Precision Bounds on Continuous-Variable State Tomography using Classical Shadows

Abstract: Shadow tomography is a framework for constructing succinct descriptions of quantum states using randomized measurement bases, called classical shadows, with powerful methods to bound the estimators used. We recast existing experimental protocols for continuous-variable quantum state tomography in the classical-shadow framework, obtaining rigorous bounds on the number of independent measurements needed for estimating density matrices from these protocols. We analyze the efficiency of homodyne, heterodyne, photon number resolving (PNR), and photon-parity protocols. To reach a desired precision on the classical shadow of an $N$-photon density matrix with a high probability, we show that homodyne detection requires an order $\mathcal{O}(N^{4+1/3})$ measurements in the worst case, whereas PNR and photon-parity detection require $\mathcal{O}(N^4)$ measurements in the worst case (both up to logarithmic corrections). We benchmark these results against numerical simulation as well as experimental data from optical homodyne experiments. We find that numerical and experimental homodyne tomography significantly outperforms our bounds, exhibiting a more typical scaling of the number of measurements that is close to linear in $N$. We extend our single-mode results to an efficient construction of multimode shadows based on local measurements.