  Exploring quantum applications of near-term quantum devices is a rapidly
growing field of quantum information science with both theoretical and
practical interests. A leading paradigm to establish such near-term quantum
applications is variational quantum algorithms (VQAs). These algorithms use a
classical optimizer to train a parameterized quantum circuit to accomplish
certain tasks, where the circuits are usually randomly initialized. In this
work, we prove that for a broad class of such random circuits, the variation
range of the cost function via adjusting any local quantum gate within the
circuit vanishes exponentially in the number of qubits with a high probability.
This result can unify the restrictions on gradient-based and gradient-free
optimizations in a natural manner and reveal extra harsh constraints on the
training landscapes of VQAs. Hence a fundamental limitation on the trainability
of VQAs is unraveled, indicating the essential mechanism of the optimization
hardness in the Hilbert space with exponential dimension. We further showcase
the validity of our results with numerical simulations of representative VQAs.
We believe that these results would deepen our understanding of the scalability
of VQAs and shed light on the search for near-term quantum applications with
advantages.
