ALGORITHMS FOR FINITE TEMPERATURE QMC

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1 ALGORITHMS FOR FINITE TEMPERATURE QMC Bryan Clark Station Q QMC INT Conference: June 12, 2013

2 Current de-facto standard for fermions at finite temperature Restricted Path Integral Monte Carlo! * see PIMC++

3 Current de-facto standard for fermions at finite temperature Restricted Path Integral Monte Carlo! * see PIMC++ RPIMC is a black box that (approximately) gives out samples R with probability (R, R) =hr exp[ H] Ri 1. Pick bead i at random R3 2. Move it to spot with probability Ri new hr i 1 exp[ H] Ri new ihri new exp[ H] R i+1 i hr i 1 exp[ H] R i ihr i exp[ H] R i+1 i R R2 R1

4 Current de-facto standard for fermions at finite temperature Restricted Path Integral Monte Carlo! * see PIMC++ But there are some problems Ceperley

5 Current de-facto standard for fermions at finite temperature Restricted Path Integral Monte Carlo! * see PIMC++ But there are some problems... Militzer

6 Current de-facto standard for fermions at finite temperature Restricted Path Integral Monte Carlo! * see PIMC++ But there are some problems... Need to guess a trial many body density matrix (seems harder then guessing a trial wave-function) Ergodicity at low temperature

7 The Solution: Stop using Path Integral Monte Carlo What else could we use then? ground state Projector QMC?? Variational Monte Carlo?? Fixed Node Diffusion Monte Carlo??

8 The Solution: Stop using Path Integral Monte Carlo What else could we use then? ground state Projector QMC?? Variational Monte Carlo?? Fixed Node Diffusion Monte Carlo?? CEIMC? Finite temperature protons - ground state fermions Invented because PIMC gets stuck

9 The Solution: Stop using Path Integral Monte Carlo What else could we use then? ground state Projector QMC?? Variational Monte Carlo?? Fixed Node Diffusion Monte Carlo??

10 The Solution: Stop using Path Integral Monte Carlo What else could we use then? finite temperature ground state Projector QMC?? Variational Monte Carlo?? Fixed Node Diffusion Monte Carlo?? Today s goal (a work in progress): Show you new algorithms we ve been developing for finite temperature calculations based on projector QMC.

11 The Solution: Stop using Path Integral Monte Carlo What else could we use then? finite temperature ground state Projector QMC?? Part I Variational Monte Carlo?? Part II Fixed Node Diffusion Monte Carlo?? Part III Today s goal (a work in progress): Show you new algorithms we ve been developing for finite temperature calculations based on projector QMC.

12 Part I: Projector QMC at Finite T

13 Projector QMC exp[ H] exp[ H] Ri exp[ H] exp[ H/2] Samples exp[ H/2] Ri In the limit of large beta: Samples 0i exp[ H] 3n dimensional space

14 Projector QMC Ri exp[ H] Ri Samples exp[ H/2] Ri In the limit of large beta: Samples 0i PQMC exp[ H] exp[ H] exp[ H] exp[ exp[ H] H/2] Ri 3n dimensional space

15 FT Projector QMC Ri R 0 i! Ri exp[ PQMC h /2H] Ri Why? Select R 0 i with prob hr 0 exp[ /2H] Ri 2 How? Also used by METTS

16 Why? - Proof via PIMC 1. Pick bead i at random 2. Move it to spot with probability Ri new hr i 1 exp[ H] Ri new ihri new exp[ H] R i+1 i hr i 1 exp[ H] R i ihr i exp[ H] R i+1 i R3 R R2 R1

17 Why? - Proof via PIMC 1. Pick bead i at random 2. Move it to spot with probability Ri new hr i 1 exp[ H/2] Ri new i 2 hr i 1 exp[ H/2] R i i 2

18 Why? - Proof via PIMC 1. Move bead 0 it to spot with probability R0 new hr 1 exp[ H/2] R0 new i 2 hr 1 exp[ H/2] R 0 i 2 R new 2. Move bead 1 it to spot with 1 probability hr 0 exp[ H/2] R1 new i 2 hr 0 exp[ H/2] R 1 i 2

19 Why? - Proof via PIMC 1. Move bead 0 it to spot with probability R0 new hr 1 exp[ H/2] R0 new i 2 hr 1 exp[ H/2] R 0 i times 2. Move bead 1 it to spot with probability R1 new hr 0 exp[ H/2] R1 new i 2 hr 0 exp[ H/2] R 1 i times.

20 Why? - Proof via PIMC 1. Sample R 0 with probability hr 1 exp[ H/2] R 0 i 2 2. Sample R with probability hr 1 exp[ H/2] R 0 i 2 1

21 Why? - Proof via PIMC 1. Sample R 0 with probability hr 1 exp[ H/2] R 0 i 2 2. Sample R with probability hr 1 exp[ H/2] R 0 i 2 1 Ri R 0 i! Ri exp[ PQMC h /2H] Ri Why? Select R 0 i with prob hr 0 exp[ /2H] Ri 2

22 FT Projector QMC Ri R 0 i! Ri exp[ PQMC h /2H] Ri Why? Select R 0 i with prob hr 0 exp[ /2H] Ri 2 How?

23 How? Select R 0 i with prob hr 0 exp[ /2H] Ri 2 Option 1 Square the weight of each walker. Option 2 Overlap of two simulations. Option 3 Sample with importance sampling. h T RihR exp[ H/2] R 0 i

24 Select R 0 i with prob hr 0 exp[ /2H] Ri 2 Option 1 Psi Systematically Biased Basis Elements But not too bad when FCIQMC works Psi exp[ H] Ri Basis Elements exp[ H] exp[ H] exp[ H] exp[ H] Psi*Psi Basis Elements

25 A Test System Heisenberg Model: H = X ij i j Locked to Sz=0 sector 4x4 model Tractable exactly Like a fermion system (in second quantization) No sign problem - show algorithms

26 1 core Squaring a snapshot ~100,000 walkers

27 Psi Select R 0 i with prob hr 0 exp[ /2H] Ri 2 Option Psi 0.5 Ri Exact but lose statistical accuracy But not too bad when FCIQMC works exp[ H] exp[ H] exp[ H] exp[ H] Basis Elements Similar ideas: - Bilinear Sampling: Kalos, Shiwei - DMQMC: Foulkes - Improvements with Matt Hastings exp[ H] exp[ H] Psi*Psi Basis Elements exp[ H] exp[ H] exp[ H] 0 Basis Elements 0.1 exp[ H] Ri

29 Select R 0 i with prob hr 0 exp[ /2H] Ri 2 Option 3 Sample h T RihR exp[ H/2] R 0 i with importance sampling Just like diffusion Monte Carlo Need to choose T i exp[ H/2] R 0 i `Mixed Estimator Q: How can we pick? A: Later in the talk. (maybe) correct by forward walking..

30 How? Select R 0 i with prob hr 0 exp[ /2H] Ri 2 Option 1 Square the weight of each walker. Systematically biased but not too bad. Option 2 Overlap of two simulations. Larger statistical errors but not too bad. Option 3 Sample with importance sampling. Mixed Estimator h T RihR exp[ H/2] R 0 i

31 (DMC) Problems Time step error: Removable in the lattice population bias? There is none here. The population bias comes from different steps having an adjusted S. You can use this to fix the population bias at T=0 Squaring bias / statistical problems /mixed estimator Sign problem: Prevents big systems

32 (DMC) Problems Time step error: Removable in the lattice population bias? There is none here. The population bias comes from different steps having an adjusted S. You can use this to fix the population bias at T=0 Squaring bias / statistical problems /mixed estimator Sign problem: Prevents big systems

33 (DMC) Problems Time step error: Removable in the lattice population bias? There is none here. The population bias comes from different steps having an adjusted S. You can use this to fix the population bias at T=0 Squaring bias / statistical problems /mixed estimator Sign problem: Prevents big systems

34 (DMC) Problems Time step error: Removable in the lattice population bias? There is none here. The population bias comes from different steps having an adjusted S. You can use this to fix the population bias at T=0 Squaring bias / statistical problems /mixed estimator Sign problem: Prevents big systems

35 Part I: Projector QMC at Finite T The good No ergodicity problem No guessing trial density matrix Annihilation attenuates sign problem Everyone seems to like DMC more then PIMC The bad There s a sign problem Mixed estimator problem

36 Part II: Variational Monte Carlo at Finite T

37 VMC at Finite T Ri R 0 i! Ri h exp[ /2H] Ri How? Select R 0 i with prob hr 0 exp[ /2H] Ri 2 How?

38 exp[ Ri h /2H] Ri Stochastic nature in PQMC gives sign problem. Variational Monte Carlo: Trades sign problem for variational subspace. [ ] could be geminals, SJ, etc. Hilbert Space Ground-state VMC: [ best ] gs [ ] The variational ground state The ground state Similar in spirit to METTS.

39 exp[ Ri h /2H] Ri Stochastic nature in PQMC gives sign problem. Variational Monte Carlo: Trades sign problem for variational subspace. [ ] could be geminals, SJ, etc. Hilbert Space Ground-state VMC: [ best ] gs [ ] The variational ground state The ground state Similar in spirit to METTS.

40 exp[ Ri h /2H] Ri Stochastic nature in PQMC gives sign problem. Variational Monte Carlo: Trades sign problem for variational subspace. [ ] could be geminals, SJ, etc. Hilbert Space Ground-state VMC: [ best ] gs Want [ best] exp[ H/2] Ri [ ] High level approach exp[ H]exp[ H]exp[ H] Ri P exp[ H]P exp[ H]P exp[ H] Ri The variational ground state The ground state Low level approach: Stochastic reconfiguration Similar in spirit to METTS.

41 High level approach exp[ H]exp[ H]exp[ H] Ri P exp[ H]P exp[ H]P exp[ H] Ri Low level approach: Stochastic reconfiguration Schrodinger equation in the tangent space of local variational subspace. Tangent space of [~ ] [~ ]/@ [~ ]/@ [~ ]/@ 2,... H ij h@ [ i ] [ j]i S ij h@ [ i [ j ]i (1 HS 1 ) [ ]i Run VMC on [ ]i Measure H and S Side Note: Option 3 If we can pick T i exp[ H/2] R 0 i

42 VMC at Finite T Ri R 0 i! Ri h exp[ /2H] Ri How? Select R 0 i with prob hr 0 exp[ /2H] Ri 2 How?

43 Select R 0 i with prob hr 0 exp[ /2H] Ri 2 We have an explicit form [ ] exp[ H/2] [ ] Variational Monte Carlo is the black box that takes a wave [ ] hr 0 [ ]i 2 function and samples R with probability. Energy Imaginary Time

44 Two questions: 1. Both steps are done stochastically. Is this good enough? 2. How is the accuracy of the wave-function?

45 Surviving Stochasticity 4x1 Heisenberg Model Beta=2.0 Complete ansatz [ ] Hilbert Space [~c] = X ~ c i

46 4x4 heisenberg model How good? Huse-Elser states [~c] = X Systematically overestimated. c c c i

47 4x4 heisenberg model How good? Huse-Elser states [~c] = X Systematically overestimated. c c c i

48 Part II: Variational Monte Carlo at Finite T The good No ergodicity problem No guessing trial density matrix No exponential variance The bad Variational

49 Part III: Fixed Node Projector Monte Carlo at Finite T

50 Can we do better? FT-VMC: Approximate but statistically stable. FT-DMC: Sign problem The ground state answer to this problem is fixed node. We want a finite temperature variant. R 1 i R 2 i R 3 i R 4 i R 5 i exp[ H/2] exp[ H/2] exp[ H/2] exp[ H/2] exp[ H/2] We propagate by applying G =(1 H) which has a sign problem We need to propagate by another G which gives the same result but has no sign problem.

51 Finite Temperature Fixed Node Lattice G(R, R 0 ; k ) =G(R, R 0 ) if T (R; k ) T (R 0 ; k )G(R, R 0 ) > 0 G(R, R 0 ; k ) =0 T (R; k ) T (R 0 ; k )G(R, R 0 ) < 0 G(R, R; k ) =G(R, R)+ T (R; k ) =hr exp[ if X sign violating k H] R init i T (R 0 ; ) T (R; ) Trial wave-function depends on where you are in the path and where you started. G(R, R) ( ) In the continuum, don t cross a node defined by same trial function.

52 Proof Probability you X are in node j is Pr(j; t) = Pr(i; t 1)Pr(i! j) i [j; t] = X i6=j [i; t 1]G ij + [j; t 1]G jj Let [j; t] = [j; t] = X i6=j2good X i6=j2good [i; t 1]G ij + X G jj [t 1] = [j; t 1]G jj + i6=j2bad [i; t 1]G ij + [j; t 1]G jj [i; t 1] G ij [t 1] + [j; t 1] G jj [t 1] [i; t 1] [j; t 1] G ij

53 So far... Projector QMC which gives you finite temperature results. Variational MC which gives you finite temperature results. Fixed node MC which gives you finite temperature results. These methods may help generate imaginary time-imaginary time correlation functions.

54 Part III: Fixed Node Projector Monte Carlo at Finite T The good No ergodicity problem No guessing trial density matrix No exponential variance Less variational then FT-VMC The bad Variational (with nodes) Mixed estimator

55 Problem: Path Integral Monte Carlo has problems The Solution: Stop using Path Integral Monte Carlo

56 Problem: Path Integral Monte Carlo has problems The Solution: Stop using Path Integral Monte Carlo But... I like Path Integral Monte Carlo Can we fix it?

57 Part IV: Fix Path Integral Monte Carlo

58 One problem: Ergodicity Ceperley Militzer R1 R2 R3 Constraints (R, R 1 ; ) > 0 (R, R 2 ;2 ) > 0 (R, R 3 ;3 ) > 0 R R4 (R, R 4 ;2 ) > 0 (R, R 5 ; ) > 0 R5 Many constraints on R; can t move it.

59 One problem: Ergodicity Ri R 0 i! Ri exp[ /2H] Ri Select R 0 i with prob hr 0 exp[ /2H] Ri 2 R1 R2 R3 Constraints (R, R 1 ; ) > 0 (R, R 2 ;2 ) > 0 (R, R 3 ;3 ) > 0 R R4 (R, R 4 ;2 ) > 0 (R, R 5 ; ) > 0 R5 Many constraints on R; can t move it.

60 (Formal) Solution 1. Select R3 with probability hr 3 exp[ H/2] Ri 2 Ri R 0 i! Ri 2. R3 R Select R 0 i with prob hr 0 exp[ /2H] Ri 2 R1 R2 R3 Constraints (R, R 1 ; ) > 0 (R, R 2 ;2 ) > 0 (R, R 3 ;3 ) > 0 R R4 (R, R 4 ;2 ) > 0 (R, R 5 ; ) > 0 R5 Many constraints on R; can t move it.

61 (Formal) Solution 1. Select R3 with probability hr 3 exp[ H/2] Ri 2 Fix R Run PIMC with everything else for a long time. Pick R3 Ri R 0 i! Ri 2. R3 R Select R 0 i with prob hr 0 exp[ /2H] Ri 2 R1 R2 R3 Constraints (R, R 1 ; ) > 0 (R, R 2 ;2 ) > 0 (R, R 3 ;3 ) > 0 R R4 (R, R 4 ;2 ) > 0 (R, R 5 ; ) > 0 R5 long time: needs to mix Many constraints on R; can t move it.

62 Another problem: nodes Guessing a trial density matrix seems hard. Optimizing a trial density matrix seems hard. Previous attempts: Free fermion nodes Variational Density Matrix (Militzer and Pollock

63 Another approach We need to be able to evaluate whether hr exp[ k H] R 0 i > 0 We ve seen using VMC + stochastic reconfiguration, we start with a variational subspace [ ] and approximately generate the wave-function hr exp[ k H]. This gives us a new nodal constraint for path integrals starting only with a variational subspace. We only need to guess a variational subspace (lots of experience with this). No optimization needed! (at the level of path integrals).

64 3 New Methods Conclusions Finite Temperature Projector QMC Finite Temperature VMC Fixed Node Finite Temperature QMC 2 Improvements to Path Integral Monte Carlo Remove ergodic problems at low T Different nodal constraint Future Better access to imaginary time correlation functions Applications; AFQMC version coming soon.

Diffusion Monte Carlo

Diffusion Monte Carlo Notes for Boulder Summer School 2010 Bryan Clark July 22, 2010 Diffusion Monte Carlo The big idea: VMC is a useful technique, but often we want to sample observables of the true ground