Imaginarity as a necessary resource for trainability in QAOA

In the Quantum Approximate Optimization Algorithm (QAOA), the classical optimizer relies on gradients to tune circuit parameters. When these gradients are small, optimization becomes difficult, a problem often associated with barren plateaus. This paper investigates the fundamental quantum resources required to maintain a useful gradient signal, specifically focusing on the role of imaginarity in the final mixer angle.

The authors analyze the QAOA circuit, which alternates between cost Hamiltonian layers and mixer Hamiltonian layers. They define 'cost-basis imaginarity' as the imaginary part of the off-diagonal elements of the density matrix in the computational basis. By deriving an identity for the gradient of the cost expectation with respect to the final mixer angle, they show that only coherences between bitstrings directly connected by the mixer contribute to the gradient. They extend this analysis to noisy settings by using an adjoint-channel representation, allowing them to bound the gradient even when terminal noise is present.

The study establishes that the gradient of the QAOA cost function with respect to the final mixer angle is strictly bounded by the 'mixer-edge imaginarity'—the sum of the absolute values of the imaginary coherences between states coupled by the mixer. For the standard transverse-field mixer, this corresponds to bitstrings differing by exactly one bit. The authors demonstrate that this bound holds even under common noise models like depolarizing, phase-flip, and amplitude-damping channels, provided the adjoint-propagated cost observable remains diagonal. Numerical simulations on Max-Cut problems confirm that the gradient obeys this bound pointwise, with the saturation of the bound providing insight into the trainability of the circuit.