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Precision Bounds on Continuous-Variable State Tomography Using Classical Shadows



Srilekha Gandhari, Victor Albert, Thomas Gerrits, Jacob Taylor, Michael Gullans


Shadow tomography is a framework for constructing succinct descriptions of quantum states, called classical shadows, with powerful methods to bound the estimators used. Classical shadows are well-studied in the discrete-variable case, which consists of states of several qubits. Here, we extend this framework to continuous-variable quantum systems, such as optical modes and harmonic oscillators by adapting existing techniques for optical tomography. We analyze the efficiency of homodyne, heterodyne, photon number resolving (PNR) and photon parity experimental meth- ods from optical tomography for constructing finite-dimensional classical shadows of a continuous variable state. We provide rigorous bounds on the variance of estimating density matrices from these experimental methods. We show that, to reach a desired precision on the classical shadow of an -photon density matrix with a high probability, homodyne detection requires (^5) measure- ments in the worst case, whereas PNR and photon parity detection require (^4) measurements in the worst case. For heterodyne measurements, the upper bound effectively becomes exponential in . We benchmark these results against numerical simulation, as well as experimental data from optical homodyne experiments, finding that homodyne tomography significantly out-performs these bounds with a more typical scaling of the number of samples that is close to linear in . We further show how to extend our results to the efficient construction of multimode shadows based on local measurements.
Physical Review X - Quantum


Quantum information science, quantum tomography, continuous variable systems


Gandhari, S. , Albert, V. , Gerrits, T. , Taylor, J. and Gullans, M. (2024), Precision Bounds on Continuous-Variable State Tomography Using Classical Shadows, Physical Review X - Quantum, [online],, (Accessed April 17, 2024)
Created March 18, 2024, Updated March 27, 2024