  The Quantum Approximate Optimization Algorithm (QAOA) is an extensively
studied variational quantum algorithm utilized for solving optimization
problems on near-term quantum devices. A significant focus is placed on
determining the effectiveness of training the $n$-qubit QAOA circuit, i.e.,
whether the optimization error can converge to a constant level as the number
of optimization iterations scales polynomially with the number of qubits. In
realistic scenarios, the landscape of the corresponding QAOA objective function
is generally non-convex and contains numerous local optima. In this work,
motivated by the favorable performance of Bayesian optimization in handling
non-convex functions, we theoretically investigate the trainability of the QAOA
circuit through the lens of the Bayesian approach. This lens considers the
corresponding QAOA objective function as a sample drawn from a specific
Gaussian process. Specifically, we focus on two scenarios: the noiseless QAOA
circuit and the noisy QAOA circuit subjected to local Pauli channels. Our first
result demonstrates that the noiseless QAOA circuit with a depth of
$\tilde{\mathcal{O}}\left(\sqrt{\log n}\right)$ can be trained efficiently,
based on the widely accepted assumption that either the left or right slice of
each block in the circuit forms a local 1-design. Furthermore, we show that if
each quantum gate is affected by a $q$-strength local Pauli channel with the
noise strength range of $1/{\rm poly} (n)$ to 0.1, the noisy QAOA circuit with
a depth of $\mathcal{O}\left(\log n/\log(1/q)\right)$ can also be trained
efficiently. Our results offer valuable insights into the theoretical
performance of quantum optimization algorithms in the noisy intermediate-scale
quantum era.
