Sensing and reconstructing a quantum-classical spin bath*

We formally introduce and experimentally demonstrate a quantum sensing protocol to sample and reconstruct the noise spectrum of a single qubit sensor by using a sequence of interferometric measurements based on Walsh dynamical decoupling sequences. The Walsh sequences generate a complete basis of digital filters that directly sample the power spectrum of the fluctuating field (classical noise environment) in the sequency domain, from which we can reconstruct the autocorrelation function in the time domain – and the power spectrum in the frequency domain -- with simple linear transformations. By using deep learning algorithms, this Walsh sequency further show the ability to distinguish and reconstruct the interaction between different spin qubits (quantum environment). In comparison to typical periodic decoupling-based noise spectroscopy methods, the accuracy of our method is only limited by sampling and discretization in the time space and can be easily improved, even under limited evolution time due to decoherence and hardware limitations. Finally, we experimentally reconstruct the autocorrelation function of the effective magnetic field produced by the nuclear-spin bath on the electronic spin of a single nitrogen-vacancy center in diamond, and the hyperfine parameters with its nearby nuclear spin.