Strict Frequency Bandwidths from Causal Light Cones Enable Scalable Exact Gradient Evaluation in Quantum Optimization

Abstract

Dynamical locality is often regarded as a limitation of shallow quantum circuits, yet we show that it can reduce the number of circuit settings required for exact evaluation of gradients with respect to shared parameters. For layered ansatze generated by commuting local terms, strict Heisenberg light cones confine energy slices obtained by varying one shared layer parameter to finite Fourier bandwidths dictated by local causal geometry. This bandlimit recasts exact gradient evaluation as a local signal recovery problem and, when the relevant frequencies lie on a known frequency lattice, yields an exact multipoint parameter-shift rule based on the discrete sine transform. The quantum approximate optimization algorithm (QAOA) provides a canonical realization in which this mechanism is governed by the local structure of the graph. Specifically, when a cost angle is varied, the relevant geometry is encoded by the contact region in the interaction line graph. At fixed circuit depth on graphs of bounded degree, the number of distinct circuit settings required for the corresponding derivative collapses from $O(m)$ to $O(1)$. This shows that dynamical locality is not merely a constraint on correlation propagation, but also a computational resource that enables exact gradient evaluation with fewer circuit settings in quantum optimization.