Efficient learning of bosonic Gaussian unitaries
This work addresses a fundamental challenge in continuous-variable quantum technologies, such as quantum optics and error correction, by providing an efficient learning method for a multiparameter family of unitaries, which is incremental as it builds on existing concepts but with novel algorithmic improvements.
The authors tackled the problem of learning bosonic Gaussian unitaries efficiently, presenting the first time-efficient algorithm with rigorous analysis that achieves small worst-case error and scales polynomially with key parameters, using only experimentally friendly photonic resources.
Bosonic Gaussian unitaries are fundamental building blocks of central continuous-variable quantum technologies such as quantum-optic interferometry and bosonic error-correction schemes. In this work, we present the first time-efficient algorithm for learning bosonic Gaussian unitaries with a rigorous analysis. Our algorithm produces an estimate of the unknown unitary that is accurate to small worst-case error, measured by the physically motivated energy-constrained diamond distance. Its runtime and query complexity scale polynomially with the number of modes, the inverse target accuracy, and natural energy parameters quantifying the allowed input energy and the unitary's output-energy growth. The protocol uses only experimentally friendly photonic resources: coherent and squeezed probes, passive linear optics, and heterodyne/homodyne detection. We then employ an efficient classical post-processing routine that leverages a symplectic regularization step to project matrix estimates onto the symplectic group. In the limit of unbounded input energy, our procedure attains arbitrarily high precision using only $2m+2$ queries, where $m$ is the number of modes. To our knowledge, this is the first provably efficient learning algorithm for a multiparameter family of continuous-variable unitaries.