Efficient tomography of multiple bosonic modes with bounded total excitations

With recent advances in high-fidelity control and quantum error correction over multiple bosonic modes, it becomes an outstanding challenge to efficiently characterize quantum systems of multiple cavities. The standard Wigner tomography suffers exponential overhead with the number of bosonic modes (e.g., limiting us to characterizing only up to two or three bosonic modes). However, we may use the prior knowledge about the quantum state to enable efficient tomography. When we limit the total number of excitations, the size of the space is reduced to a polynomial of the number of bosonic modes. Based on this observation, we propose an efficient tomography protocol to characterize the projected state with polynomial measurements of some product operators.