Efficient Quantum Simulations of Yang-Mills theory with Maximal-tree Gauge

We develop a quantum algorithmic framework for the efficient simulation of Yang--Mills theories, including the $\mathrm{SU}(3)$ gauge theory in Quantum Chromodynamics (QCD). The framework uses maximal-tree gauge in terms of gauge field variables that removes all local gauge redundancies. In the resulting gauge-fixed formulation and digitization in the field-amplitude basis, we show that Hamiltonian time evolution admits an efficient implementation based on quantum singular value transformation (QSVT). We derive upper bounds on the total number of qubits and gate complexity, finding polynomial scaling with the inverse simulation precision $1/\varepsilon_s$, lattice volume $\mathcal{V}$, gauge coupling $g$, and target energy scale $E$. Our results provide a rigorous complexity-theoretic demonstration that non-Abelian Yang--Mills theories can be simulated efficiently on quantum computers, paving the way toward first-principles quantum simulations of non-perturbative QCD dynamics.