Ansätze

• Restricted Boltzmann Machine (RBM)

  • Real Restricted Boltzmann Machine (real-RBM)

  • Complex Restricted Boltzmann Machine (complex-RBM)

  • Cosine Restricted Boltzmann Machine (cos-RBM)

  • Tanh Restricted Boltzmann Machine (tanh-RBM)

  • Phase Restricted Boltzmann Machine (phase-RBM)

  • Autoregressive Restricted Boltzmann Machine (AR-RBM)

  • Ising-type Restricted Boltzmann Machine (Ising-RBM)

  • Restricted Ising-type Restricted Boltzmann Machine (RIsing-RBM)

• Recurrent Neural Network (RNN)

  • Recurrent Neural Network (RNN)

  • Gated Recurrent Unit (GRU)

  • Graph MPS(Tensor)–RNN (MPS(Tensor)–RNN)

• Transformer

• Mix-Ansatz

RBM

real(complex)-RBM

\[\begin{split}\begin{split} \psi_{\theta}(n) & = \textcolor{teal}{\exp}{\sum_{j=1}^{N_{\rm v}}a_jn_j} \times \prod_i^{N_{\rm h}}\textcolor{violet}{2\cosh}(b_i + \sum_{j=1}^{N_{\rm v}}W_{ij}n_j) \\ \text{or} & = \prod_i^{N_h}\textcolor{violet}{2\cos}(b_i + \sum_{j=1}^{N_{\rm v}}W_{ij}n_j) \quad \textbf{cos-type}\\ \text{or} & = \textcolor{teal}{\tanh}{\sum_{j=1}^{N_{\rm v}}a_jn_j} \times \prod_i^{N_{\rm h}}\textcolor{violet}{2\cosh}(b_i + \sum_{j=1}^{N_{\rm v}}W_{ij}n_j) \quad \textbf{tanh-type} \end{split}\end{split}\]

For more information, see: ./vmc/ansatz/multi/RBMWavefunction.

Transformer

use nano-chatgpt

For more information, see: ./vmc/ansatz/transformer/decoder/DecoderWaveFunction.

MPS-RNN

For more information, see: ./vmc/ansatz/rnn/graph_mpsrnn/Graph_MPS_RNN.

Mix-Ansatz

Define: \(\psi(n) = f_n\phi(n), \ket{n} \sim |\phi(n)|^2\). \(\phi(n)\) is MPS-RNN, Transformer, AR–RBM with \(|\phi(n)|^2=1\) for sampling, \(f_n\) is RBM, MLP, Jastrow Factor, Transformer and so on.

\[\begin{split}\begin{align} B & = \left\langle |f_n|^2\right\rangle_{n \sim{|\phi(n)|^2} } \\ \widetilde{f}_n & = f_n /\sqrt{B} \\ E_{\rm loc}(n) &= \dfrac{\dfrac{\sum_m f_n^* H_{nm}f_m\phi(m)}{\phi(m)}}{\langle |f_n|^2\rangle} = \dfrac{\sum_m \widetilde{f}_n^* H_{nm}\widetilde{f}_m\phi(m)}{\phi(n)} \\ \partial_\theta \langle E\rangle &= 2\Re\big\langle (\partial_\theta (\ln(f_n\phi(n)))^*)(E_{\rm loc}(n) - \langle E\rangle|\widetilde{f}_n|^2) \big\rangle_{n\sim |\phi(n)|^2} \\ \end{align}\end{split}\]

Spin-flip

see: branch spin-flip

\[\begin{split}\begin{align} B & = \bigg\langle |f_n|^2 + \eta f^*_n f_{\bar n }\frac{\phi(\bar n)}{\phi(n)}\bigg\rangle_{n \sim{\phi_n^2} } \\ \widetilde{f}_n & = f_n /\sqrt{B} \\ E_{\rm loc}(n) &= \frac{\sum_m \widetilde{f}_n^* H_{nm} (\widetilde{f}_m\phi_m + \eta\widetilde{f}_{\bar m}\phi_{\bar m})} {\phi_n} \\ C & = \frac{|f_n|^2 + \eta f^*_n f_{\bar n }\frac{\phi(\bar n)}{\phi(n)}}{B} \\ \partial_\theta E &= 2\Re\left< (\partial_\theta (\ln(\phi_n f_n))^*)(E_{\rm loc}(n) - \left\langle E \right\rangle C) \right> \end{align}\end{split}\]