Quantum Monte Carlo approaches for strongly correlated systems

The quantum many-body problem and Monte Carlo

|\psi\rangle=

Quantum Monte Carlo: sample properties without storing full wave functions

Exact simulations \(\rightarrow\) fermion sign problem (exponential decay of signal to noise ratio)

\hat{H}|\psi\rangle = E|\psi\rangle

i\frac{d |\psi\rangle}{d t} = \hat{H}|\psi\rangle

\langle \hat{O}\rangle = \frac{\langle\psi|\hat{O}|\psi\rangle}{\langle\psi|\psi\rangle}

Lattice models:

\hat{H}_{\text{Hubbard}} = -t\sum_{\langle ij\rangle,\sigma}(\hat{a}_{i\sigma}^{\dagger}\hat{a}_{j\sigma}+\hat{a}_{j\sigma}^{\dagger}\hat{a}_{i\sigma}) + U\sum_i\hat{n}_{i\uparrow}\hat{n}_{i\downarrow}

Ab initio models:

\hat{H}_{\textit{Ab initio}} = \sum t_{pr} \hat{a}_{p}^{\dagger}\hat{a}_r + \sum v_{prqs}\hat{a}_{p}^{\dagger}\hat{a}_r\hat{a}_{q}^{\dagger}\hat{a}_{s}

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

Projection QMC

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

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

Free projection: try to manage sign problem, numerically exact and exponentially scaling

Noise in QMC sampling worsens exponentially with \(\tau\) and system size

Imaginary time propagation:

Two flavors:

Constrained: use trial state to constrain projection, trial dependent bias but polynomially scaling

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}

Sampling in Auxiliary Field QMC

\hat{H} = \hat{K} + \hat{V} = t_{pr} \hat{a}_{p}^{\dagger}\hat{a}_r + \frac{1}{2}\sum_{\gamma} \left(L^{\gamma}_{pr}\hat{a}_p^{\dagger}\hat{a}_r\right)^2

Exponentiating \(\hat{K}\):

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

Exponentiating \(\hat{V}\): coupling to a scalar field

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

(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}

Free projection:

Zhang, Krakauer, Reichman, Freisner, Rubenstein, ...

Phaseless:

w_{\phi(\mathbf{x})} = \Bigg\vert\dfrac{\langle\psi_T|\phi(\mathbf{x})\rangle}{\langle\psi_T|\phi\rangle}e^{\mathbf{x}.\bar{\mathbf{x}}-\bar{\mathbf{x}}^2/2}\Bigg\vert\max(0, \cos(\Delta\theta))

CCSD as \(|\psi_r\rangle\) in free projection:

|\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

Benzene (30e, 102o)

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}

The sign problem in free projection

\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}}

Contour shift:

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

x\rightarrow x+iy

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

In AFQMC:

Baer, Head-Gordon, Neuhauser (1998)

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}}

Selected configuration interaction: put the most important configurations in the state using particle-hole excitations and diagonalize

|\psi_l\rangle = \sum_i^{N_c} c_i |\psi_i\rangle

Benzene (30e, 102o)

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}

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}

Generalized Wick's theorem: consider \(|\psi_l\rangle = c_{ptqu}\hat{a}_t^{\dagger}\hat{a}_p\hat{a}_u^{\dagger}\hat{a}_q|\psi_0\rangle\)

Benzene (30e, 102o)

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

\(\Delta E = E(\text{bis}) - E(\text{peroxo})\)

Method
DFT (UBLYP)	36.0
DFT (UB3LYP)	52.9
DFT (UMPW1K)	74.0
CCSD(T)	30.6
CR-CCSD(TQ)	33.8
DMRG-CT	27.1
ph-AFQMC (NOCI)	32.1
fp-AFQMC	24.1(6)

kcal/mol

(32e, 108o)

\(\text{H}_{50}\) (50e, 50o)

Systematically improving phaseless AFQMC

Excited states in phaseless AFQMC

Butadiene: (22e, 142o)

Nickel porphyrin: (122e, 406o)

Method
SC-NEVPT2	6.72	6.74
CCSD	6.31	7.08
AFQMC	6.46(5)	6.67(5)
TBE	6.2	6.5

Method
CASSCF (4e, 4o)	3.8
CCSD	2.55
AFQMC / sCI (50k)	3.0(1)
AFQMC / sCI (100k)	2.8(1)
Experiment	2.3-2.4

1\ ^1B_{2u}

1\ ^1B_{u}

2\ ^1A_{g}

(4e, 8o) active space