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IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Statistics in Quantum Monte Carlo
Pablo Lopez Rıos
The Towler Institute
3 August 2010
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
The need for statistical analysis
The need for statistical analysis
A QMC calculation produces millions of data
We want a single number (with its error bar) as a result:
E±σE
Serial correlation needs to be removed
How to manipulate quantities with error bars
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
The need for statistical analysis
The need for statistical analysis
A QMC calculation produces millions of data
We want a single number (with its error bar) as a result:
E±σE
Serial correlation needs to be removed
How to manipulate quantities with error bars
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
The need for statistical analysis
The need for statistical analysis
A QMC calculation produces millions of data
We want a single number (with its error bar) as a result:
E±σE
Serial correlation needs to be removed
How to manipulate quantities with error bars
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
The need for statistical analysis
The need for statistical analysis
A QMC calculation produces millions of data
We want a single number (with its error bar) as a result:
E±σE
Serial correlation needs to be removed
How to manipulate quantities with error bars
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Basic statistics
The configurations {Ri}i=Mi=1 distributed according to |Ψ(R)|2
The local energy Ei = EL(Ri) = Ψ−1(Ri)HΨ(Ri)
EL(R) forms a distribution with:
Mean
EV = 〈Ψ|H|Ψ〉〈Ψ|Ψ〉 ≈ E = ∑
Mi=1 EiM
Variance
σ2EL
= 〈Ψ|H2|Ψ〉〈Ψ|Ψ〉 −
[〈Ψ|H|Ψ〉〈Ψ|Ψ〉
]2≈ σ2
EL=
∑Mi=1(Ei−E)
2
M−1
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Basic statistics
The configurations {Ri}i=Mi=1 distributed according to |Ψ(R)|2
The local energy Ei = EL(Ri) = Ψ−1(Ri)HΨ(Ri)
EL(R) forms a distribution with:
Mean
EV = 〈Ψ|H|Ψ〉〈Ψ|Ψ〉 ≈ E = ∑
Mi=1 EiM
Variance
σ2EL
= 〈Ψ|H2|Ψ〉〈Ψ|Ψ〉 −
[〈Ψ|H|Ψ〉〈Ψ|Ψ〉
]2≈ σ2
EL=
∑Mi=1(Ei−E)
2
M−1
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Basic statistics
The configurations {Ri}i=Mi=1 distributed according to |Ψ(R)|2
The local energy Ei = EL(Ri) = Ψ−1(Ri)HΨ(Ri)
EL(R) forms a distribution with:
Mean
EV = 〈Ψ|H|Ψ〉〈Ψ|Ψ〉 ≈ E = ∑
Mi=1 EiM
Variance
σ2EL
= 〈Ψ|H2|Ψ〉〈Ψ|Ψ〉 −
[〈Ψ|H|Ψ〉〈Ψ|Ψ〉
]2≈ σ2
EL=
∑Mi=1(Ei−E)
2
M−1
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Basic statistics
The configurations {Ri}i=Mi=1 distributed according to |Ψ(R)|2
The local energy Ei = EL(Ri) = Ψ−1(Ri)HΨ(Ri)
EL(R) forms a distribution with:
Mean
EV = 〈Ψ|H|Ψ〉〈Ψ|Ψ〉 ≈ E = ∑
Mi=1 EiM
Variance
σ2EL
= 〈Ψ|H2|Ψ〉〈Ψ|Ψ〉 −
[〈Ψ|H|Ψ〉〈Ψ|Ψ〉
]2≈ σ2
EL=
∑Mi=1(Ei−E)
2
M−1
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Basic statistics
The configurations {Ri}i=Mi=1 distributed according to |Ψ(R)|2
The local energy Ei = EL(Ri) = Ψ−1(Ri)HΨ(Ri)
EL(R) forms a distribution with:
Mean
EV = 〈Ψ|H|Ψ〉〈Ψ|Ψ〉 ≈ E = ∑
Mi=1 EiM
Variance
σ2EL
= 〈Ψ|H2|Ψ〉〈Ψ|Ψ〉 −
[〈Ψ|H|Ψ〉〈Ψ|Ψ〉
]2≈ σ2
EL=
∑Mi=1(Ei−E)
2
M−1
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Basic statistics
E can be determined to a given degree of certainty→ it has an error bar σE→ it is a distribution with:
Mean
E = ∑Mi=1 EiM
Variance
σ2E ≈ σ2
E =∑
Mi=1(Ei−E)
2
M(M−1)
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Basic statistics
E can be determined to a given degree of certainty→ it has an error bar σE→ it is a distribution with:
Mean
E = ∑Mi=1 EiM
Variance
σ2E ≈ σ2
E =∑
Mi=1(Ei−E)
2
M(M−1)
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Basic statistics
E can be determined to a given degree of certainty→ it has an error bar σE→ it is a distribution with:
Mean
E = ∑Mi=1 EiM
Variance
σ2E ≈ σ2
E =∑
Mi=1(Ei−E)
2
M(M−1)
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Basic statistics
E can be determined to a given degree of certainty→ it has an error bar σE→ it is a distribution with:
Mean
E = ∑Mi=1 EiM
Variance
σ2E ≈ σ2
E =∑
Mi=1(Ei−E)
2
M(M−1)
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Basic statistics
E can be determined to a given degree of certainty→ it has an error bar σE→ it is a distribution with:
Mean
E = ∑Mi=1 EiM
Variance
σ2E ≈ σ2
E =∑
Mi=1(Ei−E)
2
M(M−1)
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Local energy and mean energy
0.5 0.55E (a.u.)
0
0.5
1P
(E)
(arb
itrar
y un
its)
EL distribution
E distribution
The local energy distribution is what we sample.The mean energy distribution is what we obtain.
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Local energy and mean energy
0.5 0.55E (a.u.)
0
0.5
1P
(E)
(arb
itrar
y un
its)
EL distribution
E distribution
The local energy distribution is what we sample.The mean energy distribution is what we obtain.
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Local energy and mean energy
0.5 0.55E (a.u.)
0
0.5
1P
(E)
(arb
itrar
y un
its)
EL distribution
E distribution
The local energy distribution is what we sample.The mean energy distribution is what we obtain.
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Sampling of configuration space
{Ri}i=Mi=1 must be distributed according to |Ψ(R)|2.
Sampling algorithm at i-th step
Start at config Ri
Propose a new config R′i
Compute the wave function ratio qi =∣∣∣Ψ(R′i)
Ψ(Ri)
∣∣∣2Generate an uniformly-distributed random numberξ ∈ [0,1)
Accept/reject step:
if ξ < qi → set Ri+1 = R′i (accept new config)if ξ > qi → set Ri+1 = Ri (reject new config)
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Sampling of configuration space
{Ri}i=Mi=1 must be distributed according to |Ψ(R)|2.
Sampling algorithm at i-th step
Start at config Ri
Propose a new config R′i
Compute the wave function ratio qi =∣∣∣Ψ(R′i)
Ψ(Ri)
∣∣∣2Generate an uniformly-distributed random numberξ ∈ [0,1)
Accept/reject step:
if ξ < qi → set Ri+1 = R′i (accept new config)if ξ > qi → set Ri+1 = Ri (reject new config)
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Sampling of configuration space
{Ri}i=Mi=1 must be distributed according to |Ψ(R)|2.
Sampling algorithm at i-th step
Start at config Ri
Propose a new config R′i
Compute the wave function ratio qi =∣∣∣Ψ(R′i)
Ψ(Ri)
∣∣∣2Generate an uniformly-distributed random numberξ ∈ [0,1)
Accept/reject step:
if ξ < qi → set Ri+1 = R′i (accept new config)if ξ > qi → set Ri+1 = Ri (reject new config)
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Sampling of configuration space
{Ri}i=Mi=1 must be distributed according to |Ψ(R)|2.
Sampling algorithm at i-th step
Start at config Ri
Propose a new config R′i
Compute the wave function ratio qi =∣∣∣Ψ(R′i)
Ψ(Ri)
∣∣∣2Generate an uniformly-distributed random numberξ ∈ [0,1)
Accept/reject step:
if ξ < qi → set Ri+1 = R′i (accept new config)if ξ > qi → set Ri+1 = Ri (reject new config)
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Sampling of configuration space
{Ri}i=Mi=1 must be distributed according to |Ψ(R)|2.
Sampling algorithm at i-th step
Start at config Ri
Propose a new config R′i
Compute the wave function ratio qi =∣∣∣Ψ(R′i)
Ψ(Ri)
∣∣∣2Generate an uniformly-distributed random numberξ ∈ [0,1)
Accept/reject step:
if ξ < qi → set Ri+1 = R′i (accept new config)if ξ > qi → set Ri+1 = Ri (reject new config)
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Sampling of configuration space
{Ri}i=Mi=1 must be distributed according to |Ψ(R)|2.
Sampling algorithm at i-th step
Start at config Ri
Propose a new config R′i
Compute the wave function ratio qi =∣∣∣Ψ(R′i)
Ψ(Ri)
∣∣∣2Generate an uniformly-distributed random numberξ ∈ [0,1)
Accept/reject step:
if ξ < qi → set Ri+1 = R′i (accept new config)if ξ > qi → set Ri+1 = Ri (reject new config)
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Sampling of configuration space
{Ri}i=Mi=1 must be distributed according to |Ψ(R)|2.
Sampling algorithm at i-th step
Start at config Ri
Propose a new config R′i
Compute the wave function ratio qi =∣∣∣Ψ(R′i)
Ψ(Ri)
∣∣∣2Generate an uniformly-distributed random numberξ ∈ [0,1)
Accept/reject step:
if ξ < qi → set Ri+1 = R′i (accept new config)if ξ > qi → set Ri+1 = Ri (reject new config)
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Sampling of configuration space
{Ri}i=Mi=1 must be distributed according to |Ψ(R)|2.
Sampling algorithm at i-th step
Start at config Ri
Propose a new config R′i
Compute the wave function ratio qi =∣∣∣Ψ(R′i)
Ψ(Ri)
∣∣∣2Generate an uniformly-distributed random numberξ ∈ [0,1)
Accept/reject step:
if ξ < qi → set Ri+1 = R′i (accept new config)if ξ > qi → set Ri+1 = Ri (reject new config)
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Proposing Ri→ R′i
If R′i proposed at random:→ chances of landing in a reasonable region of configurationspaces are slim→ qi will be small→ most moves are rejected→ poor sampling
If R′i is Ri plus a small displacement:→ R′i similar to Ri
→ EL(R′i) similar to EL(Ri)→ Serial correlation
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Proposing Ri→ R′i
If R′i proposed at random:→ chances of landing in a reasonable region of configurationspaces are slim→ qi will be small→ most moves are rejected→ poor sampling
If R′i is Ri plus a small displacement:→ R′i similar to Ri
→ EL(R′i) similar to EL(Ri)→ Serial correlation
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Proposing Ri→ R′i
If R′i proposed at random:→ chances of landing in a reasonable region of configurationspaces are slim→ qi will be small→ most moves are rejected→ poor sampling
If R′i is Ri plus a small displacement:→ R′i similar to Ri
→ EL(R′i) similar to EL(Ri)→ Serial correlation
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Proposing Ri→ R′i
If R′i proposed at random:→ chances of landing in a reasonable region of configurationspaces are slim→ qi will be small→ most moves are rejected→ poor sampling
If R′i is Ri plus a small displacement:→ R′i similar to Ri
→ EL(R′i) similar to EL(Ri)→ Serial correlation
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Proposing Ri→ R′i
If R′i proposed at random:→ chances of landing in a reasonable region of configurationspaces are slim→ qi will be small→ most moves are rejected→ poor sampling
If R′i is Ri plus a small displacement:→ R′i similar to Ri
→ EL(R′i) similar to EL(Ri)→ Serial correlation
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Proposing Ri→ R′i
If R′i proposed at random:→ chances of landing in a reasonable region of configurationspaces are slim→ qi will be small→ most moves are rejected→ poor sampling
If R′i is Ri plus a small displacement:→ R′i similar to Ri
→ EL(R′i) similar to EL(Ri)→ Serial correlation
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Proposing Ri→ R′i
If R′i proposed at random:→ chances of landing in a reasonable region of configurationspaces are slim→ qi will be small→ most moves are rejected→ poor sampling
If R′i is Ri plus a small displacement:→ R′i similar to Ri
→ EL(R′i) similar to EL(Ri)→ Serial correlation
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Proposing Ri→ R′i
If R′i proposed at random:→ chances of landing in a reasonable region of configurationspaces are slim→ qi will be small→ most moves are rejected→ poor sampling
If R′i is Ri plus a small displacement:→ R′i similar to Ri
→ EL(R′i) similar to EL(Ri)→ Serial correlation
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Proposing Ri→ R′i
If R′i proposed at random:→ chances of landing in a reasonable region of configurationspaces are slim→ qi will be small→ most moves are rejected→ poor sampling
If R′i is Ri plus a small displacement:→ R′i similar to Ri
→ EL(R′i) similar to EL(Ri)→ Serial correlation
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Effect of serial correlation
Consider an uncorrelated set of energies {E1,E2,E3, . . . ,EM}Generate a new set with artificial serial correlation:
{E1, . . . ,E1︸ ︷︷ ︸τ
,E2, . . . ,E2︸ ︷︷ ︸τ
,E3, . . . ,E3︸ ︷︷ ︸τ
, . . . ,EM, . . . ,EM︸ ︷︷ ︸τ
}
No new information → mean and error bar should beunchanged
Computed mean of new set is E′ = E
Computed error bar of new set is σ ′E = σE/√
τ
→ error bar underestimated!
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Effect of serial correlation
Consider an uncorrelated set of energies {E1,E2,E3, . . . ,EM}Generate a new set with artificial serial correlation:
{E1, . . . ,E1︸ ︷︷ ︸τ
,E2, . . . ,E2︸ ︷︷ ︸τ
,E3, . . . ,E3︸ ︷︷ ︸τ
, . . . ,EM, . . . ,EM︸ ︷︷ ︸τ
}
No new information → mean and error bar should beunchanged
Computed mean of new set is E′ = E
Computed error bar of new set is σ ′E = σE/√
τ
→ error bar underestimated!
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Effect of serial correlation
Consider an uncorrelated set of energies {E1,E2,E3, . . . ,EM}Generate a new set with artificial serial correlation:
{E1, . . . ,E1︸ ︷︷ ︸τ
,E2, . . . ,E2︸ ︷︷ ︸τ
,E3, . . . ,E3︸ ︷︷ ︸τ
, . . . ,EM, . . . ,EM︸ ︷︷ ︸τ
}
No new information → mean and error bar should beunchanged
Computed mean of new set is E′ = E
Computed error bar of new set is σ ′E = σE/√
τ
→ error bar underestimated!
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Effect of serial correlation
Consider an uncorrelated set of energies {E1,E2,E3, . . . ,EM}Generate a new set with artificial serial correlation:
{E1, . . . ,E1︸ ︷︷ ︸τ
,E2, . . . ,E2︸ ︷︷ ︸τ
,E3, . . . ,E3︸ ︷︷ ︸τ
, . . . ,EM, . . . ,EM︸ ︷︷ ︸τ
}
No new information → mean and error bar should beunchanged
Computed mean of new set is E′ = E
Computed error bar of new set is σ ′E = σE/√
τ
→ error bar underestimated!
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Effect of serial correlation
Consider an uncorrelated set of energies {E1,E2,E3, . . . ,EM}Generate a new set with artificial serial correlation:
{E1, . . . ,E1︸ ︷︷ ︸τ
,E2, . . . ,E2︸ ︷︷ ︸τ
,E3, . . . ,E3︸ ︷︷ ︸τ
, . . . ,EM, . . . ,EM︸ ︷︷ ︸τ
}
No new information → mean and error bar should beunchanged
Computed mean of new set is E′ = E
Computed error bar of new set is σ ′E = σE/√
τ
→ error bar underestimated!
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Effect of serial correlation
Consider an uncorrelated set of energies {E1,E2,E3, . . . ,EM}Generate a new set with artificial serial correlation:
{E1, . . . ,E1︸ ︷︷ ︸τ
,E2, . . . ,E2︸ ︷︷ ︸τ
,E3, . . . ,E3︸ ︷︷ ︸τ
, . . . ,EM, . . . ,EM︸ ︷︷ ︸τ
}
No new information → mean and error bar should beunchanged
Computed mean of new set is E′ = E
Computed error bar of new set is σ ′E = σE/√
τ
→ error bar underestimated!
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Removing serial correlation
In this example we can remove the serial correlation byignoring τ−1 of every τ consecutive energies
For real data the correlation time τ varies during the run→ would need to ignore τmax−1 of each τmax data to be safe→ lots of relevant data discarded→ inefficiency
However we may assume that
σ ′E = σE/√
τ
still holds, where τ is the average correlation time.
This is an alternative approach to the reblocking algorithm
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Removing serial correlation
In this example we can remove the serial correlation byignoring τ−1 of every τ consecutive energies
For real data the correlation time τ varies during the run→ would need to ignore τmax−1 of each τmax data to be safe→ lots of relevant data discarded→ inefficiency
However we may assume that
σ ′E = σE/√
τ
still holds, where τ is the average correlation time.
This is an alternative approach to the reblocking algorithm
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Removing serial correlation
In this example we can remove the serial correlation byignoring τ−1 of every τ consecutive energies
For real data the correlation time τ varies during the run→ would need to ignore τmax−1 of each τmax data to be safe→ lots of relevant data discarded→ inefficiency
However we may assume that
σ ′E = σE/√
τ
still holds, where τ is the average correlation time.
This is an alternative approach to the reblocking algorithm
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Removing serial correlation
In this example we can remove the serial correlation byignoring τ−1 of every τ consecutive energies
For real data the correlation time τ varies during the run→ would need to ignore τmax−1 of each τmax data to be safe→ lots of relevant data discarded→ inefficiency
However we may assume that
σ ′E = σE/√
τ
still holds, where τ is the average correlation time.
This is an alternative approach to the reblocking algorithm
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Removing serial correlation
In this example we can remove the serial correlation byignoring τ−1 of every τ consecutive energies
For real data the correlation time τ varies during the run→ would need to ignore τmax−1 of each τmax data to be safe→ lots of relevant data discarded→ inefficiency
However we may assume that
σ ′E = σE/√
τ
still holds, where τ is the average correlation time.
This is an alternative approach to the reblocking algorithm
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Removing serial correlation
In this example we can remove the serial correlation byignoring τ−1 of every τ consecutive energies
For real data the correlation time τ varies during the run→ would need to ignore τmax−1 of each τmax data to be safe→ lots of relevant data discarded→ inefficiency
However we may assume that
σ ′E = σE/√
τ
still holds, where τ is the average correlation time.
This is an alternative approach to the reblocking algorithm
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Removing serial correlation
In this example we can remove the serial correlation byignoring τ−1 of every τ consecutive energies
For real data the correlation time τ varies during the run→ would need to ignore τmax−1 of each τmax data to be safe→ lots of relevant data discarded→ inefficiency
However we may assume that
σ ′E = σE/√
τ
still holds, where τ is the average correlation time.
This is an alternative approach to the reblocking algorithm
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
The reblocking algorithm
Consider the following operation on data, where the itemunder each brace is the average of the two numbers above:
E(0)1 E(0)
2︸ ︷︷ ︸ E(0)3 E(0)
4︸ ︷︷ ︸ E(0)5 E(0)
6︸ ︷︷ ︸ E(0)7 E(0)
8︸ ︷︷ ︸E(1)
1 E(1)2︸ ︷︷ ︸ E(1)
3 E(1)4︸ ︷︷ ︸
. . . . . .
If we succesively apply transformations until τmax of theoriginal data are averaged together→ resulting (smaller) data set is not serially correlated
Cannot compute τmax directly. Must find out how manyreblocking transformations to apply in some other way
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
The reblocking algorithm
Consider the following operation on data, where the itemunder each brace is the average of the two numbers above:
E(0)1 E(0)
2︸ ︷︷ ︸ E(0)3 E(0)
4︸ ︷︷ ︸ E(0)5 E(0)
6︸ ︷︷ ︸ E(0)7 E(0)
8︸ ︷︷ ︸E(1)
1 E(1)2︸ ︷︷ ︸ E(1)
3 E(1)4︸ ︷︷ ︸
. . . . . .
If we succesively apply transformations until τmax of theoriginal data are averaged together→ resulting (smaller) data set is not serially correlated
Cannot compute τmax directly. Must find out how manyreblocking transformations to apply in some other way
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
The reblocking algorithm
Consider the following operation on data, where the itemunder each brace is the average of the two numbers above:
E(0)1 E(0)
2︸ ︷︷ ︸ E(0)3 E(0)
4︸ ︷︷ ︸ E(0)5 E(0)
6︸ ︷︷ ︸ E(0)7 E(0)
8︸ ︷︷ ︸E(1)
1 E(1)2︸ ︷︷ ︸ E(1)
3 E(1)4︸ ︷︷ ︸
. . . . . .
If we succesively apply transformations until τmax of theoriginal data are averaged together→ resulting (smaller) data set is not serially correlated
Cannot compute τmax directly. Must find out how manyreblocking transformations to apply in some other way
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
The reblocking algorithm
Consider the following operation on data, where the itemunder each brace is the average of the two numbers above:
E(0)1 E(0)
2︸ ︷︷ ︸ E(0)3 E(0)
4︸ ︷︷ ︸ E(0)5 E(0)
6︸ ︷︷ ︸ E(0)7 E(0)
8︸ ︷︷ ︸E(1)
1 E(1)2︸ ︷︷ ︸ E(1)
3 E(1)4︸ ︷︷ ︸
. . . . . .
If we succesively apply transformations until τmax of theoriginal data are averaged together→ resulting (smaller) data set is not serially correlated
Cannot compute τmax directly. Must find out how manyreblocking transformations to apply in some other way
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
The reblocking algorithm
Consider the following operation on data, where the itemunder each brace is the average of the two numbers above:
E(0)1 E(0)
2︸ ︷︷ ︸ E(0)3 E(0)
4︸ ︷︷ ︸ E(0)5 E(0)
6︸ ︷︷ ︸ E(0)7 E(0)
8︸ ︷︷ ︸E(1)
1 E(1)2︸ ︷︷ ︸ E(1)
3 E(1)4︸ ︷︷ ︸
. . . . . .
If we succesively apply transformations until τmax of theoriginal data are averaged together→ resulting (smaller) data set is not serially correlated
Cannot compute τmax directly. Must find out how manyreblocking transformations to apply in some other way
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
The reblocking algorithm
Consider the following operation on data, where the itemunder each brace is the average of the two numbers above:
E(0)1 E(0)
2︸ ︷︷ ︸ E(0)3 E(0)
4︸ ︷︷ ︸ E(0)5 E(0)
6︸ ︷︷ ︸ E(0)7 E(0)
8︸ ︷︷ ︸E(1)
1 E(1)2︸ ︷︷ ︸ E(1)
3 E(1)4︸ ︷︷ ︸
. . . . . .
If we succesively apply transformations until τmax of theoriginal data are averaged together→ resulting (smaller) data set is not serially correlated
Cannot compute τmax directly. Must find out how manyreblocking transformations to apply in some other way
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
The reblocking algorithm
Consider the following operation on data, where the itemunder each brace is the average of the two numbers above:
E(0)1 E(0)
2︸ ︷︷ ︸ E(0)3 E(0)
4︸ ︷︷ ︸ E(0)5 E(0)
6︸ ︷︷ ︸ E(0)7 E(0)
8︸ ︷︷ ︸E(1)
1 E(1)2︸ ︷︷ ︸ E(1)
3 E(1)4︸ ︷︷ ︸
. . . . . .
If we succesively apply transformations until τmax of theoriginal data are averaged together→ resulting (smaller) data set is not serially correlated
Cannot compute τmax directly. Must find out how manyreblocking transformations to apply in some other way
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Error estimator after reblocking
At the k-th iteration in this procedure:
σ(k+1)2E ≈ σ
(k)2E +
2∑M(k)/2i=1
(E(k)
2i−1− E)(
E(k)2i − E
)M(k)(M(k)−2)
If there is no serial correlation, the last term tends to zero
If there is serial correlation, the last term is positive
Hence σ(k)
E will increase until it reaches the true error bar atk ≈ log2(τmax)
Plateau in σ(k)
E signals convergence of reblocking algorithm
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Error estimator after reblocking
At the k-th iteration in this procedure:
σ(k+1)2E ≈ σ
(k)2E +
2∑M(k)/2i=1
(E(k)
2i−1− E)(
E(k)2i − E
)M(k)(M(k)−2)
If there is no serial correlation, the last term tends to zero
If there is serial correlation, the last term is positive
Hence σ(k)
E will increase until it reaches the true error bar atk ≈ log2(τmax)
Plateau in σ(k)
E signals convergence of reblocking algorithm
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Error estimator after reblocking
At the k-th iteration in this procedure:
σ(k+1)2E ≈ σ
(k)2E +
2∑M(k)/2i=1
(E(k)
2i−1− E)(
E(k)2i − E
)M(k)(M(k)−2)
If there is no serial correlation, the last term tends to zero
If there is serial correlation, the last term is positive
Hence σ(k)
E will increase until it reaches the true error bar atk ≈ log2(τmax)
Plateau in σ(k)
E signals convergence of reblocking algorithm
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Error estimator after reblocking
At the k-th iteration in this procedure:
σ(k+1)2E ≈ σ
(k)2E +
2∑M(k)/2i=1
(E(k)
2i−1− E)(
E(k)2i − E
)M(k)(M(k)−2)
If there is no serial correlation, the last term tends to zero
If there is serial correlation, the last term is positive
Hence σ(k)
E will increase until it reaches the true error bar atk ≈ log2(τmax)
Plateau in σ(k)
E signals convergence of reblocking algorithm
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Error estimator after reblocking
At the k-th iteration in this procedure:
σ(k+1)2E ≈ σ
(k)2E +
2∑M(k)/2i=1
(E(k)
2i−1− E)(
E(k)2i − E
)M(k)(M(k)−2)
If there is no serial correlation, the last term tends to zero
If there is serial correlation, the last term is positive
Hence σ(k)
E will increase until it reaches the true error bar atk ≈ log2(τmax)
Plateau in σ(k)
E signals convergence of reblocking algorithm
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Reblock plot
0 5 10 15 20 25k
0.00000
0.00001
0.00002
0.00003
0.00004
~ σ E (k)
(a.
u.)
log 2(τ
)
log 2(τ
max
)
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
How to run efficient VMC calculations
Reducing serial correlation, by
Choosing an appropriate timestepUsing electron-by-electron samplingSkipping the right number of steps between every twocalculations of expectation values
Reducing the intrinsic variance/expense, by
Using appropriate trial wave functions
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
How to run efficient VMC calculations
Reducing serial correlation, by
Choosing an appropriate timestepUsing electron-by-electron samplingSkipping the right number of steps between every twocalculations of expectation values
Reducing the intrinsic variance/expense, by
Using appropriate trial wave functions
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
How to run efficient VMC calculations
Reducing serial correlation, by
Choosing an appropriate timestepUsing electron-by-electron samplingSkipping the right number of steps between every twocalculations of expectation values
Reducing the intrinsic variance/expense, by
Using appropriate trial wave functions
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
How to run efficient VMC calculations
Reducing serial correlation, by
Choosing an appropriate timestepUsing electron-by-electron samplingSkipping the right number of steps between every twocalculations of expectation values
Reducing the intrinsic variance/expense, by
Using appropriate trial wave functions
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
How to run efficient VMC calculations
Reducing serial correlation, by
Choosing an appropriate timestepUsing electron-by-electron samplingSkipping the right number of steps between every twocalculations of expectation values
Reducing the intrinsic variance/expense, by
Using appropriate trial wave functions
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
How to run efficient VMC calculations
Reducing serial correlation, by
Choosing an appropriate timestepUsing electron-by-electron samplingSkipping the right number of steps between every twocalculations of expectation values
Reducing the intrinsic variance/expense, by
Using appropriate trial wave functions
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
The VMC timestep
The “timestep” T is the variance of the distribution used togenerate the random displacements when proposing moves
It is actually a squared length, but can be regarded a time ifconsidering a diffusion process
T does not enter the VMC formalism→ can be chosen so that statistics are improved
T small → R′i very similar to Ri→ serial correlation increasedT large → R′i very dissimilar from Ri→ most moves are rejected→ serial correlation increased
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
The VMC timestep
The “timestep” T is the variance of the distribution used togenerate the random displacements when proposing moves
It is actually a squared length, but can be regarded a time ifconsidering a diffusion process
T does not enter the VMC formalism→ can be chosen so that statistics are improved
T small → R′i very similar to Ri→ serial correlation increasedT large → R′i very dissimilar from Ri→ most moves are rejected→ serial correlation increased
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
The VMC timestep
The “timestep” T is the variance of the distribution used togenerate the random displacements when proposing moves
It is actually a squared length, but can be regarded a time ifconsidering a diffusion process
T does not enter the VMC formalism→ can be chosen so that statistics are improved
T small → R′i very similar to Ri→ serial correlation increasedT large → R′i very dissimilar from Ri→ most moves are rejected→ serial correlation increased
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
The VMC timestep
The “timestep” T is the variance of the distribution used togenerate the random displacements when proposing moves
It is actually a squared length, but can be regarded a time ifconsidering a diffusion process
T does not enter the VMC formalism→ can be chosen so that statistics are improved
T small → R′i very similar to Ri→ serial correlation increasedT large → R′i very dissimilar from Ri→ most moves are rejected→ serial correlation increased
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
The VMC timestep
The “timestep” T is the variance of the distribution used togenerate the random displacements when proposing moves
It is actually a squared length, but can be regarded a time ifconsidering a diffusion process
T does not enter the VMC formalism→ can be chosen so that statistics are improved
T small → R′i very similar to Ri→ serial correlation increasedT large → R′i very dissimilar from Ri→ most moves are rejected→ serial correlation increased
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
The VMC timestep
The “timestep” T is the variance of the distribution used togenerate the random displacements when proposing moves
It is actually a squared length, but can be regarded a time ifconsidering a diffusion process
T does not enter the VMC formalism→ can be chosen so that statistics are improved
T small → R′i very similar to Ri→ serial correlation increasedT large → R′i very dissimilar from Ri→ most moves are rejected→ serial correlation increased
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
The VMC timestep
The “timestep” T is the variance of the distribution used togenerate the random displacements when proposing moves
It is actually a squared length, but can be regarded a time ifconsidering a diffusion process
T does not enter the VMC formalism→ can be chosen so that statistics are improved
T small → R′i very similar to Ri→ serial correlation increasedT large → R′i very dissimilar from Ri→ most moves are rejected→ serial correlation increased
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
The VMC timestep
The “timestep” T is the variance of the distribution used togenerate the random displacements when proposing moves
It is actually a squared length, but can be regarded a time ifconsidering a diffusion process
T does not enter the VMC formalism→ can be chosen so that statistics are improved
T small → R′i very similar to Ri→ serial correlation increasedT large → R′i very dissimilar from Ri→ most moves are rejected→ serial correlation increased
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
The 50% rule
The 50% rule
Choose T such that the acceptance ratio a = 50%
0.001 0.01 0.1 1T (a.u.)
0
25
50
τ (s
teps
)
0
25
50
75
100
a (%
)
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
The 50% rule
There are better strategies for some systems, e.g. the electrongas
0.01 0.1 1 10T (a.u.)
0
10
20
30
τ (s
teps
)
0
25
50
75
100
a (%
)
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Electron-by-electron sampling
QMC sampling usually described using configuration moves,→ Configuration-by-configuration sampling (CBCS)Usual implementation: move each electron and accept/rejecttheir moves individually→ Electron-by-electron sampling (EBES)Set T to the same value in CBCS and EBES→ aC would equal aN
E→ EBES winsSet T so that aC = aE = a→ the chances of Ri+1 = Ri in the CBCS is a,while in the EBES it is aN
→ EBES wins
The EBES is more efficient
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Electron-by-electron sampling
QMC sampling usually described using configuration moves,→ Configuration-by-configuration sampling (CBCS)Usual implementation: move each electron and accept/rejecttheir moves individually→ Electron-by-electron sampling (EBES)Set T to the same value in CBCS and EBES→ aC would equal aN
E→ EBES winsSet T so that aC = aE = a→ the chances of Ri+1 = Ri in the CBCS is a,while in the EBES it is aN
→ EBES wins
The EBES is more efficient
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Electron-by-electron sampling
QMC sampling usually described using configuration moves,→ Configuration-by-configuration sampling (CBCS)Usual implementation: move each electron and accept/rejecttheir moves individually→ Electron-by-electron sampling (EBES)Set T to the same value in CBCS and EBES→ aC would equal aN
E→ EBES winsSet T so that aC = aE = a→ the chances of Ri+1 = Ri in the CBCS is a,while in the EBES it is aN
→ EBES wins
The EBES is more efficient
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Electron-by-electron sampling
QMC sampling usually described using configuration moves,→ Configuration-by-configuration sampling (CBCS)Usual implementation: move each electron and accept/rejecttheir moves individually→ Electron-by-electron sampling (EBES)Set T to the same value in CBCS and EBES→ aC would equal aN
E→ EBES winsSet T so that aC = aE = a→ the chances of Ri+1 = Ri in the CBCS is a,while in the EBES it is aN
→ EBES wins
The EBES is more efficient
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Electron-by-electron sampling
QMC sampling usually described using configuration moves,→ Configuration-by-configuration sampling (CBCS)Usual implementation: move each electron and accept/rejecttheir moves individually→ Electron-by-electron sampling (EBES)Set T to the same value in CBCS and EBES→ aC would equal aN
E→ EBES winsSet T so that aC = aE = a→ the chances of Ri+1 = Ri in the CBCS is a,while in the EBES it is aN
→ EBES wins
The EBES is more efficient
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Electron-by-electron sampling
QMC sampling usually described using configuration moves,→ Configuration-by-configuration sampling (CBCS)Usual implementation: move each electron and accept/rejecttheir moves individually→ Electron-by-electron sampling (EBES)Set T to the same value in CBCS and EBES→ aC would equal aN
E→ EBES winsSet T so that aC = aE = a→ the chances of Ri+1 = Ri in the CBCS is a,while in the EBES it is aN
→ EBES wins
The EBES is more efficient
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Electron-by-electron sampling
QMC sampling usually described using configuration moves,→ Configuration-by-configuration sampling (CBCS)Usual implementation: move each electron and accept/rejecttheir moves individually→ Electron-by-electron sampling (EBES)Set T to the same value in CBCS and EBES→ aC would equal aN
E→ EBES winsSet T so that aC = aE = a→ the chances of Ri+1 = Ri in the CBCS is a,while in the EBES it is aN
→ EBES wins
The EBES is more efficient
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Electron-by-electron sampling
QMC sampling usually described using configuration moves,→ Configuration-by-configuration sampling (CBCS)Usual implementation: move each electron and accept/rejecttheir moves individually→ Electron-by-electron sampling (EBES)Set T to the same value in CBCS and EBES→ aC would equal aN
E→ EBES winsSet T so that aC = aE = a→ the chances of Ri+1 = Ri in the CBCS is a,while in the EBES it is aN
→ EBES wins
The EBES is more efficient
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Electron-by-electron sampling
QMC sampling usually described using configuration moves,→ Configuration-by-configuration sampling (CBCS)Usual implementation: move each electron and accept/rejecttheir moves individually→ Electron-by-electron sampling (EBES)Set T to the same value in CBCS and EBES→ aC would equal aN
E→ EBES winsSet T so that aC = aE = a→ the chances of Ri+1 = Ri in the CBCS is a,while in the EBES it is aN
→ EBES wins
The EBES is more efficient
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Electron-by-electron sampling
QMC sampling usually described using configuration moves,→ Configuration-by-configuration sampling (CBCS)Usual implementation: move each electron and accept/rejecttheir moves individually→ Electron-by-electron sampling (EBES)Set T to the same value in CBCS and EBES→ aC would equal aN
E→ EBES winsSet T so that aC = aE = a→ the chances of Ri+1 = Ri in the CBCS is a,while in the EBES it is aN
→ EBES wins
The EBES is more efficient
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Choosing the right wave function
The better the wave function, the lower the variance→ fewer data required to achieve a given errorbar→ lower cost
Generally, better wave functions are more expensive toevaluate→ higher cost
Important!
Regarding accuracy, the best trial wave function for aproblem need not be the most complicated
There is a risk of over-parametrization!
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Choosing the right wave function
The better the wave function, the lower the variance→ fewer data required to achieve a given errorbar→ lower cost
Generally, better wave functions are more expensive toevaluate→ higher cost
Important!
Regarding accuracy, the best trial wave function for aproblem need not be the most complicated
There is a risk of over-parametrization!
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Choosing the right wave function
The better the wave function, the lower the variance→ fewer data required to achieve a given errorbar→ lower cost
Generally, better wave functions are more expensive toevaluate→ higher cost
Important!
Regarding accuracy, the best trial wave function for aproblem need not be the most complicated
There is a risk of over-parametrization!
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Choosing the right wave function
The better the wave function, the lower the variance→ fewer data required to achieve a given errorbar→ lower cost
Generally, better wave functions are more expensive toevaluate→ higher cost
Important!
Regarding accuracy, the best trial wave function for aproblem need not be the most complicated
There is a risk of over-parametrization!
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Choosing the right wave function
The better the wave function, the lower the variance→ fewer data required to achieve a given errorbar→ lower cost
Generally, better wave functions are more expensive toevaluate→ higher cost
Important!
Regarding accuracy, the best trial wave function for aproblem need not be the most complicated
There is a risk of over-parametrization!
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Choosing the right wave function
The better the wave function, the lower the variance→ fewer data required to achieve a given errorbar→ lower cost
Generally, better wave functions are more expensive toevaluate→ higher cost
Important!
Regarding accuracy, the best trial wave function for aproblem need not be the most complicated
There is a risk of over-parametrization!
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
DefinitionsSampling and serial correlationStatistical efficiency
Wave functions and the local energy distribution
0.5 0.55E
L (a.u.)
0
0.5
1
P(E
L)
(arb
. uni
ts)
Slater-JastrowSlater-Jastrow-backflow
Local energy distributions for two different wave functionsPablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Definitions
The projection method
Time-dependent Schrodinger equation
H(R)Φ(R,x) = i ∂Φ(R,x)∂x
Imaginary time (ix = t) and energy shift(H(R)−ET
)Φ(R, t) = ∂Φ(R,t)
∂ t
Eigenstate expansion
Φ(R, t) = ∑∞n=0 cnΦn(R)e−(En−ET )t
If we adjust ET ∼ E0, excited eigenstates decay exponentiallyas t→ ∞ and only ground state remains
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Definitions
The projection method
Time-dependent Schrodinger equation
H(R)Φ(R,x) = i ∂Φ(R,x)∂x
Imaginary time (ix = t) and energy shift(H(R)−ET
)Φ(R, t) = ∂Φ(R,t)
∂ t
Eigenstate expansion
Φ(R, t) = ∑∞n=0 cnΦn(R)e−(En−ET )t
If we adjust ET ∼ E0, excited eigenstates decay exponentiallyas t→ ∞ and only ground state remains
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Definitions
The projection method
Time-dependent Schrodinger equation
H(R)Φ(R,x) = i ∂Φ(R,x)∂x
Imaginary time (ix = t) and energy shift(H(R)−ET
)Φ(R, t) = ∂Φ(R,t)
∂ t
Eigenstate expansion
Φ(R, t) = ∑∞n=0 cnΦn(R)e−(En−ET )t
If we adjust ET ∼ E0, excited eigenstates decay exponentiallyas t→ ∞ and only ground state remains
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Definitions
The projection method
Time-dependent Schrodinger equation
H(R)Φ(R,x) = i ∂Φ(R,x)∂x
Imaginary time (ix = t) and energy shift(H(R)−ET
)Φ(R, t) = ∂Φ(R,t)
∂ t
Eigenstate expansion
Φ(R, t) = ∑∞n=0 cnΦn(R)e−(En−ET )t
If we adjust ET ∼ E0, excited eigenstates decay exponentiallyas t→ ∞ and only ground state remains
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
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Definitions
Projection using discrete walkers
Consider the equation
A(R)f (R, t) =− ∂ f (R,t)∂ t
Given f (R, t) at time t and Green’s function G(R′← R,T):
f (R, t + T) =∫
G(R′← R,T)f (R, t)dRIf f (R, t) probability distribution → can represent discretely:
f (R, t)≈ ∑Pp=1 wp(t)δ [R−Rp(t)]
Therefore:
f (R, t + T) = ∑Pp=1 wp(t)G[R′← Rp(t),T]≈
≈ ∑Pp=1 wp(t + T)δ [R−Rp(t + T)]
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Definitions
Projection using discrete walkers
Consider the equation
A(R)f (R, t) =− ∂ f (R,t)∂ t
Given f (R, t) at time t and Green’s function G(R′← R,T):
f (R, t + T) =∫
G(R′← R,T)f (R, t)dRIf f (R, t) probability distribution → can represent discretely:
f (R, t)≈ ∑Pp=1 wp(t)δ [R−Rp(t)]
Therefore:
f (R, t + T) = ∑Pp=1 wp(t)G[R′← Rp(t),T]≈
≈ ∑Pp=1 wp(t + T)δ [R−Rp(t + T)]
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Definitions
Projection using discrete walkers
Consider the equation
A(R)f (R, t) =− ∂ f (R,t)∂ t
Given f (R, t) at time t and Green’s function G(R′← R,T):
f (R, t + T) =∫
G(R′← R,T)f (R, t)dRIf f (R, t) probability distribution → can represent discretely:
f (R, t)≈ ∑Pp=1 wp(t)δ [R−Rp(t)]
Therefore:
f (R, t + T) = ∑Pp=1 wp(t)G[R′← Rp(t),T]≈
≈ ∑Pp=1 wp(t + T)δ [R−Rp(t + T)]
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Definitions
Projection using discrete walkers
Consider the equation
A(R)f (R, t) =− ∂ f (R,t)∂ t
Given f (R, t) at time t and Green’s function G(R′← R,T):
f (R, t + T) =∫
G(R′← R,T)f (R, t)dRIf f (R, t) probability distribution → can represent discretely:
f (R, t)≈ ∑Pp=1 wp(t)δ [R−Rp(t)]
Therefore:
f (R, t + T) = ∑Pp=1 wp(t)G[R′← Rp(t),T]≈
≈ ∑Pp=1 wp(t + T)δ [R−Rp(t + T)]
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
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Definitions
Projection using discrete walkers
Green’s function in DMC
The Green’s function can be used as a transition probability todetermine the evolution of walkers in a stochastic approach
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
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Definitions
Diffusion Monte Carlo
DMC consists in choosing f (R, t) = Φ(R, t)Ψ(R), where Ψ isthe trial wave function and Φ is called the DMC wavefunction
For small T, Green’s function can be split into drift, diffusionand branching
f (R, t) is not a probability distribution unless Φ and Ψ havethe same sign everywhere in space
The fixed-node approximation
Φ is constrained to have the same nodes as ΨThe resulting Φ is the lowest energy wave function among
those with the same nodal structure as ΨTherefore DMC is always better than VMC
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Definitions
Diffusion Monte Carlo
DMC consists in choosing f (R, t) = Φ(R, t)Ψ(R), where Ψ isthe trial wave function and Φ is called the DMC wavefunction
For small T, Green’s function can be split into drift, diffusionand branching
f (R, t) is not a probability distribution unless Φ and Ψ havethe same sign everywhere in space
The fixed-node approximation
Φ is constrained to have the same nodes as ΨThe resulting Φ is the lowest energy wave function among
those with the same nodal structure as ΨTherefore DMC is always better than VMC
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Definitions
Diffusion Monte Carlo
DMC consists in choosing f (R, t) = Φ(R, t)Ψ(R), where Ψ isthe trial wave function and Φ is called the DMC wavefunction
For small T, Green’s function can be split into drift, diffusionand branching
f (R, t) is not a probability distribution unless Φ and Ψ havethe same sign everywhere in space
The fixed-node approximation
Φ is constrained to have the same nodes as ΨThe resulting Φ is the lowest energy wave function among
those with the same nodal structure as ΨTherefore DMC is always better than VMC
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Definitions
Diffusion Monte Carlo
DMC consists in choosing f (R, t) = Φ(R, t)Ψ(R), where Ψ isthe trial wave function and Φ is called the DMC wavefunction
For small T, Green’s function can be split into drift, diffusionand branching
f (R, t) is not a probability distribution unless Φ and Ψ havethe same sign everywhere in space
The fixed-node approximation
Φ is constrained to have the same nodes as ΨThe resulting Φ is the lowest energy wave function among
those with the same nodal structure as ΨTherefore DMC is always better than VMC
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Definitions
The DMC algorithm
Start from P walkers {R0,α}Pα=1 distributed according to
|Ψ(R)|2 (from VMC)DMC evolution of the walkers:
Drift-diffusion: move Ri,α → R′i,αBranching: define weight wi,α→ configurations breed/die according to branching factorw′i,α/wi,α→ variable number of walkers Pi
Equilibrate the walkers until we reach infinite-time limit→ look at Ei = ∑
Piα=1 wα,iEα,i/∑
Piα=1 wα,i
Accumulate data after equilibration to improve statistics ofresult
DMC mixed estimator
〈A〉DMC = limt→∞〈Ψ|A|Φ(t)〉/〈Ψ|Φ(t)〉Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Definitions
The DMC algorithm
Start from P walkers {R0,α}Pα=1 distributed according to
|Ψ(R)|2 (from VMC)DMC evolution of the walkers:
Drift-diffusion: move Ri,α → R′i,αBranching: define weight wi,α→ configurations breed/die according to branching factorw′i,α/wi,α→ variable number of walkers Pi
Equilibrate the walkers until we reach infinite-time limit→ look at Ei = ∑
Piα=1 wα,iEα,i/∑
Piα=1 wα,i
Accumulate data after equilibration to improve statistics ofresult
DMC mixed estimator
〈A〉DMC = limt→∞〈Ψ|A|Φ(t)〉/〈Ψ|Φ(t)〉Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Definitions
The DMC algorithm
Start from P walkers {R0,α}Pα=1 distributed according to
|Ψ(R)|2 (from VMC)DMC evolution of the walkers:
Drift-diffusion: move Ri,α → R′i,αBranching: define weight wi,α→ configurations breed/die according to branching factorw′i,α/wi,α→ variable number of walkers Pi
Equilibrate the walkers until we reach infinite-time limit→ look at Ei = ∑
Piα=1 wα,iEα,i/∑
Piα=1 wα,i
Accumulate data after equilibration to improve statistics ofresult
DMC mixed estimator
〈A〉DMC = limt→∞〈Ψ|A|Φ(t)〉/〈Ψ|Φ(t)〉Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Definitions
The DMC algorithm
Start from P walkers {R0,α}Pα=1 distributed according to
|Ψ(R)|2 (from VMC)DMC evolution of the walkers:
Drift-diffusion: move Ri,α → R′i,αBranching: define weight wi,α→ configurations breed/die according to branching factorw′i,α/wi,α→ variable number of walkers Pi
Equilibrate the walkers until we reach infinite-time limit→ look at Ei = ∑
Piα=1 wα,iEα,i/∑
Piα=1 wα,i
Accumulate data after equilibration to improve statistics ofresult
DMC mixed estimator
〈A〉DMC = limt→∞〈Ψ|A|Φ(t)〉/〈Ψ|Φ(t)〉Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Definitions
The DMC algorithm
Start from P walkers {R0,α}Pα=1 distributed according to
|Ψ(R)|2 (from VMC)DMC evolution of the walkers:
Drift-diffusion: move Ri,α → R′i,αBranching: define weight wi,α→ configurations breed/die according to branching factorw′i,α/wi,α→ variable number of walkers Pi
Equilibrate the walkers until we reach infinite-time limit→ look at Ei = ∑
Piα=1 wα,iEα,i/∑
Piα=1 wα,i
Accumulate data after equilibration to improve statistics ofresult
DMC mixed estimator
〈A〉DMC = limt→∞〈Ψ|A|Φ(t)〉/〈Ψ|Φ(t)〉Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Definitions
The DMC algorithm
Start from P walkers {R0,α}Pα=1 distributed according to
|Ψ(R)|2 (from VMC)DMC evolution of the walkers:
Drift-diffusion: move Ri,α → R′i,αBranching: define weight wi,α→ configurations breed/die according to branching factorw′i,α/wi,α→ variable number of walkers Pi
Equilibrate the walkers until we reach infinite-time limit→ look at Ei = ∑
Piα=1 wα,iEα,i/∑
Piα=1 wα,i
Accumulate data after equilibration to improve statistics ofresult
DMC mixed estimator
〈A〉DMC = limt→∞〈Ψ|A|Φ(t)〉/〈Ψ|Φ(t)〉Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Definitions
The DMC algorithm
Start from P walkers {R0,α}Pα=1 distributed according to
|Ψ(R)|2 (from VMC)DMC evolution of the walkers:
Drift-diffusion: move Ri,α → R′i,αBranching: define weight wi,α→ configurations breed/die according to branching factorw′i,α/wi,α→ variable number of walkers Pi
Equilibrate the walkers until we reach infinite-time limit→ look at Ei = ∑
Piα=1 wα,iEα,i/∑
Piα=1 wα,i
Accumulate data after equilibration to improve statistics ofresult
DMC mixed estimator
〈A〉DMC = limt→∞〈Ψ|A|Φ(t)〉/〈Ψ|Φ(t)〉Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Definitions
The DMC algorithm
Start from P walkers {R0,α}Pα=1 distributed according to
|Ψ(R)|2 (from VMC)DMC evolution of the walkers:
Drift-diffusion: move Ri,α → R′i,αBranching: define weight wi,α→ configurations breed/die according to branching factorw′i,α/wi,α→ variable number of walkers Pi
Equilibrate the walkers until we reach infinite-time limit→ look at Ei = ∑
Piα=1 wα,iEα,i/∑
Piα=1 wα,i
Accumulate data after equilibration to improve statistics ofresult
DMC mixed estimator
〈A〉DMC = limt→∞〈Ψ|A|Φ(t)〉/〈Ψ|Φ(t)〉Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Definitions
The DMC algorithm
Start from P walkers {R0,α}Pα=1 distributed according to
|Ψ(R)|2 (from VMC)DMC evolution of the walkers:
Drift-diffusion: move Ri,α → R′i,αBranching: define weight wi,α→ configurations breed/die according to branching factorw′i,α/wi,α→ variable number of walkers Pi
Equilibrate the walkers until we reach infinite-time limit→ look at Ei = ∑
Piα=1 wα,iEα,i/∑
Piα=1 wα,i
Accumulate data after equilibration to improve statistics ofresult
DMC mixed estimator
〈A〉DMC = limt→∞〈Ψ|A|Φ(t)〉/〈Ψ|Φ(t)〉Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
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Definitions
Calculation of the energy in DMC
0 500 1000 1500
Number of moves
-55.8
-55.7
-55.6
-55.5
-55.4
Local energy (Ha)Reference energyBest estimate
0 500 1000 15001000
1100
1200
1300
1400
1500
POPULATION
ED ≈ E =∑
Mi=1 WiEi
∑Mi=1 Wi
; σ2E ≈ σ
2E =
∑Mi=1 Wi
(Ei− E
)2
M(
∑Mi=1 Wi− ∑
Mi=1 W2
i
∑Mi=1 Wi
)Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Definitions
Sources of error in DMC
Timestep: we have assumed that T is small→ must extrapolate to zero timestep to obtain a reliable result→ cannot use timestep to improve statistics
Population: Φ is represented by set of configurations→ must use sufficient configurations to represent it accurately→ possible to extrapolate to infinite population
Fixed-node error: only limitation of DMC→ ED is still variational (very important!)→ can be reduced by using Ψ with better nodes
Locality approximation: from pseudopotentials→ ED non-variational→ goes away with good Ψ
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Definitions
Sources of error in DMC
Timestep: we have assumed that T is small→ must extrapolate to zero timestep to obtain a reliable result→ cannot use timestep to improve statistics
Population: Φ is represented by set of configurations→ must use sufficient configurations to represent it accurately→ possible to extrapolate to infinite population
Fixed-node error: only limitation of DMC→ ED is still variational (very important!)→ can be reduced by using Ψ with better nodes
Locality approximation: from pseudopotentials→ ED non-variational→ goes away with good Ψ
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Definitions
Sources of error in DMC
Timestep: we have assumed that T is small→ must extrapolate to zero timestep to obtain a reliable result→ cannot use timestep to improve statistics
Population: Φ is represented by set of configurations→ must use sufficient configurations to represent it accurately→ possible to extrapolate to infinite population
Fixed-node error: only limitation of DMC→ ED is still variational (very important!)→ can be reduced by using Ψ with better nodes
Locality approximation: from pseudopotentials→ ED non-variational→ goes away with good Ψ
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Definitions
Sources of error in DMC
Timestep: we have assumed that T is small→ must extrapolate to zero timestep to obtain a reliable result→ cannot use timestep to improve statistics
Population: Φ is represented by set of configurations→ must use sufficient configurations to represent it accurately→ possible to extrapolate to infinite population
Fixed-node error: only limitation of DMC→ ED is still variational (very important!)→ can be reduced by using Ψ with better nodes
Locality approximation: from pseudopotentials→ ED non-variational→ goes away with good Ψ
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Definitions
Sources of error in DMC
Timestep: we have assumed that T is small→ must extrapolate to zero timestep to obtain a reliable result→ cannot use timestep to improve statistics
Population: Φ is represented by set of configurations→ must use sufficient configurations to represent it accurately→ possible to extrapolate to infinite population
Fixed-node error: only limitation of DMC→ ED is still variational (very important!)→ can be reduced by using Ψ with better nodes
Locality approximation: from pseudopotentials→ ED non-variational→ goes away with good Ψ
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Definitions
Sources of error in DMC
Timestep: we have assumed that T is small→ must extrapolate to zero timestep to obtain a reliable result→ cannot use timestep to improve statistics
Population: Φ is represented by set of configurations→ must use sufficient configurations to represent it accurately→ possible to extrapolate to infinite population
Fixed-node error: only limitation of DMC→ ED is still variational (very important!)→ can be reduced by using Ψ with better nodes
Locality approximation: from pseudopotentials→ ED non-variational→ goes away with good Ψ
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Definitions
Sources of error in DMC
Timestep: we have assumed that T is small→ must extrapolate to zero timestep to obtain a reliable result→ cannot use timestep to improve statistics
Population: Φ is represented by set of configurations→ must use sufficient configurations to represent it accurately→ possible to extrapolate to infinite population
Fixed-node error: only limitation of DMC→ ED is still variational (very important!)→ can be reduced by using Ψ with better nodes
Locality approximation: from pseudopotentials→ ED non-variational→ goes away with good Ψ
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Definitions
Sources of error in DMC
Timestep: we have assumed that T is small→ must extrapolate to zero timestep to obtain a reliable result→ cannot use timestep to improve statistics
Population: Φ is represented by set of configurations→ must use sufficient configurations to represent it accurately→ possible to extrapolate to infinite population
Fixed-node error: only limitation of DMC→ ED is still variational (very important!)→ can be reduced by using Ψ with better nodes
Locality approximation: from pseudopotentials→ ED non-variational→ goes away with good Ψ
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Definitions
Sources of error in DMC
Timestep: we have assumed that T is small→ must extrapolate to zero timestep to obtain a reliable result→ cannot use timestep to improve statistics
Population: Φ is represented by set of configurations→ must use sufficient configurations to represent it accurately→ possible to extrapolate to infinite population
Fixed-node error: only limitation of DMC→ ED is still variational (very important!)→ can be reduced by using Ψ with better nodes
Locality approximation: from pseudopotentials→ ED non-variational→ goes away with good Ψ
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Definitions
Sources of error in DMC
Timestep: we have assumed that T is small→ must extrapolate to zero timestep to obtain a reliable result→ cannot use timestep to improve statistics
Population: Φ is represented by set of configurations→ must use sufficient configurations to represent it accurately→ possible to extrapolate to infinite population
Fixed-node error: only limitation of DMC→ ED is still variational (very important!)→ can be reduced by using Ψ with better nodes
Locality approximation: from pseudopotentials→ ED non-variational→ goes away with good Ψ
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Definitions
Sources of error in DMC
Timestep: we have assumed that T is small→ must extrapolate to zero timestep to obtain a reliable result→ cannot use timestep to improve statistics
Population: Φ is represented by set of configurations→ must use sufficient configurations to represent it accurately→ possible to extrapolate to infinite population
Fixed-node error: only limitation of DMC→ ED is still variational (very important!)→ can be reduced by using Ψ with better nodes
Locality approximation: from pseudopotentials→ ED non-variational→ goes away with good Ψ
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Definitions
Sources of error in DMC
Timestep: we have assumed that T is small→ must extrapolate to zero timestep to obtain a reliable result→ cannot use timestep to improve statistics
Population: Φ is represented by set of configurations→ must use sufficient configurations to represent it accurately→ possible to extrapolate to infinite population
Fixed-node error: only limitation of DMC→ ED is still variational (very important!)→ can be reduced by using Ψ with better nodes
Locality approximation: from pseudopotentials→ ED non-variational→ goes away with good Ψ
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
The normal distribution
The normal distribution is D(E; E,σ) = 1√2πσ
exp[− (E−E)2
2σ2
]The probability of the E being in an interval (A,B) is
P(A < E < B) = f(
B−Eσ
)− f(
A−Eσ
)f (x) = 1√
2π
∫ x−∞
exp(−y2/2
)dy
One-sigma interval (E−σ , E + σ) → 68.3% → unreliable
Two-sigma interval (E−2σ , E + 2σ) → 95.4% → reliable
Three-sigma interval (E−3σ , E + 3σ) → 99.7% → veryreliable
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
The normal distribution
The normal distribution is D(E; E,σ) = 1√2πσ
exp[− (E−E)2
2σ2
]The probability of the E being in an interval (A,B) is
P(A < E < B) = f(
B−Eσ
)− f(
A−Eσ
)f (x) = 1√
2π
∫ x−∞
exp(−y2/2
)dy
One-sigma interval (E−σ , E + σ) → 68.3% → unreliable
Two-sigma interval (E−2σ , E + 2σ) → 95.4% → reliable
Three-sigma interval (E−3σ , E + 3σ) → 99.7% → veryreliable
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
The normal distribution
The normal distribution is D(E; E,σ) = 1√2πσ
exp[− (E−E)2
2σ2
]The probability of the E being in an interval (A,B) is
P(A < E < B) = f(
B−Eσ
)− f(
A−Eσ
)f (x) = 1√
2π
∫ x−∞
exp(−y2/2
)dy
One-sigma interval (E−σ , E + σ) → 68.3% → unreliable
Two-sigma interval (E−2σ , E + 2σ) → 95.4% → reliable
Three-sigma interval (E−3σ , E + 3σ) → 99.7% → veryreliable
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
The normal distribution
The normal distribution is D(E; E,σ) = 1√2πσ
exp[− (E−E)2
2σ2
]The probability of the E being in an interval (A,B) is
P(A < E < B) = f(
B−Eσ
)− f(
A−Eσ
)f (x) = 1√
2π
∫ x−∞
exp(−y2/2
)dy
One-sigma interval (E−σ , E + σ) → 68.3% → unreliable
Two-sigma interval (E−2σ , E + 2σ) → 95.4% → reliable
Three-sigma interval (E−3σ , E + 3σ) → 99.7% → veryreliable
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
The normal distribution
The normal distribution is D(E; E,σ) = 1√2πσ
exp[− (E−E)2
2σ2
]The probability of the E being in an interval (A,B) is
P(A < E < B) = f(
B−Eσ
)− f(
A−Eσ
)f (x) = 1√
2π
∫ x−∞
exp(−y2/2
)dy
One-sigma interval (E−σ , E + σ) → 68.3% → unreliable
Two-sigma interval (E−2σ , E + 2σ) → 95.4% → reliable
Three-sigma interval (E−3σ , E + 3σ) → 99.7% → veryreliable
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
The normal distribution
The normal distribution is D(E; E,σ) = 1√2πσ
exp[− (E−E)2
2σ2
]The probability of the E being in an interval (A,B) is
P(A < E < B) = f(
B−Eσ
)− f(
A−Eσ
)f (x) = 1√
2π
∫ x−∞
exp(−y2/2
)dy
One-sigma interval (E−σ , E + σ) → 68.3% → unreliable
Two-sigma interval (E−2σ , E + 2σ) → 95.4% → reliable
Three-sigma interval (E−3σ , E + 3σ) → 99.7% → veryreliable
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
The normal distribution
The normal distribution is D(E; E,σ) = 1√2πσ
exp[− (E−E)2
2σ2
]The probability of the E being in an interval (A,B) is
P(A < E < B) = f(
B−Eσ
)− f(
A−Eσ
)f (x) = 1√
2π
∫ x−∞
exp(−y2/2
)dy
One-sigma interval (E−σ , E + σ) → 68.3% → unreliable
Two-sigma interval (E−2σ , E + 2σ) → 95.4% → reliable
Three-sigma interval (E−3σ , E + 3σ) → 99.7% → veryreliable
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
The normal distribution
Comparison of a Gaussian and the local energy distribution
I I
I I
I I
0.5 0.55E (a.u.)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
P(E
) (a
rbitr
aty
units
)
GaussianVMC histogram
68.3%95.4%99.7%
-σ +σ-2σ
-3σ+2σ
+3σ
From Central Limit Theorem:
The mean energy is exactly normal
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
How to compare quantities with errorbars
Want to find distribution of difference, denoted(E−±σ−
)=(E1±σ1
)−(E2±σ2
)Results in
E− = E1− E2σ2− = σ2
1 + σ22
Example:
Ψ1 gives E1 =−14.66728(2) a.u.Ψ2 gives E2 =−14.66733(7) a.u.Comparison: E− = 0.00005(7) a.u. → 76% chance of E2 < E1→ unreliable!If E2 =−14.66733(2) a.u. instead → E− = 0.00005(3) a.u.→ 95% chance of E2 < E1 → reliable
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
How to compare quantities with errorbars
Want to find distribution of difference, denoted(E−±σ−
)=(E1±σ1
)−(E2±σ2
)Results in
E− = E1− E2σ2− = σ2
1 + σ22
Example:
Ψ1 gives E1 =−14.66728(2) a.u.Ψ2 gives E2 =−14.66733(7) a.u.Comparison: E− = 0.00005(7) a.u. → 76% chance of E2 < E1→ unreliable!If E2 =−14.66733(2) a.u. instead → E− = 0.00005(3) a.u.→ 95% chance of E2 < E1 → reliable
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
How to compare quantities with errorbars
Want to find distribution of difference, denoted(E−±σ−
)=(E1±σ1
)−(E2±σ2
)Results in
E− = E1− E2σ2− = σ2
1 + σ22
Example:
Ψ1 gives E1 =−14.66728(2) a.u.Ψ2 gives E2 =−14.66733(7) a.u.Comparison: E− = 0.00005(7) a.u. → 76% chance of E2 < E1→ unreliable!If E2 =−14.66733(2) a.u. instead → E− = 0.00005(3) a.u.→ 95% chance of E2 < E1 → reliable
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
How to compare quantities with errorbars
Want to find distribution of difference, denoted(E−±σ−
)=(E1±σ1
)−(E2±σ2
)Results in
E− = E1− E2σ2− = σ2
1 + σ22
Example:
Ψ1 gives E1 =−14.66728(2) a.u.Ψ2 gives E2 =−14.66733(7) a.u.Comparison: E− = 0.00005(7) a.u. → 76% chance of E2 < E1→ unreliable!If E2 =−14.66733(2) a.u. instead → E− = 0.00005(3) a.u.→ 95% chance of E2 < E1 → reliable
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
How to compare quantities with errorbars
Want to find distribution of difference, denoted(E−±σ−
)=(E1±σ1
)−(E2±σ2
)Results in
E− = E1− E2σ2− = σ2
1 + σ22
Example:
Ψ1 gives E1 =−14.66728(2) a.u.Ψ2 gives E2 =−14.66733(7) a.u.Comparison: E− = 0.00005(7) a.u. → 76% chance of E2 < E1→ unreliable!If E2 =−14.66733(2) a.u. instead → E− = 0.00005(3) a.u.→ 95% chance of E2 < E1 → reliable
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
How to compare quantities with errorbars
Want to find distribution of difference, denoted(E−±σ−
)=(E1±σ1
)−(E2±σ2
)Results in
E− = E1− E2σ2− = σ2
1 + σ22
Example:
Ψ1 gives E1 =−14.66728(2) a.u.Ψ2 gives E2 =−14.66733(7) a.u.Comparison: E− = 0.00005(7) a.u. → 76% chance of E2 < E1→ unreliable!If E2 =−14.66733(2) a.u. instead → E− = 0.00005(3) a.u.→ 95% chance of E2 < E1 → reliable
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
How to compare quantities with errorbars
Want to find distribution of difference, denoted(E−±σ−
)=(E1±σ1
)−(E2±σ2
)Results in
E− = E1− E2σ2− = σ2
1 + σ22
Example:
Ψ1 gives E1 =−14.66728(2) a.u.Ψ2 gives E2 =−14.66733(7) a.u.Comparison: E− = 0.00005(7) a.u. → 76% chance of E2 < E1→ unreliable!If E2 =−14.66733(2) a.u. instead → E− = 0.00005(3) a.u.→ 95% chance of E2 < E1 → reliable
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
How to compare quantities with errorbars
Want to find distribution of difference, denoted(E−±σ−
)=(E1±σ1
)−(E2±σ2
)Results in
E− = E1− E2σ2− = σ2
1 + σ22
Example:
Ψ1 gives E1 =−14.66728(2) a.u.Ψ2 gives E2 =−14.66733(7) a.u.Comparison: E− = 0.00005(7) a.u. → 76% chance of E2 < E1→ unreliable!If E2 =−14.66733(2) a.u. instead → E− = 0.00005(3) a.u.→ 95% chance of E2 < E1 → reliable
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
How to compare quantities with errorbars
Want to find distribution of difference, denoted(E−±σ−
)=(E1±σ1
)−(E2±σ2
)Results in
E− = E1− E2σ2− = σ2
1 + σ22
Example:
Ψ1 gives E1 =−14.66728(2) a.u.Ψ2 gives E2 =−14.66733(7) a.u.Comparison: E− = 0.00005(7) a.u. → 76% chance of E2 < E1→ unreliable!If E2 =−14.66733(2) a.u. instead → E− = 0.00005(3) a.u.→ 95% chance of E2 < E1 → reliable
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Summary
Reblocking algorithm applied using the reblock utility
Average correlation time τ given in VMC runs andreblock utility
VMC timestep automatically optimized to give a = 50% (donot apply on HEG)
EBEA is the default in both VMC and DMC
DMC statistics monitored using graphit utility
Timestep extrapolation carried out using theextrapolate tau utility
Pablo Lopez Rıos Statistics in Quantum Monte Carlo
IntroductionVMC statisticsDMC statistics
Normally-distributed numbersSummary
Summary
Reblocking algorithm applied using the reblock utility
Average correlation time τ given in VMC runs andreblock utility
VMC timestep automatically optimized to give a = 50% (donot apply on HEG)
EBEA is the default in both VMC and DMC
DMC statistics monitored using graphit utility
Timestep extrapolation carried out using theextrapolate tau utility
Pablo Lopez Rıos Statistics in Quantum Monte Carlo