Boson is one of the most basic types of particles and preserves the commutation
relation. An efficient way to measure a bosonic system is important not only
for simulating complex physics phenomena of bosons (such as nuclei) on a qubit
based quantum computer, but for extracting classical information from a quantum
simulator/computer that itself is built with bosons (such as a continuous variable
quantum computer). Extending the recently proposed measurement schemes for
qubits, such as shadow tomography and other local measurement schemes, here we
study efficient measurement approaches for bosonic systems. We consider truncated
qudit and continuous variable systems, corresponding to simulated bosons on a discrete
quantum computer and an inherent boson system, respectively, and propose different
measurement schemes with theoretical analyses of the variances for these two cases. We
numerically test the schemes for measuring nuclei vibrations simulated using a discrete
quantum computer and a continuous variable Gaussian state, and the simulation results
show great improvement of the performance of the proposed method compared to
conventional ones.