Deforming the Trail: Baseline Quantum Circuitry for $\text{SU(2)}_k$ Lattice Gauge Theory
Quantifying quantum resources for simulating the fundamental forces of Nature is sensitive to the mapping of gauge fields onto finite quantum computational architectures. When locally truncating lattice gauge theories in the irreducible representation basis, it has been proposed to further deform the theory via quantum groups. The purpose of this deformation is (1) to provide an infinite tower of finite-dimensional ($d = k+1$) groups systematically approximating the infinite-dimensional gauge links and (2) to restore the physical unitarity of a plaquette operator diagonalization procedure analytically derived from the field continuum by recontracting vertex pairs. For the SU(2)$_k$ Yang-Mills pure-gauge theory, we provide a constructive strategy of gauge-variant completions to extend this unitarity to the entire computational Hilbert space, leading to well-defined time evolution unitaries as targets for optimized circuit synthesis. Leveraging basic circuit decompositions and symmetries of the diagonalized plaquette operator, we report resource upper-bounds on the generalized-controlled-X two-qudit gates for arbitrary local truncation $d$, reducing estimates and scaling relative to the non-deformed theory by three polynomial powers from $O(d^8)$ to $O(d^5)$. Examining the stronger q-deformed gauge constraint, which softens the total flux at vertices, we show that the physical Hilbert space dimension of the deformed plaquette operator scales equivalently to its non-deformed counterpart with a constant factor $0.2563(5)$. Thus, despite affecting interactions at all scales as exemplified by the observed flux hierarchy inversion symmetry, q-deformation continues to pass scrutiny as a reliable truncation offering advantages in quantum circuit synthesis.