In https://medium.com/qiskit/introducing-bosonic-qiskit-a-package-for-simulating-bosonic-and-hybrid-qubit-bosonic-circuits-1e1e528287bb , could anyone explain the rationale behind the use of beamsplitter Ubs and SNAP gate Usnap ? What does it mean by n is the occupancy of site i ?

The Hamiltonian of the Bose-Hubbard model is: $$H=-J\sum_{\langle i,j\rangle}(a^\dagger_ia_j+a_ia_j^\dagger)+\frac{U}{2}\sum_i a_i^\dagger a_i^\dagger a_i a_i.$$ Here, $\langle i, j\rangle$ denotes summation over all neighboring lattice sites $i$ and $j$, while $a_i^\dagger(a_i)$ is the creation (annihilation) operators which increases (decreases) the occupancy at site $i$ by one. To study the time evolution under this Hamiltonian, we need to implement the unitary time evolution operator $e^{-iHt}$ (we have set $\hbar = 1$). To achieve this, we rely on the Trotter formula: $$e^{A+B}=\lim_{n\to\infty}(e^{A/n}e^{B/n})^n$$ Which allows us to approximate the full time evolution operator via alternation of: $$U_1=e^{-iH_1\Delta t}\mathrm{\:and\:}U_2=e^{-iH_2\Delta t}$$ Where $H_1$ and $H_2$ refer to the first (hopping) and second interaction) term of $H$, respectively.

Below, we demonstrate a simple simulation of the 1D Bose-Hubbard model using Bosonic Qiskit. In particular, we use beamsplitters of the form: $$U_\mathrm{BS}(\theta)=e^{i\theta(a_i^\dagger a_j+a_i a_j^\dagger)}$$ to realize $U_1$, and use SNAP gates, $$U_\mathrm{SNAP}(\theta_n,n)=e^{i\theta_n|n\rangle\langle n|}$$ to realize $U_2$ (i.e., one can show that $a_i^\dagger a_i^\dagger a_i a_i=\Sigma \_n(n_i(n_i-1)|n_i\rangle\langle n_i|)$ where $n$ is the occupancy of site $i$, which enables synthesis of $U_2$ through application of SNAP for each $\{n,i\}$ and an appropriate choice of the phase $\theta_n$).



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