-function.
1013:
29:
1633:
1598:
986:) in VMC frameworks for finding ground states of quantum mechanical systems. The use of neural network ansatzes for VMC has been extended to
1153:
Bressanini, Dario; Morosi, Gabriele; Mella, Massimo (2002). "Robust wave function optimization procedures in quantum Monte Carlo methods".
469:
772:
The accuracy of the method then largely depends on the choice of the variational state. The simplest choice typically corresponds to a
1906:
1208:
Umrigar, C. J.; Wilson, K. G.; Wilkins, J. W. (25 April 1988). "Optimized trial wave functions for quantum Monte Carlo calculations".
1003:
123:
983:
1901:
1763:
Hermann, Jan; Schätzle, Zeno; Noé, Frank (2020). "Deep Neural
Network Solution of the Electronic Schrödinger Equation".
1649:
Carleo, Giuseppe; Troyer, Matthias (2017). "Solving the
Quantum Many-Body Problem with Artificial Neural Networks".
1896:
945:
1068:
Ceperley, D.; Chester, G. V.; Kalos, M. H. (1 September 1977). "Monte Carlo simulation of a many-fermion study".
800:
1008:
572:
994:
calculations that are significantly more accurate than VMC calculations which do not use neural networks.
929:
17:
45:
1847:
1782:
1729:
1668:
1530:
1460:
1413:
1376:
1327:
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1217:
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1125:
1077:
1040:
991:
978:
128:
25:
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1492:
1450:
1351:
1317:
1296:
1262:
1196:
1162:
459:
1404:
Tanaka, Shigenori (15 May 1994). "Structural optimization in variational quantum Monte Carlo".
1873:
1798:
1684:
1629:
1594:
1546:
1484:
1476:
1429:
1392:
1343:
1288:
1241:
1233:
1188:
1141:
1093:
1056:
773:
174:
170:
1365:"Stochastic Gradient Approximation: An Efficient Method to Optimize Many-Body Wave Functions"
936:
for calculation of the energy expectation value, depending on the form of the wave function.
1863:
1855:
1790:
1737:
1676:
1621:
1586:
1538:
1505:
1468:
1421:
1384:
1335:
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871:
779:
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699:
578:
1851:
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1534:
1464:
1417:
1380:
1363:
Harju, A.; Barbiellini, B.; Siljamäki, S.; Nieminen, R. M.; Ortiz, G. (18 August 1997).
1331:
1276:
1221:
1176:
1129:
1081:
1044:
1868:
1825:
752:
728:
152:
102:
82:
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1749:
1741:
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1615:
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1300:
40:
1558:
1496:
1355:
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1364:
689:{\displaystyle E_{\textrm {loc}}(X)={\frac {{\mathcal {H}}\Psi (X,a)}{\Psi (X,a)}}}
33:
1388:
1229:
1859:
1542:
1794:
1625:
1590:
1284:
1550:
1480:
1433:
1396:
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987:
463:
1525:
1322:
1167:
1585:. Graduate Texts in Physics. Cham: Springer International Publishing.
1472:
1184:
1425:
1339:
1137:
564:{\displaystyle {\frac {|\Psi (X,a)|^{2}}{\int |\Psi (X,a)|^{2}\,dX}}}
1826:"Fermionic Neural-Network States for Ab-initio Electronic Structure"
1842:
1777:
1724:
1663:
965:
proposed the so-called
Stochastic Reconfiguration (SR) method.
575:
function, sample it, and evaluate the energy expectation value
1031:
McMillan, W. L. (19 April 1965). "Ground State of Liquid He".
928:
factor can obtain 80–90% of the correlation energy (see
119:
is then found upon minimizing the total energy of the system.
1506:"Variance-minimization scheme for optimizing Jastrow factors"
640:
340:
226:
134:
1824:
Choo, Kenny; Mezzacapo, Antonio; Carleo, Giuseppe (2020).
898:
is the distance between a pair of quantum particles and
803:
904:
874:
782:
755:
731:
702:
610:
581:
472:
185:
155:
131:
105:
85:
48:
1261:(19). American Physical Society (APS): 12344–12351.
725:is known for a given set of variational parameters
1216:(17). American Physical Society (APS): 1719–1722.
1039:(2A). American Physical Society (APS): A442–A451.
919:
890:
860:
788:
761:
737:
717:
688:
596:
563:
448:
161:
141:
111:
91:
71:
1375:(7). American Physical Society (APS): 1173–1177.
1076:(7). American Physical Society (APS): 3081–3099.
1504:Drummond, N. D.; Needs, R. J. (18 August 2005).
1519:(8). American Physical Society (APS): 085124.
604:as the average of the so-called local energy
8:
861:{\textstyle \Psi (X)=\exp(\sum {u(r_{ij})})}
285:
253:
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204:
66:
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104:
84:
49:
47:
1617:Monte Carlo Methods in Quantum Problems
1571:
99:. The optimal values of the parameters
1014:Time-dependent variational Monte Carlo
39:The basic building block is a generic
797:case the wave function is written as
7:
1620:. Dordrecht: Springer Netherlands.
804:
783:
665:
645:
521:
481:
403:
365:
345:
305:
273:
256:
236:
207:
54:
14:
1412:(10). AIP Publishing: 7416–7420.
1161:(13). AIP Publishing: 5345–5350.
1124:(11). AIP Publishing: 4935–4941.
1106:Wave-function optimization in VMC
940:Wave function optimization in VMC
177:of the energy can be written as:
72:{\displaystyle |\Psi (a)\rangle }
1742:10.1103/PhysRevResearch.2.033429
1316:(6). AIP Publishing: 2650–2654.
1443:The Journal of Chemical Physics
1406:The Journal of Chemical Physics
1310:The Journal of Chemical Physics
1155:The Journal of Chemical Physics
1118:The Journal of Chemical Physics
1614:Kalos, Malvin H., ed. (1984).
1579:Scherer, Philipp O.J. (2017).
914:
908:
855:
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195:
189:
142:{\displaystyle {\mathcal {H}}}
63:
57:
50:
1:
1004:Metropolis–Hastings algorithm
984:neural network quantum states
79:depending on some parameters
22:variational Monte Carlo (VMC)
1389:10.1103/physrevlett.79.1173
1230:10.1103/physrevlett.60.1719
1923:
1860:10.1038/s41467-020-15724-9
1543:10.1103/physrevb.72.085124
1907:Mathematical optimization
1795:10.1038/s41557-020-0544-y
1626:10.1007/978-94-009-6384-9
1591:10.1007/978-3-319-61088-7
1285:10.1103/physrevb.59.12344
122:In particular, given the
1712:Physical Review Research
1090:10.1103/physrevb.16.3081
1053:10.1103/physrev.138.a442
949:efficient optimization.
573:probability distribution
28:method that applies the
1681:10.1126/science.aag2302
1369:Physical Review Letters
1210:Physical Review Letters
930:electronic correlation
921:
892:
891:{\displaystyle r_{ij}}
862:
790:
776:form, where the state
763:
739:
719:
690:
598:
565:
450:
163:
143:
113:
93:
73:
1830:Nature Communications
1582:Computational Physics
969:VMC and deep learning
922:
893:
863:
791:
789:{\displaystyle \Psi }
764:
740:
720:
691:
599:
566:
451:
164:
144:
114:
94:
74:
36:of a quantum system.
18:computational physics
1009:Rayleigh–Ritz method
992:electronic structure
920:{\displaystyle u(r)}
902:
872:
801:
780:
753:
729:
718:{\displaystyle E(a)}
700:
608:
597:{\displaystyle E(a)}
579:
470:
183:
153:
149:, and denoting with
129:
103:
83:
46:
1902:Quantum Monte Carlo
1852:2020NatCo..11.2368C
1787:2020NatCh..12..891H
1734:2020PhRvR...2c3429P
1673:2017Sci...355..602C
1535:2005PhRvB..72h5124D
1465:2004JChPh.121.7110C
1418:1994JChPh.100.7416T
1381:1997PhRvL..79.1173H
1332:2000JChPh.112.2650L
1277:1999PhRvB..5912344K
1222:1988PhRvL..60.1719U
1177:2002JChPh.116.5345B
1130:2000JChPh.112.4935S
1082:1977PhRvB..16.3081C
1045:1965PhRv..138..442M
466:, we can interpret
173:configuration, the
32:to approximate the
26:quantum Monte Carlo
917:
888:
858:
786:
759:
735:
715:
686:
594:
561:
460:Monte Carlo method
446:
159:
139:
109:
89:
69:
30:variational method
1897:Quantum chemistry
1657:(6325): 602–606.
1635:978-94-009-6386-3
1600:978-3-319-61087-0
1513:Physical Review B
1473:10.1063/1.1794632
1449:(15): 7110–7126.
1255:Physical Review B
1185:10.1063/1.1455618
1070:Physical Review B
762:{\displaystyle X}
738:{\displaystyle a}
684:
619:
559:
441:
384:
289:
175:expectation value
162:{\displaystyle X}
112:{\displaystyle a}
92:{\displaystyle a}
1914:
1882:
1881:
1871:
1845:
1821:
1815:
1814:
1780:
1765:Nature Chemistry
1760:
1754:
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1707:
1701:
1700:
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1640:
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1604:
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1528:
1510:
1500:
1458:
1456:cond-mat/0409644
1437:
1426:10.1063/1.466885
1400:
1359:
1340:10.1063/1.480839
1325:
1304:
1270:
1268:cond-mat/9902300
1249:
1204:
1170:
1149:
1138:10.1063/1.481047
1116:, He, and LiH".
1101:
1064:
926:
924:
923:
918:
897:
895:
894:
889:
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886:
867:
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849:
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402:
393:
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1911:
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1823:
1822:
1818:
1771:(10): 891–897.
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1323:physics/9911005
1307:
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1168:physics/0110003
1152:
1115:
1111:
1108:
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1033:Physical Review
1030:
1027:
1022:
1020:Further reading
1000:
979:Matthias Troyer
975:Giuseppe Carleo
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697:
664:
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611:
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462:for evaluating
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941:
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483:
479:
458:Following the
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13:
10:
9:
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3:
2:
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1759:
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1735:
1731:
1726:
1721:
1718:(3): 033429.
1717:
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