Quantum-Enhanced Learning of Continuous-Variable Quantum States

Quantum-Enhanced Learning of Continuous-Variable Quantum States

Ya-Dong Wu, Giulio Chiribella, Nana Liu
QICI Quantum Information and Computation Initiative, Computer Science Department, The University of Hong Kong




Efficient characterization of continuous-variable quantum states is important for quantum communication, quantum sensing, quantum simulation and quantum computing. However, conventional quantum state tomography and recently proposed classical shadow tomography require truncation of the Hilbert space or phase space and the resulting sample complexity scales exponentially with the number of modes. In this paper, we propose a quantum-enhanced learning strategy for continuous-variable states overcoming the previous shortcomings. We use this to estimate the point values of a state characteristic function, which is useful for quantum state tomography and inferring physical properties like quantum fidelity, nonclassicality and quantum non-Gaussianity. We show that for any continuous-variable quantum states $rho$ with reflection symmetry - for example Gaussian states with zero mean values, Fock states, Gottesman-Kitaev-Preskill states, Schr"odinger cat states and binomial code states - on practical quantum devices we only need a constant number of copies of state $rho$ to accurately estimate the square of its characteristic function at arbitrary phase-space points. This is achieved by performinig a balanced beam splitter on two copies of $rho$ followed by homodyne measurements. Based on this result