Using a qubit to probe non-Gaussian noise environments is theoretically studied in the context of classical random telegraph processes. Protocols for control pulses are developed to effectively scan higher noise correlations, offering valuable information on the charge environment of the qubit. Specifically, the noise power spectrum and trispectrum are reconstructed simultaneously for a wide range of qubit-fluctuator coupling strengths, demonstrating the method's robustness. These protocols are readily testable in various qubit systems with well-developed quantum control, including quantum dot spins, superconducting qubits and NV centers in diamond.

13 pages, 7 figures