Provable avoidance of barren plateaus for GM-QAOA

Abstract: We analyze the dynamical Lie algebras (DLAs) associated with the Grover-mixer variant of the Quantum Approximate Optimization Algorithm (GM-QAOA). When the initial state is the uniform superposition of computational basis states, we show that the corresponding DLA is isomorphic to either $\mathfrak{su}{r} \oplus \mathfrak{u}{1}^{\oplus 2}$ or $\mathfrak{su}{r} \oplus \mathfrak{u}{1}$, where (r) denotes the number of distinct values of the objective function. We also establish an analogous classification for other choices of initial states and Grover-type mixers. Furthermore, we prove that the DLA of GM-QAOA has the largest possible commutant among all QAOA variants initialized with the same state $|\xi\rangle$, corresponding physically to the maximal set of conserved quantities. In addition, we derive an explicit formula for the variance of the GM-QAOA loss function in terms of the objective function values, and we show that, for a broad class of optimization problems, GM-QAOA with sufficiently many layers avoids barren plateaus.