Quantum Monte Carlo approaches for strongly correlated systems

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

Convergence of phaseless errors

Jastrow symmetry projected states in VMC

Auxiliary field QMC

Variational MC

Ongoing work

Free 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

|\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}

Mixed energy estimator:

Trial states: Selected CI, CCSD, Jastrow, MPS, ...

Noise in QMC sampling 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}

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

Exponentiating \(\hat{K}\): orbital transformation

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}}

\(x_{\gamma}\): 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}

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}

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

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}

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}}

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

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

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}

If \(|\psi_l\rangle\) is a Slater determinant: \(|\psi_l\rangle = |\psi_0\rangle\)

O(N^4)

If \(|\psi_l\rangle\) is a selected CI wave function: \(|\psi_l\rangle = \sum_i^{N_d} c_i |\psi_i\rangle\)

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

Generalized Wick's theorem

consider \(|\psi_l\rangle = c_{ptqu}\hat{a}_t^{\dagger}\hat{a}_p\hat{a}_u^{\dagger}\hat{a}_q|\psi_0\rangle\) (double excitations)

\(O(N^4 + N_dN)\)

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

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

kcal/mol

Phaseless AFQMC

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

Gets rid of the sign problem, but has a systematic trial dependent bias

Active space trial states

\(\text{H}_{10}\) (10e, 50o)

Outline

Sampling and the sign problem in AFQMC

Reducing noise using selected CI wave functions

Convergence of phaseless errors

Jastrow symmetry projected states in VMC

Auxiliary field QMC

Variational MC

Ongoing work

Variational Monte Carlo (VMC)

E = \dfrac{\langle \psi|H|\psi\rangle }{\langle \psi|\psi\rangle}

Strategy:

Parametrize the wave function: \(|\psi(\mathbf{p})\rangle\), choose initial \(\mathbf{p}\)

Calculate energy and gradient: Markov chain Monte Carlo

\ \dfrac{\langle \psi(\mathbf{p})|H|\psi(\mathbf{p})\rangle }{\langle \psi(\mathbf{p})|\psi(\mathbf{p})\rangle} = \sum_{\mathbf{n}} \rho_{\mathbf{n}} E_L[\mathbf{n}]