  We introduce and experimentally demonstrate a quantum sensing protocol to
sample and reconstruct the auto-correlation of a noise process using a
single-qubit sensor under digital control modulation. This Walsh noise
spectroscopy method exploits simple sequences of spin-flip pulses to generate a
complete basis of digital filters that directly sample the power spectrum of
the target noise in the sequency domain -- from which the auto-correlation
function in the time domain, as well as the power spectrum in the frequency
domain, can be reconstructed using linear transformations. Our method, which
can also be seen as an implementation of frame-based noise spectroscopy, solves
the fundamental difficulty in sampling continuous functions with digital
filters by introducing a transformation that relates the arithmetic and logical
time domains. In comparison to standard, frequency-based dynamical-decoupling
noise spectroscopy protocols, the accuracy of our method is only limited by the
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 auto-correlation function of the
effective magnetic field produced by the nuclear-spin bath on the electronic
spin of a single nitrogen-vacancy center in diamond, discuss practical
limitations of the method, and avenues for further improving the reconstruction
accuracy.
