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
- 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−τ(H^−E0)∣ψ⟩=c0∣Ψ0⟩+c1e−ΔE1τ∣Ψ1⟩+…
e^{-\tau (\hat{H}-E_0)}|\psi\rangle = c_0|\Psi_0\rangle + c_1 e^{-\Delta E_1\tau}|\Psi_1\rangle+\dots
∣ψ⟩=c0∣Ψ0⟩+c1∣Ψ1⟩+…
|\psi\rangle = c_0|\Psi_0\rangle + c_1|\Psi_1\rangle +\dots
- Better ∣ψ⟩ approximates ∣Ψ0⟩, faster the convergence with τ
E(τ)=⟨ψl∣e−τH^∣ψr⟩⟨ψl∣H^e−τH^∣ψr⟩
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 τ
Sampling in AFQMC
H^=K^+V^=tpra^p†a^r+21vprqsa^p†a^ra^q†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 H^: [K^,V^]=0
e−τH^≈(e−Nτ2K^e−NτV^e−Nτ2K^)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 K^: orbital transformation
etpra^p†a^r∣ϕ⟩=∣ϕ′⟩
e^{t_{pr}\hat{a}_p^{\dagger}\hat{a}_r}|\phi\rangle=|\phi'\rangle
where ∣ϕ⟩ and ∣ϕ′⟩ are nonorthogonal determinants.
- Exponentiating V^=21∑γ(Lprγa^p†a^r)2:
e−2L^γ2=∫2πdxγ e2−xγ2eixγL^γ
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}}
xγ: auxiliary field
Motta and Zhang (2017), 1711.02242
(Thouless, 1960)
(Stratonovich, 1957)
E(τ)=⟨ψl∣e−τH^∣ψr⟩⟨ψl∣H^e−τH^∣ψr⟩≈∫ dXp(X)⟨ψl∣B^(X)∣ψr⟩∫ dXp(X)⟨ψl∣H^B^(X)∣ψr⟩
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(τ)≈∑i⟨ψl∣ϕi⟩∑i⟨ψl∣H^∣ϕi⟩
E(\tau)\approx\dfrac{\sum_i\langle\psi_l|\hat{H}|\phi_i\rangle}{\sum_i\langle\psi_l|\phi_i\rangle}
CCSD as ∣ψr⟩: sampling Slater determinants from CCSD
∣ψr⟩=exp(tikjla^i†a^ka^j†a^l)exp(tika^i†a^k)∣ϕ0⟩
|\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 → no Trotter error
(H2O)2, (16e, 80o)
The sign problem
Var(DN)≈D2Var(N)+D4N2Var(D)−2D3NCov(N,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}
E(τ)≈∑i⟨ψl∣ϕi⟩∑i⟨ψl∣H^∣ϕi⟩=DN
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−2y2=∫2πdx e2−x2+ixy
e^{-\frac{y^2}{2}} = \int \frac{dx}{\sqrt{2\pi}}\ e^{\frac{-x^2}{2}+ixy}
x→x+iy
x\rightarrow x+iy
xγ→xγ+iτ⟨L^γ⟩
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 ∣ψl⟩
Zero variance principle: If ∣ψl⟩ is the exact ground state, then N and D are perfectly correlated, ⟨ψ0∣H^∣ϕi⟩=E0⟨ψ0∣ϕi⟩, and the energy estimator has zero variance.
More accurate ∣ψl⟩ → higher Cov(N,D)
E(τ)≈∑i⟨ψl∣ϕi⟩∑i⟨ψl∣H^∣ϕi⟩=DN
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}}
Var(DN)≈D2Var(N)+D4N2Var(D)−2D3NCov(N,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}
(H2O)2, (16e, 80o)
Selected CI local energy algorithm
EL[ϕ]=⟨ψl∣ϕ⟩⟨ψl∣H^∣ϕ⟩, two-body part: LprγLqsγ⟨ψl∣ϕ⟩⟨ψl∣a^p†a^q†a^sa^r∣ϕ⟩
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 ∣ψl⟩ is a Slater determinant: ∣ψl⟩=∣ϕ0⟩
O(N4)
O(N^4)
If ∣ψl⟩ is a selected CI wave function: ∣ψl⟩=∑iNdci∣ϕi⟩
Naive way: calculating local energy of each Slater determinant as above costs O(NdN4)
One of the terms:
Consider doubly excited determinants: cjkila^j†a^ka^i†a^l∣ϕ0⟩
store intermediate
Overall scaling: O(N4+NdN)
factorizable term
(H2O)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
[Cu2O2]2+ isomerization
kcal/mol
Converging phaseless bias in ph-AFQMC
FeO (22e, 76o)
Symmetry projection in VMC
Symmetry breaking → more variational freedom
Break the symmetry under a projector, to retain good quantum numbers
∣ψ⟩=P^∣ϕ⟩
|\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 H2 near dissociation
∣RHF⟩=(a1↑†+a2↑†)(a1↓†+a2↓†)∣0⟩=(a1↑†a1↓†+a1↑†a2↓†+a2↑†a1↓†+a2↑†a2↓†)∣0⟩
|\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
∣cRHF⟩=(eiπ/4a1↑†+e−iπ/4a2↑†)(eiπ/4a1↓†+e−iπ/4a2↓†)∣0⟩=(ia1↑†a1↓†+a1↑†a2↓†+a2↑†a1↓†−ia2↑†a2↓†)∣0⟩
|\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
K^∣cRHF⟩=(a1↑†a2↓†+a2↑†a1↓†)∣0⟩
\hat{K}|\text{cRHF}\rangle = (a_{1\uparrow}^{\dagger}a_{2\downarrow}^{\dagger}+a_{2\uparrow}^{\dagger}a_{1\downarrow}^{\dagger})|0\rangle
P^J^∣ψ⟩=P^exp(∑pσ,qγvpσ,qγn^pσn^qγ)∣ψ⟩
\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:
N2
| 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 |
H50 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
Thank you!
Quantum Monte Carlo approaches for strongly correlated systems
afqmc_vmc
By Ankit Mahajan
afqmc_vmc
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