We develop an efficient and robust approach for quantum measurement of nearly sparse many-body quantum Hamiltonians based on the method of compressive sensing. This work demonstrates that with only $O(s\ln (d))$ experimental configurations, consisting of random local preparations and measurements, one can estimate the Hamiltonian of a $d$-dimensional system, provided that the Hamiltonian is nearly $s$ sparse in a known basis. The classical postprocessing is a convex optimization problem on the total Hilbert space which is generally not scalable. We numerically simulate the performance of this algorithm for three- and four-body interactions in spin-coupled quantum dots and atoms in optical lattices. Furthermore, we apply the algorithm to characterize Hamiltonian fine structure and unknown system-bath interactions.

• Received 1 March 2010

DOI:https://doi.org/10.1103/PhysRevA.84.012107