Quantum DeepONet gets conformal calibration: near-linear hidden layers, coverage near the 90% target

The hidden-layer running time of a quantum DeepONet can be near-linear rather than quadratic as in dense layers, but only inside an explicit resource regime. Earlier quantum DeepONet work quoted a quantum feedforward cost of O(n/δ²) against the O(n²) cost of a classical dense layer without accounting for classical parallelism. Counting P classical processors, the quantum depth-shot cost is O(n log n/δ²), which grows more slowly when P = o(nδ²/log n); if P grows proportionally with n, the classical bound becomes O(n) and the advantage disappears.

After conformal calibration, ensemble predictions carry a finite-sample lower bound on query-averaged coverage for a new input-output function pair. Conventional split conformal prediction treats each query point as a separate example, while queries under one input function may be strongly dependent. Taking complete function pairs as exchangeable units and assuming nothing about within-unit queries gives the lower bound r/((K+1)M); for the antiderivative experiment with K = 50, M = 30, and α = 0.1 the bound is 88.24%.

Under ideal simulation and device-calibrated noise simulation, prediction accuracy and empirical coverage both stay near the target. Earlier quantum operator learning lacked an evaluation of calibration under noisy conditions. The antiderivative and advection benchmarks reach relative L2 errors of 0.46% and 2.28% with coverage of 92.13% and 88.99%; power-system offline voltage-to-active-power, offline voltage-to-voltage, and online voltage-to-voltage tasks reach 4.08%, 12.49%, and 5.49% error with coverage of 89.74%, 89.66%, and 90.11%.

The result applies to square quantum orthogonal hidden transformations under an idealized depth-based execution model, so it is a layer-level asymptotic comparison rather than an end-to-end runtime or a measured hardware speedup. The evaluation rests on ideal and noisy circuit simulations covering synthetic antiderivative and advection benchmarks plus offline and online power-system tasks, with compact networks and small ensembles. The conformal guarantee requires jointly exchangeable calibration and test units with equal-length score vectors, and the online experiment treats complete parent trajectories as the exchangeable units.

The finite-sample lower bound can fall below the nominal 1−α target when the number of calibration units is small, approaching it only as the calibration set grows. Under depolarizing noise the error first rises then falls, coverage climbs to roughly 99% while average interval width stays nearly constant, and this trend is described as preliminary with its mechanism unexamined. The multiplexed circuit is validated only at L = 4, leaving depth, sampling, and runtime scaling at larger ensemble sizes unestablished.