Estimating many local expectation values over time is a central measurement bottleneck in quantum simulation and device characterization. We study the task of reconstructing the Pauli-signal matrix Sij=Tr(Oiρ(tj)) for a collection of M low-weight Pauli observables {Oi}i=1M over N timesteps {tj}j=1N, while minimizing the total number of device shots. We propose a Compressed Sensing Shadow Tomography (CSST) protocol that combines two complementary reductions. First, local classical shadows reduce the observable dimension by enabling many Pauli expectation values to be estimated from the same randomized snapshots at a fixed time. Second, compressed sensing reduces the time dimension by exploiting the fact that many expectation-value traces are spectrally sparse or compressible in a unitary (e.g., Fourier) transform basis. Operationally, CSST samples m≪N timesteps uniformly at random, collects shadows only at those times, and then reconstructs each length-N signal via standard ℓ1-based recovery in the unitary transform domain. We provide end-to-end guarantees that explicitly combine shadow estimation error with compressed sensing recovery bounds. For exactly s-sparse signals in a unitary transform basis, we show that m=O(slog2slogN) random timesteps suffice (with high probability), leading to total-shot savings scaling as Θ(N/s) (i.e., up to polylogarithmic factors) relative to collecting shadows at all N timesteps. For approximately sparse signals, the reconstruction error decomposes into a compressibility (tail) term plus a noise term. We present numerical experiments on noisy many-qubit dynamics that support strong Fourier compressibility of Pauli traces and demonstrate substantial shot reductions with accurate reconstruction.
In engineered quantum systems, the Hamiltonian is often not completely known and needs to be determined experimentally with accuracy and efficiency. We show that this may be done at temperatures that are higher than the characteristic interaction energies, but not too much higher. The condition for this is that there are not too many multiparticle interactions: the Hamiltonian is sparse in a well-defined sense. The protocol that accomplishes this is related to compressed sensing methods of classical signal processing, in this case applied to sparse rather than low-rank matrices.
Optical imaging of quantum emitters is essential for a wide range of quantum applications. Conventional confocal imaging relies on point-by-point raster scanning, which is inherently time-consuming and photon-inefficient, particularly for sparse emitter distributions and photon-limited samples. Here, we demonstrate a compressive sensing-based imaging approach, where spatially structured wide-field excitation replaces raster scanning, enabling reconstruction of sparse emitters. In our implementation, random binary patterns are used to acquire compressive measurements, from which the spatial fluorescence distribution is reconstructed using a GPSR-BB algorithm. We experimentally demonstrate this approach using nitrogen-vacancy (NV) centers in diamond as a representative platform, with high-fidelity image reconstruction achieved using only approximately 20% of the measurements required for conventional raster scanning. In addition to intensity reconstruction, we extend this framework to reconstruct spatial maps of the second-order correlation function g(2)(0) from compressive measurements. This enables identification of single-photon emitters through antibunching signatures using significantly reduced data.