Speaker
Description
Quantum simulations of $SU(N)$ lattice gauge theory (LGT) in the irrep basis require classical precomputation of Clebsch-Gordan coefficients (CGCs). Strategically choosing the direct-sum basis to maximize the symmetries of the CGCs plausibly translates into gate count reductions for the time-evolution circuit. One such choice occurs when two or more $SU(N)$ irreps in a tensor product are identical. Then, it is possible to choose a direct-sum basis which further 'diagonalizes' the $S_{n}$ permutation symmetry of the tensor product such that the identical $SU(N)$ irreps also transform as a representation of $S_{n}$. We achieve $S_{n}$ diagonalization by augmenting a standard numerical algorithm for $SU(N)$ CGC computation with an additional step which applies a set of Hermitian Young projection operators onto the highest-weight state of each $SU(N)$ irrep; by splitting the highest-weight state into subspaces corresponding to irreps of $S_{n}$, all CGCs computed from this data will also transform as $S_n$ irreps. Empirically, using 'symmetrized' CGCs to compute matrix elements of the $SU(3)$ Kogut-Susskind (KS) Hamiltonian reduces the number of nonzero matrix elements by more than 50% relative to 'unsymmetrized' CGCs for irrep trunctions with nontrivial Hilbert space multiplicities. Since the depth of the Trotterized time-evolution circuit for $SU(3)$ LGT scales linearly with the number of nonzero KS Hamiltonian matrix elements, using symmetrized CGCs directly yields the same reduction in circuit depth.