Efficient learning of bosonic Gaussian unitaries

2025-10-06

AI Summary

This work addresses the challenge of efficiently reconstructing bosonic Gaussian unitary transformations in continuous-variable quantum technologies. Methodologically, it employs coherent and squeezed states as probes, combined with passive linear optical elements and homodyne/heterodyne detection, augmented by symplectic regularization in classical post-processing to enforce strict adherence to the symplectic structure of the estimated transformation matrix. Under an energy constraint, the algorithm achieves high-precision estimation in diamond-norm distance using polynomial resources in mode number $m$, precision $varepsilon$, and energy. In the unbounded-energy limit, it attains arbitrary precision with only $2m+2$ queries. This constitutes the first Gaussian unitary learning scheme with provably polynomial time and query complexity in $m$, $varepsilon$, and energy—establishing a scalable characterization tool for quantum optical interferometry and bosonic quantum error correction.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Trust and reliance of crowd workers and data experts on GenAI

Abstract

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.

Problem

• Efficiently learning bosonic Gaussian unitaries with rigorous analysis
• Achieving accurate estimation with energy-constrained diamond distance
• Using photonic resources and classical post-processing for implementation

Innovation

• Efficient algorithm for learning bosonic Gaussian unitaries
• Uses coherent and squeezed probes with linear optics
• Classical post-processing with symplectic regularization step