  The quantum approximate optimization algorithm~(QAOA) first proposed by Farhi
et al. promises near-term applications based on its simplicity, universality,
and provable optimality. A depth-p QAOA consists of p interleaved unitary
transformations induced by two mutually non-commuting Hamiltonians. A
long-standing question concerning the performance of QAOA is the dependence of
its success probability as a function of circuit depth p. We make initial
progress by analyzing the success probability of QAOA for realizing state
transfer in a one-dimensional qubit chain using two-qubit XY Hamiltonians and
single-qubit Hamiltonians. We provide analytic state transfer success
probability dependencies on p in both low and large p limits by leveraging the
unique spectral property of the XY Hamiltonian. We support our proof under a
given QAOA ansatz with numerical optimizations of QAOA for up to \(N\)=20
qubits. We show that the optimized QAOA can achieve the well-known quadratic
speedup, Grover speedup, over the classical alternatives. Treating QAOA
optimization as a quantum control problem, we also provide numerical evidence
of how the circuit depth determines the controllability of the QAOA ansatz.
