Speaker
Description
The quantum approximate optimization algorithm (QAOA) prepares candidate solutions to combinatorial optimization problems by alternating between operations that encode the objective function and a mixing operation that shift probability between candidate solutions. Gradient-based training of such circuits requires estimating how the measured cost changes when a circuit angle is varied. We focus on the gradient with respect to the final mixing angle and ask which entries of the quantum state can contribute to it. For any diagonal cost Hamiltonian and any mixer that is real in the cost basis, this gradient is determined only by imaginary coherences between basis states connected by the mixer.
These mixer-edge coherences therefore give an upper bound on the gradient magnitude and are necessary for a nonzero gradient, although different edge contributions can cancel. For the common QAOA mixing operation that flips one bit at a time, the relevant coherences connect bitstrings that differ by one bit. We illustrate the bound on Max-Cut, a standard graph-partitioning problem, using exact simulations of small random regular graphs with common terminal noise channels. In this setting, the gradient is governed by how the noise changes the effective cost observable; coherence left in the final noisy state provides a separate diagnostic.