Algorithm 2 Quantum-Encodable Bayesian PINNs trained via Classical Ensemble Kalman Inversion (QEKI)

Require: Observations $y_{\mathrm{obs}}$, initial J ensemble states ${\left\{{\mathit{\xi }}_0^j\right\}}_{j=1}^J$, observation covariance R, parameter covariance Q, training data points $\mathit{x}$, iteration index $i=0$.     while not converge do         for $j=1,2,\dots ,J$ do             Sample ${\mathit{ϵ}}_i^j$ from $\mathcal{N}(0,Q)$             Update each ensemble state ${\mathit{\xi }}_i^j\leftarrow {\mathit{\xi }}_i^j+{\mathit{ϵ}}_i^j$             $({\mathit{\theta }}_i^j,{\mathit{\lambda }}_i^j)\leftarrow {\mathit{\xi }}_i^j$             Apply quantum circuit $|\widehat{\mathit{x}}\rangle \leftarrow U(\mathit{x},\mathit{\theta })|0\rangle$             Evaluate the expectation value ${\langle O\rangle }_i^j\leftarrow \langle \widehat{\mathit{x}}\left|O\right|\widehat{\mathit{x}}\rangle$             ${\mathit{y}}_i^j\leftarrow {\mathcal{N}}_{\mathit{x}}({\langle O\rangle }_i^j;{\mathit{\lambda }}_i^j)$         end for         $C_i^{yy}\leftarrow {\displaystyle \frac{1}{J-1}}{\sum }_{j=1}^J\left({\mathit{y}}_i^j-{\overline{\mathit{y}}}_i\right){\left({\mathit{y}}_i^j-{\overline{\mathit{y}}}_i\right)}^T$         $C_i^{\xi y}\leftarrow {\displaystyle \frac{1}{J-1}}{\sum }_{j=1}^J\left({\mathit{\xi }}_i^j-{\overline{\mathit{\xi }}}_i\right){\left({\mathit{y}}_i^j-{\overline{\mathit{y}}}_i\right)}^T$         for $j=1,2,\dots ,J$ do             Sample ${\mathit{\eta }}_i^j$ from $\mathcal{N}(0,R)$             ${\mathit{\xi }}_{i+1}^j\leftarrow {\mathit{\xi }}_i^j+C_i^{\xi y}{\left(C_i^{yy}+R\right)}^{-1}\left({\mathit{y}}_{\mathrm{obs}}+{\mathit{\eta }}_i^j-{\mathit{y}}_i^j\right)$         end for     end while      $\widehat{I}\leftarrow i+1$     Return:  ${\mathit{\xi }}_{\widehat{I}}^1,\dots ,{\mathit{\xi }}_{\widehat{I}}^J$