Diagnosing Barren Plateaus with Tools from Quantum Optimal Control

Variational Quantum Algorithms (VQAs) have received considerable attention due to their potential for achieving near-term quantum advantage. However, more work is needed understand scalability. One known scaling result VQAs barren plateaus, where certain circumstances lead exponentially vanishing gradients. It common folklore that problem-inspired ansatzes avoid but in fact, very little about gradient scaling. In this we employ tools from optimal control develop a framework can diagnose the presence or absence of plateaus ansatzes. Such include Alternating Operator Ansatz (QAOA), Hamiltonian (HVA), and others. With our framework, prove avoiding these not always guaranteed. Specifically, show VQA depends on degree controllability system, hence be diagnosed through dynamical Lie algebra $\mathfrak{g}$ obtained generators ansatz. We analyze existence QAOA HVA ansatzes, highlight role input state, as different initial states plateaus. Taken together, results provide trainability-aware ansatz design strategies do come at cost extra resources. Moreover, no-go obtaining ground with variational controllable system such spin glasses. Our establishes link between dimension $\mathfrak{g}$.