importance sampling and variational Monte Carlo for Rimu.jl
Gutzwiller.jl is not yet registered. To install it, run
import Pkg; Pkg.add("https://github.com/RimuQMC/Gutzwiller.jl")using Rimu
using Gutzwiller
using CairoMakie
using LaTeXStringsFirst, we set up a starting address and a Hamiltonian
addr = near_uniform(BoseFS{10,10})
H = HubbardReal1D(addr; u=2.0)HubbardReal1D(fs"|1 1 1 1 1 1 1 1 1 1⟩"; u=2.0, t=1.0)
In this example, we'll set up a Gutzwiller ansatz to importance-sample the Hamiltonian.
ansatz = GutzwillerAnsatz(H)Gutzwiller.GutzwillerAnsatz{Rimu.BitStringAddresses.BoseFS{10, 10, Rimu.BitStringAddresses.BitString{19, 1, UInt32}}, Float64, Rimu.Hamiltonians.HubbardReal1D{Float64, Rimu.BitStringAddresses.BoseFS{10, 10, Rimu.BitStringAddresses.BitString{19, 1, UInt32}}, 2.0, 1.0}}(HubbardReal1D(fs"|1 1 1 1 1 1 1 1 1 1⟩"; u=2.0, t=1.0))
An ansatz is a struct that given a set of parameters and an address, produces the value it would have if it was a vector.
ansatz(addr, [1.0])1.0
In addition, the function val_and_grad can be used to compute both the value and its
gradient with respect to the parameters.
val_and_grad(ansatz, addr, [1.0])(1.0, [-0.0])
For effective importance sampling, we want the ansatz to be as good of an approximation to the ground state of the Hamiltonian as possible. As the value of the Rayleigh quotient of a given ansatz is always larger than the Hamiltonian's ground state energy, we can use an optimization algorithm to find the paramters that minimize its energy.
When the basis of the Hamiltonian is small enough to fit into memory, it's best to use the
LocalEnergyEvaluator
le = LocalEnergyEvaluator(H, ansatz)LocalEnergyEvaluator(HubbardReal1D(fs"|1 1 1 1 1 1 1 1 1 1⟩"; u=2.0, t=1.0), Gutzwiller.GutzwillerAnsatz{Rimu.BitStringAddresses.BoseFS{10, 10, Rimu.BitStringAddresses.BitString{19, 1, UInt32}}, Float64, Rimu.Hamiltonians.HubbardReal1D{Float64, Rimu.BitStringAddresses.BoseFS{10, 10, Rimu.BitStringAddresses.BitString{19, 1, UInt32}}, 2.0, 1.0}}(HubbardReal1D(fs"|1 1 1 1 1 1 1 1 1 1⟩"; u=2.0, t=1.0)))
which can be used to evaulate the value of the Rayleigh quotient (or its gradient) for given parameters. In the case of the Gutzwiller ansatz, there is only one parameter.
le([1.0])-7.825819465047312
Like before, we can use val_and_grad to also evaluate its gradient.
val_and_grad(le, [1.0])(-7.825819465047312, [10.614147776143358])
Now, let's plot the energy landscape for this particular case
begin
fig = Figure()
ax = Axis(fig[1, 1]; xlabel=L"p", ylabel=L"E")
ps = range(0, 2; length=100)
Es = [le([p]) for p in ps]
lines!(ax, ps, Es)
fig
end
To find the minimum, pass le to optimize from Optim.jl
using Optim
opt_nelder = optimize(le, [1.0]) * Status: success
* Candidate solution
Final objective value: -1.300521e+01
* Found with
Algorithm: Nelder-Mead
* Convergence measures
√(Σ(yᵢ-ȳ)²)/n ≤ 1.0e-08
* Work counters
Seconds run: 1 (vs limit Inf)
Iterations: 11
f(x) calls: 25
To take advantage of the gradients, wrap the evaluator in Optim.only_fg!. This will
usually reduce the number of steps needed to reach the minimum.
opt_lbgfs = optimize(Optim.only_fg!(le), [1.0]) * Status: success
* Candidate solution
Final objective value: -1.300521e+01
* Found with
Algorithm: L-BFGS
* Convergence measures
|x - x'| = 4.19e-05 ≰ 0.0e+00
|x - x'|/|x'| = 1.26e-04 ≰ 0.0e+00
|f(x) - f(x')| = 5.04e-08 ≰ 0.0e+00
|f(x) - f(x')|/|f(x')| = 3.87e-09 ≰ 0.0e+00
|g(x)| = 4.80e-09 ≤ 1.0e-08
* Work counters
Seconds run: 1 (vs limit Inf)
Iterations: 5
f(x) calls: 16
∇f(x) calls: 16
We can inspect the parameters and the value at the minimum as
opt_lbgfs.minimizer, opt_lbgfs.minimum([0.3331889106855026], -13.00520818638074)
When the Hamiltonian is too large to store its full basis in memory, we can use
variational QMC to sample addresses from the Hilbert space and evaluate their energy
at the same time. An important paramter we have tune is the number steps. More steps
will give us a better approximation of the energy, but take longer to evaluate.
Not taking enough samples can also result in producing a biased result.
Consider the following.
p0 = [1.0]
@time kinetic_vqmc(H, ansatz, p0; steps=1e2)KineticVQMCResult
walkers: 1
samples: 100
local energy: -7.3254 ± 0.33167
@time kinetic_vqmc(H, ansatz, p0; steps=1e5)KineticVQMCResult
walkers: 1
samples: 100000
local energy: -7.8369 ± 0.016555
@time kinetic_vqmc(H, ansatz, p0; steps=1e7)KineticVQMCResult
walkers: 1
samples: 10000000
local energy: -7.825 ± 0.0016978
For this simple example, 1e2 steps gives an energy that is significantly higher, while
1e7 takes too long. 1e5 seems to work well enough. For more convenient evaluation, we
wrap VQMC into a struct that behaves much like the LocalEnergyEvaluator.
qmc = KineticVQMC(H, ansatz; samples=1e4)KineticVQMC(
HubbardReal1D(fs"|1 1 1 1 1 1 1 1 1 1⟩"; u=2.0, t=1.0),
Gutzwiller.GutzwillerAnsatz{Rimu.BitStringAddresses.BoseFS{10, 10, Rimu.BitStringAddresses.BitString{19, 1, UInt32}}, Float64, Rimu.Hamiltonians.HubbardReal1D{Float64, Rimu.BitStringAddresses.BoseFS{10, 10, Rimu.BitStringAddresses.BitString{19, 1, UInt32}}, 2.0, 1.0}}(HubbardReal1D(fs"|1 1 1 1 1 1 1 1 1 1⟩"; u=2.0, t=1.0));
steps=10000,
walkers=1,
)
qmc([1.0]), le([1.0])(-7.828851494141587, -7.825819465047312)
Because the output of this procedure is noisy, optimizing it with Optim.jl will not work. However, we can use a stochastic gradient descent (in this case AMSGrad).
grad_result = amsgrad(qmc, [1.0])GradientDescentResult
iterations: 101
converged: false (iterations)
last value: -13.054226066948441
last params: [0.33407952896097876]
While amsgrad attempts to determine if the optimization converged, it will generally not
detect convergence due to the noise in the QMC evaluation. The best way to determine
convergence is to plot the results. grad_result can be converted to a DataFrame.
grad_df = DataFrame(grad_result)101×10 DataFrame
Row │ α β1 β2 iter param value gradient first_moment second_moment param_delta
│ Float64 Float64 Float64 Int64 SArray… Float64 SArray… SArray… SArray… SArray…
─────┼─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
1 │ 0.01 0.1 0.01 1 [1.0] -7.86947 [10.6335] [10.6726] [1.13072] [-0.100367]
2 │ 0.01 0.1 0.01 2 [0.899633] -8.86381 [10.4539] [10.6507] [2.21224] [-0.0716081]
3 │ 0.01 0.1 0.01 3 [0.828025] -9.61802 [9.98253] [10.5839] [3.18663] [-0.0592897]
4 │ 0.01 0.1 0.01 4 [0.768735] -10.2245 [10.1458] [10.5401] [4.18414] [-0.0515277]
5 │ 0.01 0.1 0.01 5 [0.717207] -10.5959 [8.97968] [10.384] [4.94865] [-0.0466792]
6 │ 0.01 0.1 0.01 6 [0.670528] -11.1979 [9.08252] [10.2539] [5.72408] [-0.0428584]
7 │ 0.01 0.1 0.01 7 [0.62767] -11.5211 [8.55523] [10.084] [6.39876] [-0.0398645]
8 │ 0.01 0.1 0.01 8 [0.587805] -11.8079 [7.71817] [9.84744] [6.93047] [-0.037406]
9 │ 0.01 0.1 0.01 9 [0.550399] -11.9961 [6.90477] [9.55317] [7.33793] [-0.0352664]
10 │ 0.01 0.1 0.01 10 [0.515133] -12.3262 [6.60435] [9.25829] [7.70072] [-0.033363]
11 │ 0.01 0.1 0.01 11 [0.48177] -12.5107 [5.36565] [8.86902] [7.91162] [-0.0315314]
12 │ 0.01 0.1 0.01 12 [0.450238] -12.7398 [5.00355] [8.48248] [8.08286] [-0.029836]
13 │ 0.01 0.1 0.01 13 [0.420402] -12.8374 [3.90748] [8.02498] [8.15471] [-0.0281021]
14 │ 0.01 0.1 0.01 14 [0.3923] -12.9241 [2.93999] [7.51648] [8.1596] [-0.0263136]
15 │ 0.01 0.1 0.01 15 [0.365987] -13.0258 [1.86548] [6.95138] [8.1596] [-0.0243353]
16 │ 0.01 0.1 0.01 16 [0.341651] -13.1445 [0.578173] [6.31406] [8.1596] [-0.0221042]
17 │ 0.01 0.1 0.01 17 [0.319547] -13.0476 [-0.471392] [5.63551] [8.1596] [-0.0197287]
18 │ 0.01 0.1 0.01 18 [0.299819] -13.005 [-2.23219] [4.84874] [8.1596] [-0.0169744]
19 │ 0.01 0.1 0.01 19 [0.282844] -12.9864 [-2.92041] [4.07183] [8.16329] [-0.0142514]
20 │ 0.01 0.1 0.01 20 [0.268593] -12.9353 [-4.70996] [3.19365] [8.3035] [-0.011083]
21 │ 0.01 0.1 0.01 21 [0.25751] -12.8462 [-5.47252] [2.32703] [8.51995] [-0.0079723]
22 │ 0.01 0.1 0.01 22 [0.249537] -12.6467 [-6.60268] [1.43406] [8.8707] [-0.00481491]
23 │ 0.01 0.1 0.01 23 [0.244723] -12.7247 [-6.33227] [0.657428] [9.18297] [-0.00216948]
24 │ 0.01 0.1 0.01 24 [0.242553] -12.8298 [-6.70496] [-0.0788115] [9.54071] [0.000255152]
25 │ 0.01 0.1 0.01 25 [0.242808] -12.6414 [-6.96349] [-0.767279] [9.9302] [0.00243486]
26 │ 0.01 0.1 0.01 26 [0.245243] -12.6024 [-7.14236] [-1.40479] [10.341] [0.00436846]
27 │ 0.01 0.1 0.01 27 [0.249612] -12.6806 [-6.39765] [-1.90407] [10.6469] [0.00583541]
28 │ 0.01 0.1 0.01 28 [0.255447] -12.8766 [-5.42621] [-2.25629] [10.8349] [0.0068546]
29 │ 0.01 0.1 0.01 29 [0.262302] -12.8672 [-4.98699] [-2.52936] [10.9752] [0.0076349]
30 │ 0.01 0.1 0.01 30 [0.269936] -12.69 [-5.36535] [-2.81296] [11.1534] [0.00842287]
31 │ 0.01 0.1 0.01 31 [0.278359] -12.8472 [-3.85241] [-2.9169] [11.1902] [0.00871971]
32 │ 0.01 0.1 0.01 32 [0.287079] -13.003 [-2.67633] [-2.89284] [11.1902] [0.0086478]
33 │ 0.01 0.1 0.01 33 [0.295727] -12.897 [-2.41077] [-2.84464] [11.1902] [0.00850369]
34 │ 0.01 0.1 0.01 34 [0.304231] -12.9895 [-1.61242] [-2.72141] [11.1902] [0.00813533]
35 │ 0.01 0.1 0.01 35 [0.312366] -13.047 [-0.86709] [-2.53598] [11.1902] [0.007581]
36 │ 0.01 0.1 0.01 36 [0.319947] -13.0287 [-0.666515] [-2.34904] [11.1902] [0.00702215]
37 │ 0.01 0.1 0.01 37 [0.326969] -13.0253 [-0.155275] [-2.12966] [11.1902] [0.00636635]
38 │ 0.01 0.1 0.01 38 [0.333335] -13.0209 [0.128243] [-1.90387] [11.1902] [0.00569138]
39 │ 0.01 0.1 0.01 39 [0.339027] -12.9271 [0.0294629] [-1.71054] [11.1902] [0.00511343]
40 │ 0.01 0.1 0.01 40 [0.34414] -12.9156 [0.367215] [-1.50276] [11.1902] [0.00449232]
41 │ 0.01 0.1 0.01 41 [0.348633] -12.986 [0.659557] [-1.28653] [11.1902] [0.00384592]
42 │ 0.01 0.1 0.01 42 [0.352478] -13.0485 [1.09632] [-1.04824] [11.1902] [0.0031336]
43 │ 0.01 0.1 0.01 43 [0.355612] -12.9853 [1.28838] [-0.814582] [11.1902] [0.00243509]
44 │ 0.01 0.1 0.01 44 [0.358047] -12.9752 [1.5468] [-0.578444] [11.1902] [0.00172918]
45 │ 0.01 0.1 0.01 45 [0.359776] -13.0076 [1.541] [-0.366499] [11.1902] [0.0010956]
46 │ 0.01 0.1 0.01 46 [0.360872] -12.9594 [1.35066] [-0.194784] [11.1902] [0.000582281]
47 │ 0.01 0.1 0.01 47 [0.361454] -12.956 [1.4433] [-0.0309752] [11.1902] [9.25964e-5]
48 │ 0.01 0.1 0.01 48 [0.361547] -12.956 [1.60799] [0.132921] [11.1902] [-0.00039735]
49 │ 0.01 0.1 0.01 49 [0.361149] -13.0046 [1.42665] [0.262294] [11.1902] [-0.000784094]
50 │ 0.01 0.1 0.01 50 [0.360365] -13.0956 [1.82696] [0.41876] [11.1902] [-0.00125183]
51 │ 0.01 0.1 0.01 51 [0.359114] -12.9629 [1.36866] [0.51375] [11.1902] [-0.00153579]
52 │ 0.01 0.1 0.01 52 [0.357578] -12.9446 [1.60631] [0.623007] [11.1902] [-0.0018624]
53 │ 0.01 0.1 0.01 53 [0.355715] -13.0111 [0.884743] [0.64918] [11.1902] [-0.00194064]
54 │ 0.01 0.1 0.01 54 [0.353775] -12.9621 [0.692179] [0.65348] [11.1902] [-0.0019535]
55 │ 0.01 0.1 0.01 55 [0.351821] -13.0818 [1.19488] [0.70762] [11.1902] [-0.00211534]
56 │ 0.01 0.1 0.01 56 [0.349706] -13.0126 [0.808823] [0.717741] [11.1902] [-0.0021456]
57 │ 0.01 0.1 0.01 57 [0.34756] -13.0746 [0.809506] [0.726917] [11.1902] [-0.00217303]
58 │ 0.01 0.1 0.01 58 [0.345387] -13.0823 [0.780243] [0.73225] [11.1902] [-0.00218897]
59 │ 0.01 0.1 0.01 59 [0.343198] -13.0186 [0.499421] [0.708967] [11.1902] [-0.00211937]
60 │ 0.01 0.1 0.01 60 [0.341079] -13.0456 [0.526001] [0.69067] [11.1902] [-0.00206467]
61 │ 0.01 0.1 0.01 61 [0.339014] -13.0493 [0.341787] [0.655782] [11.1902] [-0.00196038]
62 │ 0.01 0.1 0.01 62 [0.337054] -12.8872 [-0.478001] [0.542404] [11.1902] [-0.00162145]
63 │ 0.01 0.1 0.01 63 [0.335432] -12.9422 [0.243911] [0.512554] [11.1902] [-0.00153222]
64 │ 0.01 0.1 0.01 64 [0.3339] -12.9122 [-0.1211] [0.449189] [11.1902] [-0.00134279]
65 │ 0.01 0.1 0.01 65 [0.332557] -13.0425 [0.377574] [0.442027] [11.1902] [-0.00132139]
66 │ 0.01 0.1 0.01 66 [0.331236] -12.9999 [-0.0737422] [0.39045] [11.1902] [-0.0011672]
67 │ 0.01 0.1 0.01 67 [0.330069] -13.0888 [0.252431] [0.376648] [11.1902] [-0.00112594]
68 │ 0.01 0.1 0.01 68 [0.328943] -12.9758 [-0.122331] [0.32675] [11.1902] [-0.00097678]
69 │ 0.01 0.1 0.01 69 [0.327966] -13.0302 [-0.257161] [0.268359] [11.1902] [-0.000802227]
70 │ 0.01 0.1 0.01 70 [0.327164] -12.8849 [-0.604668] [0.181057] [11.1902] [-0.000541246]
71 │ 0.01 0.1 0.01 71 [0.326623] -12.948 [-0.474788] [0.115472] [11.1902] [-0.00034519]
72 │ 0.01 0.1 0.01 72 [0.326277] -12.9255 [-0.293566] [0.0745683] [11.1902] [-0.000222913]
73 │ 0.01 0.1 0.01 73 [0.326054] -12.9125 [-0.179361] [0.0491754] [11.1902] [-0.000147004]
74 │ 0.01 0.1 0.01 74 [0.325907] -13.0816 [-0.162379] [0.02802] [11.1902] [-8.37624e-5]
75 │ 0.01 0.1 0.01 75 [0.325824] -12.9071 [-0.817104] [-0.0564923] [11.1902] [0.000168877]
76 │ 0.01 0.1 0.01 76 [0.325993] -13.0681 [-0.239916] [-0.0748347] [11.1902] [0.000223709]
77 │ 0.01 0.1 0.01 77 [0.326216] -13.0101 [-0.502281] [-0.117579] [11.1902] [0.000351489]
78 │ 0.01 0.1 0.01 78 [0.326568] -13.1136 [0.0883187] [-0.0969895] [11.1902] [0.000289938]
79 │ 0.01 0.1 0.01 79 [0.326858] -13.0495 [0.0585228] [-0.0814383] [11.1902] [0.00024345]
80 │ 0.01 0.1 0.01 80 [0.327101] -13.0158 [-0.62517] [-0.135811] [11.1902] [0.000405991]
81 │ 0.01 0.1 0.01 81 [0.327507] -12.981 [0.289529] [-0.0932774] [11.1902] [0.000278841]
82 │ 0.01 0.1 0.01 82 [0.327786] -13.0029 [-0.115089] [-0.0954586] [11.1902] [0.000285361]
83 │ 0.01 0.1 0.01 83 [0.328071] -12.9877 [-0.181564] [-0.104069] [11.1902] [0.000311102]
84 │ 0.01 0.1 0.01 84 [0.328382] -12.9942 [-0.170822] [-0.110744] [11.1902] [0.000331056]
85 │ 0.01 0.1 0.01 85 [0.328714] -13.0454 [-0.264889] [-0.126159] [11.1902] [0.000377136]
86 │ 0.01 0.1 0.01 86 [0.329091] -13.0744 [0.0157432] [-0.111969] [11.1902] [0.000334716]
87 │ 0.01 0.1 0.01 87 [0.329425] -13.0618 [-0.218752] [-0.122647] [11.1902] [0.000366638]
88 │ 0.01 0.1 0.01 88 [0.329792] -12.9788 [-0.40391] [-0.150773] [11.1902] [0.000450718]
89 │ 0.01 0.1 0.01 89 [0.330243] -13.0376 [-0.0941277] [-0.145109] [11.1902] [0.000433784]
90 │ 0.01 0.1 0.01 90 [0.330677] -13.0262 [0.101687] [-0.120429] [11.1902] [0.000360008]
91 │ 0.01 0.1 0.01 91 [0.331037] -12.914 [-0.424442] [-0.15083] [11.1902] [0.000450889]
92 │ 0.01 0.1 0.01 92 [0.331487] -13.0794 [-0.00773753] [-0.136521] [11.1902] [0.000408113]
93 │ 0.01 0.1 0.01 93 [0.331896] -13.0045 [-0.222839] [-0.145153] [11.1902] [0.000433916]
94 │ 0.01 0.1 0.01 94 [0.332329] -13.0518 [-0.119833] [-0.142621] [11.1902] [0.000426347]
95 │ 0.01 0.1 0.01 95 [0.332756] -12.9658 [0.121323] [-0.116227] [11.1902] [0.000347445]
96 │ 0.01 0.1 0.01 96 [0.333103] -13.0481 [0.393995] [-0.0652044] [11.1902] [0.00019492]
97 │ 0.01 0.1 0.01 97 [0.333298] -12.9585 [-0.0774373] [-0.0664277] [11.1902] [0.000198577]
98 │ 0.01 0.1 0.01 98 [0.333497] -12.9809 [-0.215603] [-0.0813452] [11.1902] [0.000243171]
99 │ 0.01 0.1 0.01 99 [0.33374] -13.0133 [0.126927] [-0.060518] [11.1902] [0.000180911]
100 │ 0.01 0.1 0.01 100 [0.333921] -12.9609 [0.0137386] [-0.0530924] [11.1902] [0.000158713]
101 │ 0.01 0.1 0.01 101 [0.33408] -13.0542 [-0.0760604] [-0.0553892] [11.1902] [0.000165579]
To plot it, we can either work with the DataFrame or access the fields directly
begin
fig = Figure()
ax1 = Axis(fig[1, 1]; xlabel=L"i", ylabel=L"E_v")
ax2 = Axis(fig[2, 1]; xlabel=L"i", ylabel=L"p")
lines!(ax1, grad_df.iter, grad_df.value)
lines!(ax2, first.(grad_result.param))
fig
end
We see from the plot that the value of the energy is fluctiating around what appears to be
the minimum. While the parameter estimate here is probably good enough for importance
sampling, we can refine the result by creating a new KineticVQMC structure with
increased samples and use it to refine the result. Here, we can pass the previous result
grad_result in place of the initial parameters, which will continue the computation
where the previous one left off. Alternatively, this can be achieved by passing the first_moment_init and second_moment_init arguments to amsgrad.
qmc2 = KineticVQMC(H, ansatz; samples=1e6)
grad_result2 = amsgrad(qmc2, grad_result)GradientDescentResult
iterations: 101
converged: false (iterations)
last value: -13.01383286936652
last params: [0.33330557657068277]
Now, let's plot the refined result next to the minimum found by Optim.jl
begin
fig = Figure()
ax1 = Axis(fig[1, 1]; xlabel=L"i", ylabel=L"E_v")
ax2 = Axis(fig[2, 1]; xlabel=L"i", ylabel=L"p")
lines!(ax1, grad_result2.value)
hlines!(ax1, [opt_lbgfs.minimum]; linestyle=:dot)
lines!(ax2, first.(grad_result2.param))
hlines!(ax2, opt_lbgfs.minimizer; linestyle=:dot)
fig
end
Finally, we have a good estimate for the parameter to use with importance
sampling. Gutzwiller.jl provides AnsatzSampling, which is similar to
GutzwillerSampling from Rimu, but can be used with different ansatze.
p = grad_result.param[end]
G = AnsatzSampling(H, ansatz, p)Gutzwiller.AnsatzSampling{Float64, 1, Gutzwiller.GutzwillerAnsatz{Rimu.BitStringAddresses.BoseFS{10, 10, Rimu.BitStringAddresses.BitString{19, 1, UInt32}}, Float64, Rimu.Hamiltonians.HubbardReal1D{Float64, Rimu.BitStringAddresses.BoseFS{10, 10, Rimu.BitStringAddresses.BitString{19, 1, UInt32}}, 2.0, 1.0}}, Rimu.Hamiltonians.HubbardReal1D{Float64, Rimu.BitStringAddresses.BoseFS{10, 10, Rimu.BitStringAddresses.BitString{19, 1, UInt32}}, 2.0, 1.0}}(HubbardReal1D(fs"|1 1 1 1 1 1 1 1 1 1⟩"; u=2.0, t=1.0), Gutzwiller.GutzwillerAnsatz{Rimu.BitStringAddresses.BoseFS{10, 10, Rimu.BitStringAddresses.BitString{19, 1, UInt32}}, Float64, Rimu.Hamiltonians.HubbardReal1D{Float64, Rimu.BitStringAddresses.BoseFS{10, 10, Rimu.BitStringAddresses.BitString{19, 1, UInt32}}, 2.0, 1.0}}(HubbardReal1D(fs"|1 1 1 1 1 1 1 1 1 1⟩"; u=2.0, t=1.0)), [0.33407952896097876])
This can now be used with FCIQMC. Let's compare the importance sampled time series to a non-importance sampled one.
prob_standard = ProjectorMonteCarloProblem(H; target_walkers=15, last_step=2000)
sim_standard = solve(prob_standard)
shift_estimator(sim_standard; skip=1000)BlockingResult{Float64}
mean = -10.96 ± 0.5
with uncertainty of ± 0.06453926876234162
from 31 blocks after 5 transformations (k = 6).
prob_sampled = ProjectorMonteCarloProblem(G; target_walkers=15, last_step=2000)
sim_sampled = solve(prob_sampled)
shift_estimator(sim_sampled; skip=1000)BlockingResult{Float64}
mean = -12.72 ± 0.36
with uncertainty of ± 0.03299381777618156
from 62 blocks after 4 transformations (k = 5).
Note that the lower energy estimate in the sampled case is probably due to a reduced population control bias. The effect importance sampling has on the statistic can be more dramatic for larger systems and beter choices of anstaze.
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