Block Encodings of Discrete Subgroups on Quantum Computer

We introduce a block encoding method for mapping discrete subgroups to qubits on a quantum computer. This method is applicable to general discrete groups, including crystal-like subgroups such as $\mathbb{B}\mathbb{I}$ of $SU(2)$ and $\mathbb{V}$ of $SU(3)$. We detail the construction of primitive gates – the inversion gate, the group multiplication gate, the trace gate, and the group Fourier gate – utilizing this encoding method for $\mathbb{B}\mathbb{T}$ and for the first time $\mathbb{B}\mathbb{I}$ group. We also provide resource estimations to extract the gluon viscosity. The inversion gates for $\mathbb{B}\mathbb{T}$ and $\mathbb{B}\mathbb{I}$ are benchmarked on the $\mathtt{\text{Baiwang}}$ quantum computer with estimated fidelities of $40_{-4}^{+5}\mathrm{\%}$ and $4_{-3}^{+5}\mathrm{\%}$ respectively.