Quantum Gaussian processes for prediction of channel observations

Characterizing quantum device evolution is essential for quantum control, benchmarking, and variational algorithms. Fully reconstructing an unknown quantum process via process tomography requires resources that scale exponentially with system size, making it intractable for large systems. This paper investigates whether one can instead predict the expectation values of specific Pauli observables at the output of an unknown quantum channel using only a limited number of measurements and input states. While Gaussian process regression is well-established for closed-system unitary dynamics, extending these provable frameworks to open-system quantum channels remains an open challenge.

The authors extend quantum Gaussian processes (QGPs) beyond unitary dynamics to general quantum channels. By modeling the unknown channel as drawn uniformly from the convex set of quantum channels (using the Lebesgue measure), they employ Stinespring dilation and Haar integration to derive a closed-form, hyperparameter-free prior kernel. This kernel's correlations are entirely determined by the pairwise overlaps (fidelities) of the input states. Because the exact Lebesgue kernel includes a dimensional prefactor that causes exponential suppression in extensive systems, the authors also propose an empirical Bayes heuristic that replaces this prefactor with a learnable scale parameter to restore learnability.

Numerical simulations on up to 64 qubits reveal that the Lebesgue channel kernel exhibits a strong inductive bias for local channels, enabling successful extrapolation from small training sets. For global 64-qubit channels where the dimensional factor causes failure, the rescaled empirical Bayes kernel successfully restores learnability, with predictions scaling systematically with the available shot budget. Hardware implementations on a noisy IBM quantum computer confirm the robustness of the QGP regression approach under realistic experimental noise, and the framework is further validated as a Bayesian optimization surrogate for state preparation.