Efficient Truncations of SU($N_c$) Lattice Gauge Theory for Quantum Simulation

Abstract

Quantum simulations of lattice gauge theories offer the potential to directly study the non-perturbative dynamics of quantum chromodynamics, but naive analyses suggest that they require large computational resources. Large $N_c$ expansions are performed to order 1/$N_c$ to simplify the Hamiltonian of pure SU($N_c$) lattice gauge theories. A reformulation of the electric basis is introduced with a truncation strategy based on the construction of local Krylov subspaces with plaquette operators. Numerical simulations show that these truncated Hamiltonians are consistent with traditional lattice calculations at relatively small couplings. It is shown that the computational resources required for quantum simulation of time evolution generated by these Hamiltonians is 17-19 orders of magnitude smaller than previous approaches.

Figures (14)

• Figure 1: A generic link is shown equipped with a state $\ket{\mathcal{R},r_1,r_2}$. The target half-link is equipped with the vector $\ket{\mathcal{R},r_1}$, while the source half-link is equipped with the vector $\ket{\bar{\mathcal{R}},r_2}\equiv\bra{\mathcal{R},r_2}$.
• Figure 2: A generic vertex is shown in the electric basis. The linear combination \ref{['eq:vertex_factor']} of these states yields the basis of Ciavarella:2021nmj.
• Figure 3: This transition is of $\order{1/N_c}$. This can be seen because it involves the deletion of a single joined line due to the action of a plaquette operator. However, creating this field configuration from the electric vacuum requires multiple $\order{1/N_c}$ transitions. Accordingly, in this work, we do not attempt to correctly include the matrix elements associated with this transition.
• Figure 4: The transition allowed in the $(1,1,1)$ truncation.
• Figure 5: The new transition allowed in the $(1,2,1)$ truncation.