Variational Monte Carlo

Stochastic multireference perturbation theory

Auxiliary field QMC

Outline

Sampling and the sign problem in AFQMC

Reducing noise using selected CI wave functions

Benchmark results

Jastrow symmetry projected states in VMC

Auxiliary field QMC

Variational MC

Use in projection QMC

A different take on projection QMC

Projection QMC methods:

e^{-\tau (\hat{H}-E_0)}|\psi\rangle = c_0|\Psi_0\rangle + c_1 e^{-\Delta E_1\tau}|\Psi_1\rangle+\dots

e^{-\tau (\hat{H}-E_0)}|\psi\rangle = c_0|\Psi_0\rangle + c_1 e^{-\Delta E_1\tau}|\Psi_1\rangle+\dots

|\psi\rangle = c_0|\Psi_0\rangle + c_1|\Psi_1\rangle +\dots

|\psi\rangle = c_0|\Psi_0\rangle + c_1|\Psi_1\rangle +\dots

Better $|\psi\rangle$ approximates $|\Psi_0\rangle$ , faster the convergence with $\tau$

E(\tau)=\dfrac{\langle\psi_l|\hat{H}e^{-\tau \hat{H}}|\psi_r\rangle}{\langle\psi_l|e^{-\tau \hat{H}}|\psi_r\rangle}

E(\tau)=\dfrac{\langle\psi_l|\hat{H}e^{-\tau \hat{H}}|\psi_r\rangle}{\langle\psi_l|e^{-\tau \hat{H}}|\psi_r\rangle}

Mixed energy estimator:

Trial states: Multi-Slater, CCSD, Jastrow, MPS, ...

Sign problem worsens exponentially with $\tau$

Sampling in AFQMC

\hat{H} = \hat{K} + \hat{V} = t_{pr} \hat{a}_{p}^{\dagger}\hat{a}_r + \frac{1}{2}v_{prqs}\hat{a}_{p}^{\dagger}\hat{a}_r\hat{a}_{q}^{\dagger}\hat{a}_{s}

\hat{H} = \hat{K} + \hat{V} = t_{pr} \hat{a}_{p}^{\dagger}\hat{a}_r + \frac{1}{2}v_{prqs}\hat{a}_{p}^{\dagger}\hat{a}_r\hat{a}_{q}^{\dagger}\hat{a}_{s}

Exponentiating $\hat{H}$ : $[\hat{K}, \hat{V}] \neq 0$

e^{-\tau\hat{H}}\approx\left(e^{-\frac{\tau}{N}\frac{\hat{K}}{2}}e^{-\frac{\tau}{N}\hat{V}}e^{-\frac{\tau}{N}\frac{\hat{K}}{2}}\right)^N

e^{-\tau\hat{H}}\approx\left(e^{-\frac{\tau}{N}\frac{\hat{K}}{2}}e^{-\frac{\tau}{N}\hat{V}}e^{-\frac{\tau}{N}\frac{\hat{K}}{2}}\right)^N

Exponentiating $\hat{K}$ : orbital transformation

e^{t_{pr}\hat{a}_p^{\dagger}\hat{a}_r}|\phi\rangle=|\phi'\rangle

e^{t_{pr}\hat{a}_p^{\dagger}\hat{a}_r}|\phi\rangle=|\phi'\rangle

where $|\phi\rangle$ and $|\phi'\rangle$ are nonorthogonal determinants.

Exponentiating $\hat{V} = \frac{1}{2}\sum_{\gamma} \left(L^{\gamma}_{pr}\hat{a}_p^{\dagger}\hat{a}_r\right)^2$ :

e^{-\frac{\hat{L}_{\gamma}^2}{2}} = \int \frac{dx_{\gamma}}{\sqrt{2\pi}}\ e^{\frac{-x_{\gamma}^2}{2}}e^{ix_{\gamma}\hat{L}_{\gamma}}

e^{-\frac{\hat{L}_{\gamma}^2}{2}} = \int \frac{dx_{\gamma}}{\sqrt{2\pi}}\ e^{\frac{-x_{\gamma}^2}{2}}e^{ix_{\gamma}\hat{L}_{\gamma}}

$x_{\gamma}$ : auxiliary field

Motta and Zhang (2017), 1711.02242

(Thouless, 1960)

(Stratonovich, 1957)

E(\tau) = \dfrac{\langle\psi_l|\hat{H}e^{-\tau \hat{H}}|\psi_r\rangle}{\langle\psi_l|e^{-\tau \hat{H}}|\psi_r\rangle} \approx \dfrac{\int\ dX p(X)\langle\psi_l|\hat{H}\hat{\mathcal{B}}(X)|\psi_r\rangle}{\int\ dX p(X)\langle\psi_l|\hat{\mathcal{B}}(X)|\psi_r\rangle}

E(\tau) = \dfrac{\langle\psi_l|\hat{H}e^{-\tau \hat{H}}|\psi_r\rangle}{\langle\psi_l|e^{-\tau \hat{H}}|\psi_r\rangle} \approx \dfrac{\int\ dX p(X)\langle\psi_l|\hat{H}\hat{\mathcal{B}}(X)|\psi_r\rangle}{\int\ dX p(X)\langle\psi_l|\hat{\mathcal{B}}(X)|\psi_r\rangle}

Sample Gaussian auxiliary fields $X$ , propagate, and measure

E(\tau)\approx\dfrac{\sum_i\langle\psi_l|\hat{H}|\phi_i\rangle}{\sum_i\langle\psi_l|\phi_i\rangle}

E(\tau)\approx\dfrac{\sum_i\langle\psi_l|\hat{H}|\phi_i\rangle}{\sum_i\langle\psi_l|\phi_i\rangle}

CCSD as $|\psi_r\rangle$ : sampling Slater determinants from CCSD

|\psi_r\rangle = \exp\left(t_{ikjl}\hat{a}_i^{\dagger}\hat{a}_k\hat{a}_j^{\dagger}\hat{a}_l\right)\exp\left(t_{ik}\hat{a}_i^{\dagger}\hat{a}_k\right)|\phi_0\rangle

|\psi_r\rangle = \exp\left(t_{ikjl}\hat{a}_i^{\dagger}\hat{a}_k\hat{a}_j^{\dagger}\hat{a}_l\right)\exp\left(t_{ik}\hat{a}_i^{\dagger}\hat{a}_k\right)|\phi_0\rangle

commuting ph excitations $\rightarrow$ no Trotter error

$(\text{H}_2\text{O})_2$ , (16e, 80o)

The sign problem

\text{Var}\left(\dfrac{\overline{N}}{\overline{D}}\right) \approx \dfrac{\text{Var}(\overline{N})}{\overline{D}^2} + \dfrac{\overline{N}^2\text{Var}(\overline{D})}{\overline{D}^4} - 2\dfrac{\overline{N}\text{Cov}(\overline{N},\overline{D})}{\overline{D}^3}

\text{Var}\left(\dfrac{\overline{N}}{\overline{D}}\right) \approx \dfrac{\text{Var}(\overline{N})}{\overline{D}^2} + \dfrac{\overline{N}^2\text{Var}(\overline{D})}{\overline{D}^4} - 2\dfrac{\overline{N}\text{Cov}(\overline{N},\overline{D})}{\overline{D}^3}

E(\tau)\approx\dfrac{\sum_i\langle\psi_l|\hat{H}|\phi_i\rangle}{\sum_i\langle\psi_l|\phi_i\rangle} = \dfrac{\overline{N}}{\overline{D}}

E(\tau)\approx\dfrac{\sum_i\langle\psi_l|\hat{H}|\phi_i\rangle}{\sum_i\langle\psi_l|\phi_i\rangle} = \dfrac{\overline{N}}{\overline{D}}

Contour shift:

e^{-\frac{y^2}{2}} = \int \frac{dx}{\sqrt{2\pi}}\ e^{\frac{-x^2}{2}+ixy}

e^{-\frac{y^2}{2}} = \int \frac{dx}{\sqrt{2\pi}}\ e^{\frac{-x^2}{2}+ixy}

x\rightarrow x+iy

x\rightarrow x+iy

x_{\gamma} \rightarrow x_{\gamma} + i \sqrt{\tau}\langle\hat{L}_{\gamma}\rangle

x_{\gamma} \rightarrow x_{\gamma} + i \sqrt{\tau}\langle\hat{L}_{\gamma}\rangle

In AFQMC:

Baer, Head-Gordon, Neuhauser (1998)

Selected CI trial state as $|\psi_l\rangle$

Zero variance principle: If $|\psi_l\rangle$ is the exact ground state, then $N$ and $D$ are perfectly correlated, $\langle\psi_0|\hat{H}|\phi_i\rangle = E_0 \langle\psi_0|\phi_i\rangle$ , and the energy estimator has zero variance.

More accurate $|\psi_l\rangle\ \rightarrow\$ higher $\text{Cov}(N, D)$

E(\tau)\approx\dfrac{\sum_i\langle\psi_l|\hat{H}|\phi_i\rangle}{\sum_i\langle\psi_l|\phi_i\rangle} = \dfrac{\overline{N}}{\overline{D}}

E(\tau)\approx\dfrac{\sum_i\langle\psi_l|\hat{H}|\phi_i\rangle}{\sum_i\langle\psi_l|\phi_i\rangle} = \dfrac{\overline{N}}{\overline{D}}

\text{Var}\left(\dfrac{\overline{N}}{\overline{D}}\right) \approx \dfrac{\text{Var}(\overline{N})}{\overline{D}^2} + \dfrac{\overline{N}^2\text{Var}(\overline{D})}{\overline{D}^4} - 2\dfrac{\overline{N}\text{Cov}(\overline{N},\overline{D})}{\overline{D}^3}

\text{Var}\left(\dfrac{\overline{N}}{\overline{D}}\right) \approx \dfrac{\text{Var}(\overline{N})}{\overline{D}^2} + \dfrac{\overline{N}^2\text{Var}(\overline{D})}{\overline{D}^4} - 2\dfrac{\overline{N}\text{Cov}(\overline{N},\overline{D})}{\overline{D}^3}

$(\text{H}_2\text{O})_2$ , (16e, 80o)

Selected CI local energy algorithm

E_L[\phi]=\dfrac{\langle\psi_l|\hat{H}|\phi\rangle}{\langle\psi_l|\phi\rangle},\ \text{two-body part: }\ L^{\gamma}_{pr}L^{\gamma}_{qs}\dfrac{\langle\psi_l|\hat{a}_p^{\dagger}\hat{a}_q^{\dagger}\hat{a}_s\hat{a}_r|\phi\rangle}{\langle\psi_l|\phi\rangle}

E_L[\phi]=\dfrac{\langle\psi_l|\hat{H}|\phi\rangle}{\langle\psi_l|\phi\rangle},\ \text{two-body part: }\ L^{\gamma}_{pr}L^{\gamma}_{qs}\dfrac{\langle\psi_l|\hat{a}_p^{\dagger}\hat{a}_q^{\dagger}\hat{a}_s\hat{a}_r|\phi\rangle}{\langle\psi_l|\phi\rangle}

If $|\psi_l\rangle$ is a Slater determinant: $|\psi_l\rangle = |\phi_0\rangle$

O(N^4)

O(N^4)

If $|\psi_l\rangle$ is a selected CI wave function: $|\psi_l\rangle = \sum_i^{N_d} c_i |\phi_i\rangle$

Naive way: calculating local energy of each Slater determinant as above costs $O(N_dN^4)$

One of the terms:

Consider doubly excited determinants: $c_{jkil} \hat{a}_j^{\dagger} \hat{a}_k \hat{a}_i^{\dagger} \hat{a}_l |\phi_0\rangle$

store intermediate

Overall scaling: $O(N^4 + N_dN)$

factorizable term

$(\text{H}_2\text{O})_2$ , (16e, 80o)

Cyclobutadiene automerization barrier

Method	DZ (20e, 72o)	TZ (20e, 172o)
CCSD(T)	15.8	18.2
CCSDT	7.6	10.6
TCCSD (12,12)	-	9.2
MRCI+Q	-	9.2
fp-AFQMC	8.4(4)	10.2(4)

kcal/mol

$[\text{Cu}_2\text{O}_2]^{2+}$ isomerization

kcal/mol

Converging phaseless bias in ph-AFQMC

FeO (22e, 76o)

Symmetry projection in VMC

Symmetry breaking $\rightarrow$ more variational freedom

Break the symmetry under a projector, to retain good quantum numbers

|\psi\rangle=\hat{P}|\phi\rangle

|\psi\rangle=\hat{P}|\phi\rangle

Projection in VMC by restricting random walk to the symmetry sector

Symmetries: spin, number, complex conjugation, ...

Example: complex conjugation in $\text{H}_2$ near dissociation

|\text{RHF}\rangle = (a_{1\uparrow}^{\dagger}+a_{2\uparrow}^{\dagger})(a_{1\downarrow}^{\dagger}+a_{2\downarrow}^{\dagger})|0\rangle = (a_{1\uparrow}^{\dagger}a_{1\downarrow}^{\dagger}+a_{1\uparrow}^{\dagger}a_{2\downarrow}^{\dagger}+a_{2\uparrow}^{\dagger}a_{1\downarrow}^{\dagger}+a_{2\uparrow}^{\dagger}a_{2\downarrow}^{\dagger})|0\rangle

|\text{RHF}\rangle = (a_{1\uparrow}^{\dagger}+a_{2\uparrow}^{\dagger})(a_{1\downarrow}^{\dagger}+a_{2\downarrow}^{\dagger})|0\rangle = (a_{1\uparrow}^{\dagger}a_{1\downarrow}^{\dagger}+a_{1\uparrow}^{\dagger}a_{2\downarrow}^{\dagger}+a_{2\uparrow}^{\dagger}a_{1\downarrow}^{\dagger}+a_{2\uparrow}^{\dagger}a_{2\downarrow}^{\dagger})|0\rangle

|\text{cRHF}\rangle = (e^{i\pi/4}a_{1\uparrow}^{\dagger}+e^{-i\pi/4}a_{2\uparrow}^{\dagger})(e^{i\pi/4}a_{1\downarrow}^{\dagger}+e^{-i\pi/4}a_{2\downarrow}^{\dagger})|0\rangle\\ \quad = (ia_{1\uparrow}^{\dagger}a_{1\downarrow}^{\dagger}+a_{1\uparrow}^{\dagger}a_{2\downarrow}^{\dagger}+a_{2\uparrow}^{\dagger}a_{1\downarrow}^{\dagger}-ia_{2\uparrow}^{\dagger}a_{2\downarrow}^{\dagger})|0\rangle

|\text{cRHF}\rangle = (e^{i\pi/4}a_{1\uparrow}^{\dagger}+e^{-i\pi/4}a_{2\uparrow}^{\dagger})(e^{i\pi/4}a_{1\downarrow}^{\dagger}+e^{-i\pi/4}a_{2\downarrow}^{\dagger})|0\rangle\\ \quad = (ia_{1\uparrow}^{\dagger}a_{1\downarrow}^{\dagger}+a_{1\uparrow}^{\dagger}a_{2\downarrow}^{\dagger}+a_{2\uparrow}^{\dagger}a_{1\downarrow}^{\dagger}-ia_{2\uparrow}^{\dagger}a_{2\downarrow}^{\dagger})|0\rangle

\hat{K}|\text{cRHF}\rangle = (a_{1\uparrow}^{\dagger}a_{2\downarrow}^{\dagger}+a_{2\uparrow}^{\dagger}a_{1\downarrow}^{\dagger})|0\rangle

\hat{K}|\text{cRHF}\rangle = (a_{1\uparrow}^{\dagger}a_{2\downarrow}^{\dagger}+a_{2\uparrow}^{\dagger}a_{1\downarrow}^{\dagger})|0\rangle

\hat{P}\hat{\mathcal{J}}|\psi\rangle = \hat{P}\exp\left(\sum_{p\sigma,q\gamma} v_{p\sigma,q\gamma}\hat{n}_{p\sigma}\hat{n}_{q\gamma}\right)|\psi\rangle

\hat{P}\hat{\mathcal{J}}|\psi\rangle = \hat{P}\exp\left(\sum_{p\sigma,q\gamma} v_{p\sigma,q\gamma}\hat{n}_{p\sigma}\hat{n}_{q\gamma}\right)|\psi\rangle

Jastrow symmetry projected state:

$\text{N}_2$

d (Bohr)	Exact (DMRG)	Jastrow-KS_zPfaffian	Green's function MC
1.6	-0.5344	-0.5337	-0.5342
1.8	-0.5408	-0.5400	-0.5406
2.5	-0.5187	-0.5180	-0.5185

H $_{50}$ linear chain (50e, 50o)

U	Benchmark energy	Jastrow- KS_zGHF	Green's function MC
2	-1.1962	-1.1920	-1.1939
4	-0.8620	-0.8566	-0.8598
8	-0.5237	-0.5183	-0.5221

2D Hubbard: 98 sites (half filling)

Hartree/particle

Future directions

Properties and excited states

Importance sampling and constraints in AFQMC, hybrid MD-MC

Variational CCSD, other wave functions like MPS, Jastrow in AFQMC

Spin liquid states in iridates using VMC

Quantum Monte Carlo approaches for strongly correlated systems

Outline

A different take on projection QMC

Sampling in AFQMC

The sign problem

Selected CI trial state as $|\psi_l\rangle$

Selected CI local energy algorithm

$[\text{Cu}_2\text{O}_2]^{2+}$ isomerization

Converging phaseless bias in ph-AFQMC

Symmetry projection in VMC

H $_{50}$ linear chain (50e, 50o)

2D Hubbard: 98 sites (half filling)

Future directions

Thank you!