Piecewise Interaction Picture Density Matrix Quantum Monte Carlo
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Piecewise Interaction Picture Density Matrix Quantum Monte Carlo
William Van Benschoten1 and James J. Shepherd1, a)
Department of Chemistry, University of Iowa
(Dated: August 16, 2021)
The density matrix quantum Monte Carlo (DMQMC) set of methods stochastically samples the exact N-
body density matrix for interacting electrons at finite temperature. We introduce a simple modification to
the interaction picture DMQMC method (IP-DMQMC) which overcomes the limitation of only sampling one
inverse temperature point at a time. At the target inverse temperature, instead of ending the simulation,
we incorporate a change of picture away from the interaction picture. The result is a propagator that is
a piecewise function, which uses the interaction picture in the first phase of a simulation, followed by the
arXiv:2108.06252v1 [physics.chem-ph] 13 Aug 2021
application of the Bloch equation once the target inverse temperature is reached. We find that the performance
of this method is similar to or better than the DMQMC and IP-DMQMC algorithms in a variety of molecular
test systems.
I. INTRODUCTION temperature. After being developed to use the initiator
approach,65 which was also adapted from the ground-
Electrons interacting in the presence of a finite tem- state FCIQMC version66 , IP-DMQMC benchmarked the
perature play an important role in many applications in- warm dense electron gas alongside path integral Monte
cluding the study of planetary cores1,2 , plasma physics3,4 , Carlo approaches to obtain a finite-temperature local
laser experiments5 , and condensed phases of matter6,7 . density approximation functional.67–69 In addition to
There has been a recent push to take methods which these successes, IP-DMQMC showed promise in initial
are effective for solving ground state electronic struc- applications to molecular systems70 and its sign problem
ture problems, especially quantum chemical wavefunc- showed to be similar to that of FCIQMC.71
tion methods, and adapting them to treat finite temper- In this work we seek to extend IP-DMQMC by con-
ature. Examples of this include perturbation theories8–12 tinuing the simulation after the target inverse tempera-
and coupled cluster techniques13–21 . Other ab initio ture is reached. We find that continuing to apply the
methods under active development include ft-DFT22–26 Bloch equation as the propagator allows for the rest of
and various flavors of Green’s function methods27–33 the temperature-dependent energy to be found. This
such as self-consistent second-order perturbation theory if possible because IP-DMQMC reaches the exact den-
(GF2) and GW theory. Additionally, embedding the- sity matrix (on average) once it reaches a target tem-
ories, which break the calculation up into an exactly perature. This paper starts with an introduction to
treated subsystem and an approximately treated bath, DMQMC methods followed by a derivation of the new
have been proposed.34–46 There are also a variety of piecewise IP-DMQMC propagation equations (which we
quantum Monte Carlo methods which work with finite call PIP-DMQMC). Next we test PIP-DMQMC for a set
temperature ensembles of electrons, such as path in- of molecular systems making comparison with DMQMC,
tegral Monte Carlo47–54 , determinant quantum Monte IP-DMQMC, and finite temperature full configuration in-
Carlo (DQMC)55,56 , finite temperature auxiliary field teraction (ft-FCI).72 We then explore how PIP-DMQMC
quantum Monte Carlo (ft-AFQMC)57–61 , and Krylov- can be combined with the initiator approximation (i-PIP-
projected quantum Monte Carlo.62 DMQMC) and that these can be used to treat larger sys-
The method we use here, density matrix quantum tems that cannot be exactly diagonalized.
Monte Carlo (DMQMC), stochastically samples the ex-
act N-body density matrix in a finite basis.63 It is the
finite temperature equivalent to FCIQMC64 , which has II. METHODS
been very successful in treating ground-state problems to
FCI accuracy. In the original paper,63 DMQMC calcu- The DMQMC set of methods63 was a generalization
lated thermal quantities for the Heisenberg model includ- of full configuration interaction quantum Monte Carlo
ing the energy and Renyi-2 entropy. Thereafter, inter- (FCIQMC)64 to solving the N-body density matrix at fi-
action picture DMQMC (IP-DMQMC) was introduced
nite temperature: ρ̂(β) = e−β Ĥ (where β = 1/kT ). The
which introduced a change of picture allowing for the
density matrix is written in a finite basis in imaginary
diagonal (and trace) of the density matrix to be sam-
time: ρ̂(β) → fˆ(τ ) = ij fij (τ )|Di ihDj |, where Di are
P
pled much more accurately than DMQMC.65 It does so
by starting by simulating one temperature at a time the orthogonal Slater determinants. One option is to di-
and starting at an approximate density matrix for that agonalize the Hamiltonian to find these coefficients from
a sum over the full configuration interaction states.72 In-
stead, DMQMC proceeds by Monte Carlo which starts by
having walkers, which reside on a given site within a sim-
a) Electronic mail: james-shepherd@uiowa.edu ulation. The goal is to find E(β) = Tr(Ĥ ρ̂(β)/ Tr(ρ̂(β))2
by taking an average over Nβ separate simulations (these a single site. This introduces a population-and-system-
are termed β loops). dependent systematic error to the simulation that is re-
In DMQMC, the simulation is started from a random moved in the limit of an infinite total walker population
distribution of walkers along the diagonal of the density (Nw → ∞). This is typically referred to as being system-
matrix, which represents the exact density matrix at high atically improvable.66,73–78
temperature:
f (τ = 0) = 1 (1) III. RESULTS AND DISCUSSION
Then the Bloch equation is applied in the following form, The method we present here seeks to overcome the
limitation of IP-DMQMC that only one βT can be found
∆τ X at a time. This is significant because this means that
∆fij (τ ) = − (Hik fkj (τ ) + fik (τ )Hkj ), (2)
2 more time is spent in the calculation trying to reach βT
k
than collecting statistics for the calculation.
and then the shift (S) is applied through a transformation Noting that the density matrix in an IP-DMQMC sim-
of Ĥ → T̂ = −(Ĥ − S1). This shift is dynamically ulation is exact at τ = βT :
updated throughout the simulation to control the walker
fij (τ = βT ) = ρij (βT ) (5)
population growth. At every time step for DMQMC,
fij (τ ) is equivalent to sampling ρij (β) at β = τ . Application of the Bloch equation on the IP-DMQMC
In IP-DMQMC, the simulation starts with the known density matrix would allow us to determine ρ̂(βT + δβ).
density matrix based on mean-field theory: Repeated application of the Bloch equation will generally
0
allow the density matrix at β > βT to be found.
f (τ = 0) = e−βT H (3) In piecewise IP-DMQMC (PIP-DMQMC) we therefore
start in the same place as IP-DMQMC:
where H 0 is the diagonal of H. Here, a target βT must be 0
specified. The algorithm to initializes IP-DMQMC was f (τ = 0) = e−βT H (6)
modified slightly for this manuscript and is described in
the Supporting Information. The propagator is then: and the propagator is now combined from IP-DMQMC
with ongoing propagation with the Bloch equation in a
X
0 piecewise fashion:
∆fij (τ ) = −∆τ (−Hik fkj (τ ) + fik (τ )Hkj ), (4) (
k 0
P
−∆τ k (−Hik fkj (τ ) + fik (τ )Hkj ) τ < βT
∆fij (τ ) = ∆τ
P
and then the shift is applied through a transformation of − 2 k (Hik fkj (τ ) + fik (τ )Hkj ) τ ≥ βT
Ĥ → T̂ = −(Ĥ − S1). The matrix H 0 is not shifted. In (7)
IP-DMQMC, fij (τ ) is equivalent to sampling ρij (β) only As before, the shift is applied through a transformation
at at τ = βT . of Ĥ → T̂ = −(Ĥ − S1) (again excluding H 0 in the
DMQMC and IP-DMQMC interpret the steps in the IP-DMQMC phase of propagation).
propagator through spawning, cloning/death, and annhi- An alternative formulation of Eq. (7) is to perform the
lation steps. This is key for the computational effi- propagation in the second phase only asymmetrically:
ciency of the method and are described in more detail (
0
P
in the supplementary −∆τ k (−Hik fkj (τ ) + fik (τ )Hkj ) τ < βT
P information. Spawning stochasti- ∆fij (τ ) = P
cally samples the k taking advantage of the sparsity of −∆τ k fik (τ )Hkj τ ≥ βT
Ĥ and the death/cloning steps control the walker popu- (8)
lation/memory cost. As in the original DMQMC algorithm, symmetric prop-
The initiator approximation was developed in agation should generally result in less stochastic noise
FCIQMC as a way to stabilize the sign problem in the but may also raise the plateau.71 This is consistent with
method,66 and subsequently developed for DMQMC.73 what we found in preliminary calculations: the asym-
In DMQMC, two parameters are introduced: a spawned metric propagation is effective without the initiator ap-
walker threshhold, nadd , and a excitation number cut- proximation and symmetric propagation is effective with
off, nex . The population of sites occupied in the sim- the initiator approximation. In PIP-DMQMC, fij (τ ) is
ulation (the ij indices) is divided into initiators which equivalent to sampling ρij (β) at τ ≥ βT . The key re-
have a population that is ≥ nadd or an excitation num- sult here is that piecewise propagation has the potential
ber ≤ nex . To find the excitation number for the ij site, benefits of IP-DMQMC – skipping initialization on the
the excitations between Di and Dj in |Di ihDj | are used identity matrix – while also allowing for continued prop-
(i.e. if Di was a double excitation of Dj , this would agation using the Bloch equation to higher values of β.
count as 2). Spawning events to sites without walkers We implemented the PIP-DMQMC algorithm in the
are then allowed only if they come from initiator sites or HANDE-QMC79 package. In order to validate PIP-
two spawning events with the same sign arrive at once to DMQMC, we set a target accuracy of 1 millihartree,3
−2.0
0.05
plementary information and come from a variety of
sources.64,71,80–85 Molecular integrals files are generated
Ecorr (β) (Ha)
0.00
with Molpro.86 These are systems which can be exactly
−7.0 −0.05 diagonalized, with a number of density matrix elements
E(β) (Ha)
−0.10 (the square of the determinants in the space) ranging
−12.0 −0.15 from 105 to 108 . As well as PIP-DMQMC, four other
0 5 10 15 20 25 calculations were performed: DMQMC, IP-DMQMC, ft-
β (Ha−1 )
FCI, and a thermal Hartree-Fock like mean-field (THF)73
−18.0
0.003 where the orbitals and eigenvalues are frozen. The cal-
0.002
culation of the THF energy is performed by rewriting
0
Eq. (4) as a mean field density matrix ρ0 (β) = e−βH ,
∆E(β) (Ha)
0.001 then ETHF (β)=Tr(H 0 ρ0 (β))/ Tr(ρ0 (β)). In each of the
DMQMC-type calculations, the walker population was
0.000
set above the plateau for DMQMC71 and the initiator
−0.001 adaptation was not used. The walker numbers used
ranged from 5 × 105 to 107 depending on the approxi-
−0.002
0 5 10 15 20 25 mate systems plateau estimated by running a single cal-
β (Ha−1 ) culation in fixed shift mode. Energies are reported as
ft-FCI ±1 mHa PIP-DMQMC averages over 100 β-loops (Nβ = 100). Other simulation
THF DMQMC IP-DMQMC parameters are included in our data repository and sup-
porting information. Preliminary investigations showed
Figure 1. Finite temperature energies for BeH2 Be/cc-pVDZ that systematic errors were slightly smaller in magnitude
H/DZ from a variety of methods. In the top panel, total ener- for asymmetric propagation (Eq. (8)) in PIP-DMQMC
gies are shown plotted against β. Energies in the inset are cal-
(with no initiator approximation) compared with sym-
culated by taking the difference to THF, which is found with
0 metric propagation (Eq. (7)), though we did not study
ETHF (β) = Tr(H 0 ρ0 (β))/ Tr(ρ0 (β)), where ρ0 (β) = e−βH .
The inset DMQMC and PIP-DMQMC data are re-sampled
this systematically. For these tests, we used asymmetric
every 50 points starting from β = 1. In the bottom panel, propagation.
energy differences with respect to ft-FCI are shown (when A representative example of our set, the BeH2
∆E is positive, ft-FCI is lower in energy). Data here show molecule, is analyzed in Fig. 1. In Fig. 1 (top panel), PIP-
asymmetric PIP-DMQMC propagation (Eq. (8)). DMQMC is found to agree with DMQMC, IP-DMQMC,
and ft-FCI by visual inspection. This was true across
0.002 the whole data set but only corresponds to an accuracy
ft-FCI
±1 mHa
of ∼ 0.2 Ha because the scale is so large. In Fig. 1
0.000 Be (bottom panel), we take a closer look at the system-
∆E(β) (Ha)
BeH2
atic energy differences by subtracting the ft-FCI energy.
−0.002 CO
equilibrium H4 For the range of β values studied, PIP-DMQMC con-
equilibrium H8 sistently achieves equivalent or improved accuracy com-
−0.004 stretched H8
HCN
pared to DMQMC and IP-DMQMC, and its systematic
−0.006 LiF error lies within 1mHa. In particular, in common with
0.002 N2
β (Ha−1 ) IP-DMQMC, it is able to improve upon the ‘shouldering’
Be
0.000 BeH2 seen in this DMQMC line – an increase in stochastic error
at intermediate β values – which comes from loss of in-
∆E(β) (Ha)
CO
equilibrium H4
−0.002 equilibrium H8
formation from the diagonal of the density in DMQMC
stretched H8 (this is which was what prompted the development of
−0.004 HCN IP-DMQMC)65 . PIP-DMQMC generally performs com-
LiF
N2 parably with IP-DMQMC with the error for both meth-
−0.006 ods falling well within the stated 1mHa accuracy target,
0 5 10 15 20 25
though there is a difference in performance at β values
Figure 2. Finite temperature energy differences to ft-FCI between 5 and 10.
are shown for a variety of test systems for PIP-DMQMC (in Figure 2 compares IP-DMQMC and PIP-DMQMC by
the top panel) and IP-DMQMC (in the bottom panel). Data
looking at all of the test set data. In each case, differ-
here show asymmetric PIP-DMQMC propagation (Eq. (8)).
ences were found between the method and ft-FCI and we
which is the standard for high accuracy approaches are paying particular attention to the mHa error thresh-
in QMC for ground state. Our test systems were: hold. Here, we do find that IP-DMQMC has a drift to-
Be/aug-cc-pVQZ, BeH2 Be/cc-pVDZ H/DZ, equilib- wards lower values at intermediate β regimes in most
rium H8 /STO-3G, equilibrium H4 /cc-pVDZ, stretched cases which recovers at large β. Similar to DMQMC, this
H8 /STO-3G, N2 /STO-3G, LiF/STO-3G, CO/STO-3G, appears to be related to how the simulation is initialized;
and HCN/STO-3G. Geometries are provided in the sup- if a particular state is unlikely to be chosen there can4
−60 tems because the critical walker population (also known
0.0050
as a plateau height) rapidly grows beyond what we can
∆E(β) (Ha)
0.0025
−61 store. From our experience and preliminary calculations,
0.0000
E(β) (Ha)
we found that adding to the initiator space when the site
−62 −0.0025
has a population of 3 or greater (nadd = 3) and hav-
−0.0050
0 10 20 30 ing states be initiators if the bra and ket only differ by
−63
β (Ha−1 ) a double excitation (nex = 2) gives reasonable results.
−64 This is consistent with observations in the uniform elec-
−69
0.02 β (Ha−1 ) tron gas.73 As nex causes the simulation to have a plateau
again, we also must make sure we are above the plateau
∆E(β) (Ha)
−71 0.01
0.00 to overcome the sign problem.
E(β) (Ha)
−73 −0.01 In Fig. 3, we show the results of the initiator adapta-
−0.02 tion with PIP-DMQMC (i-PIP-DMQMC) on three sys-
0 10 20 30
−75
β (Ha−1 )
tems: HBCH2 /STO-3G, H2 O/cc-pVDZ, and CH4 /cc-
pVDZ. These calculations were run with a variety of
−77
−33 walker numbers so that we could check for initiator con-
0.050 β (Ha−1 )
vergence, which occurs when the walker number is in-
∆E(β) (Ha)
−35 0.025 creased without the energy changing.66,76,87 In prelimi-
E(β) (Ha)
0.000 nary calculations, we noted relatively little difference be-
−37
tween the two different modes of propagation and decided
−0.025
0 10 20 30 to use symmetric propagation (Eq. (7)) as this had been
−39
β (Ha−1 ) shown to reduce stochastic noise in previous studies.88
−41
In general, we see that the plots for different systems at
0 5 10 15 20 25 30 different walker numbers generally overlay one another.
β (Ha−1 ) The exception to this is low walker populations for CH4 ,
ft-FCI Nw = 5 × 105 Nw = 2 × 107 and this is an example where the population falls below
±1 mHa Nw = 5 × 106 Nw = 5 × 107
the nex = 2 plateau height. Each plot includes an in-
i-FCIQMC Nw = 1 × 107 Nw = 5 × 108
Nw = 5 × 104 set where the difference is taken to the largest walker
population for H2 O and CH4 and to ft-FCI for HBCH2 .
Figure 3. Graphs to show initiator convergence for symmet- For HBCH2 , we show the convergence to ft-FCI in the
ric i-PIP-DMQMC (Eq. (7)) with increasing walker popula- inset. At the highest walker population, the systematic
tions for HBCH2 /STO-3G (top panel), H2 O/cc-pVDZ (mid- error generally falls to ∼ 1mHa for most β values. At
dle panel) and CH4 /cc-pVDZ (bottom panel). Not all walker its peak, the systematic error reaches ∼ 2mHa at β at
populations were run for all systems. Except for methane,
the highest population. For H2 O and CH4 , where we
the walker numbers fully overlap on the main graph. To show
initiator convergence, the inset shows the energy calculated do not have ft-FCI results available, the inset instead
as a difference to the simulation with the largest walker pop- shows the energy difference to the largest walker number
ulation for CH4 and H2 O, and ft-FCI for HBCH2 . The inset (which is taken to be the best estimate we have). We
only shows the two largest walker populations to make trends find that, for high β, the initiator error is well converged.
more visible. Inset data are re-sampled every 50 points start- Between Nw = 5 × 107 and Nw = 5 × 108 , the energy
ing from β = 1. The largest walker population was 5 × 108 for difference is submillihartree above β > 5 and β > 10
H2 O and CH4 . Nβ was adjusted between population to give for H2 O and CH4 respectively. For the smaller β val-
comparable error bars, the details of which are found in the ues for both systems, the stochastic error rises sharply
supplementary information. The Nw = 5 × 107 for HBCH2 to an order of magnitude higher (i.e. ∼ 0.01Ha). Any
and Nw = 5 × 108 for H2 O contains one less β-loop compared
systematic initiator error is difficult to estimate due to
to the other systems matching Nw data sets.
this increase in stochastic error. In general, this shows
be errors due to under-sampling. This would ultimately significant promise for PIP-DMQMC and highlights the
be remedied if enough β loops were run. PIP-DMQMC advantage of having so much data over the β range avail-
appears to mostly remedy this, though we note that PIP- able.
DMQMC does tend to have a dip in energy near its
crossover point (where it switches propagator). Exam-
ples of this are CO, found in the supporting information, IV. CONCLUSIONS
and BeH2 (Fig. 1) respectively. Overall, therefore, we can
conclude that PIP-DMQMC achieves just as good if not
In summary, we have introduced a piecewise general-
better energies than DMQMC and IP-DMQMC across
ization of the interaction picture propagator in density
our test set.
matrix quantum Monte Carlo, which leads to our being
Our next test is to find out whether the initiator ap- able to sample a wide range of temperatures with a single
proximation works well with PIP-DMQMC. The initia- calculation. In proof of concept calculations on a variety
tor approximation is important for treating larger sys- of molecular systems, PIP-DMQMC is generally at least5
as accurate as IP-DMQMC and DMQMC. The result is 5 R. Ernstorfer, M. Harb, C. T. Hebeisen, G. Sciaini, T. Dartiga-
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11 P. K. Jha and S. Hirata, Physical Review E 101, 022106 (2020).
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V. ACKNOWLEDGEMENTS
Computation 15, 6137 (2019).
18 P. Shushkov and T. F. Miller, The Journal of Chemical Physics
Research was supported by the U.S. Department of 151, 134107 (2019).
19 A. F. White and G. Kin-Lic Chan, The Journal of Chemical
Energy, Office of Science, Office of Basic Energy Sciences
Physics 152, 224104 (2020).
Early Career Research Program (ECRP) under Award 20 R. Peng, A. F. White, H. Zhai, and G. Kin-Lic Chan, The
Number DE-SC0021317. Journal of Chemical Physics 155, 044103 (2021).
21 G. Harsha, Y. Xu, T. M. Henderson,
This research also used resources from the University of and G. E. Scuseria,
Iowa and the resources of the National Energy Research arXiv:2107.07922 [cond-mat, physics:physics] (2021), arXiv:
2107.07922.
Scientific Computing Center, a DOE Office of Science 22 V. V. Karasiev, T. Sjostrom, and S. B. Trickey, Physical Review
User Facility supported by the Office of Science of the B 86, 115101 (2012).
U.S. Department of Energy under Contract No. DE- 23 J. A. Ellis, L. Fiedler, G. A. Popoola, N. A. Modine, J. A.
AC02-05CH11231 (computer time for calculations only). Stephens, A. P. Thompson, A. Cangi, and S. Rajamanickam,
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24 S. Pittalis, C. R. Proetto, A. Floris, A. Sanna, C. Bersier,
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be queued for public release after the manuscript is pub- 163001 (2011).
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26 A. Pribram-Jones, P. E. Grabowski, and K. Burke, Physical
the calculations used, files will be deposited with Iowa
Review Letters 116, 233001 (2016).
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inserted at production]. Journal of Chemical Theory and Computation 12, 2250 (2016).
28 A. R. Welden, A. A. Rusakov, and D. Zgid, The Journal of
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29 J. Kas and J. Rehr, Physical Review Letters 119, 176403 (2017).
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31 D. Neuhauser, R. Baer, and D. Zgid, Journal of Chemical Theory
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32 J. Gu, J. Chen, Y. Wang, and X.-G. Zhang, Computer Physics
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33 J. Li, M. Wallerberger, N. Chikano, C.-N. Yeh, E. Gull, and
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