Scaling Laws for Parameter Setting in Digitized Counterdiabatic QAOA

Abstract

Optimizing parameters separately for each problem instance limits the reuse of variational quantum algorithms. We develop empirical scaling laws for parameter setting in the digitized counterdiabatic quantum approximate optimization algorithm (DC-QAOA) for weighted MaxCut. The prescription combines exact covariance under uniform edge-weight rescaling with power laws in mean degree fitted separately to offline-optimized cost and counterdiabatic angles. After the weight and degree scales are removed, the optimized angles of different training graphs cluster around common amplitudes, so a small coefficient table suffices. Frozen coefficients assign angles from a target graph's mean degree and root-mean-square edge weight without target-instance variational optimization. In ideal-circuit simulations across 1050 test graphs spanning ten graph families and five positive edge-weight distributions, mean optimal-solution probability is higher than that of the fixed-angle quantum approximate optimization algorithm (QAOA) at each depth from one to four layers, reaching 19.36% versus 13.11% at four layers. A six-layer extension on 600 graphs also shows a mean probability advantage. Hardware execution on the IBM Quantum processor ibm_fez for 21 unit-weight 3-regular graphs shows a mean probability advantage at one layer, while the ordering reverses at two layers. In separate energy-based experiments, direct assignment gives a higher mean approximation ratio than random-start optimization reaches within 500 evaluations, and refinement started from the formula angles improves it further. These results show how a scaling-based representation of optimized angles can guide parameter reuse across graph instances, enabling direct parameter assignment without target-instance variational optimization and providing initial angles for local refinement.