Anderson localization for the multi-frequency quasi-periodic CMV matrices and quantum walks

In this paper we prove Anderson localization for multi-frequency quasi-periodic extended CMV matrices with analytic Verblunsky coefficients in the regime of positive Lyapunov exponents. By constructing a suitable semialgebraic set and combining the Avalanche Principle with a Large Deviation Theorem, we overcome the key obstruction of eliminating double resonances along the orbit, where multi-frequency potentials introduce significant challenges compared to the single-frequency case. As a direct application, we establish Anderson localization for corresponding analytic multi-frequency quasi-periodic quantum walks via unitary equivalence.