# Efficient learning of bosonic Gaussian unitaries

arXiv: https://arxiv.org/abs/2510.05531 · HTML: https://mlsanity.com/paper/2510.05531/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 115 references that are in the map.
- First posted on arXiv 7 Oct 2025.

## Builds on (most cited first, 26 of 26)

- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Quantum advantage in learning from experiments](https://mlsanity.com/paper/2112.00778/) (2021)
- [Provably efficient machine learning for quantum many-body problems](https://mlsanity.com/paper/2106.12627/) (2021)
- [Efficient quantum tomography II](https://mlsanity.com/paper/1612.00034/) (2016)
- [Exponential separations between learning with and without quantum memory](https://mlsanity.com/paper/2111.05881/) (2021)
- [Sub-Gaussian estimators of the mean of a random vector](https://mlsanity.com/paper/1702.00482/) (2017)
- [A survey on the complexity of learning quantum states](https://mlsanity.com/paper/2305.20069/) (2023)
- [Entanglement is Necessary for Optimal Quantum Property Testing](https://mlsanity.com/paper/2004.07869/) (2020)
- [Learning shallow quantum circuits](https://mlsanity.com/paper/2401.10095/) (2024)
- [Sample Efficient Algorithms for Learning Quantum Channels in PAC Model and the Approximate State Discrimination Problem](https://mlsanity.com/paper/1810.10938/) (2018)
- [Toward Instance-Optimal State Certification With Incoherent Measurements](https://mlsanity.com/paper/2102.13098/) (2021)
- [When Does Adaptivity Help for Quantum State Learning?](https://mlsanity.com/paper/2206.05265/) (2022)
- [Tight Bounds for Quantum State Certification with Incoherent Measurements](https://mlsanity.com/paper/2204.07155/) (2022)
- [Improved Stabilizer Estimation via Bell Difference Sampling](https://mlsanity.com/paper/2304.13915/) (2023)
- [Query-optimal estimation of unitary channels in diamond distance](https://mlsanity.com/paper/2302.14066/) (2023)
- [Efficient Tomography of Non-Interacting Fermion States](https://mlsanity.com/paper/2102.10458/) (2021)
- [Tight bounds on Pauli channel learning without entanglement](https://mlsanity.com/paper/2309.13461/) (2023)
- [Efficient Pauli channel estimation with logarithmic quantum memory](https://mlsanity.com/paper/2309.14326/) (2023)
- [Stabilizer bootstrapping: A recipe for efficient agnostic tomography and magic estimation](https://mlsanity.com/paper/2408.06967/) (2024)
- [Learning finitely correlated states: stability of the spectral reconstruction](https://mlsanity.com/paper/2312.07516/) (2023)
- [Agnostic Tomography of Stabilizer Product States](https://mlsanity.com/paper/2404.03813/) (2024)
- [A learning theory for quantum photonic processors and beyond](https://mlsanity.com/paper/2209.03075/) (2022)
- [Information-Computation Gaps in Quantum Learning via Low-Degree Likelihood](https://mlsanity.com/paper/2505.22743/) (2025)
- [Predicting quantum channels over general product distributions](https://mlsanity.com/paper/2409.03684/) (2024)
- [Efficient Hamiltonian, structure and trace distance learning of Gaussian states](https://mlsanity.com/paper/2411.03163/) (2024)
- [Mildly-Interacting Fermionic Unitaries are Efficiently Learnable](https://mlsanity.com/paper/2504.11318/) (2025)

## Built on by (most cited first, 1 of 1)

- [Parameterised graph theory for tensor networks: entanglement rerouting, structural simplification, and agnostic tomography](https://mlsanity.com/paper/2609.04165/) (2026)

## Abstract (arXiv)

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.