# Mildly-Interacting Fermionic Unitaries are Efficiently Learnable

arXiv: https://arxiv.org/abs/2504.11318 · HTML: https://mlsanity.com/paper/2504.11318/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 44 references that are in the map.
- First posted on arXiv 15 Apr 2025.

## Builds on (most cited first, 9 of 9)

- [Learning quantum states and unitaries of bounded gate complexity](https://mlsanity.com/paper/2310.19882/) (2023)
- [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)
- [Efficient Learning of Quantum States Prepared With Few Non-Clifford Gates](https://mlsanity.com/paper/2305.13409/) (2023)
- [Stabilizer bootstrapping: A recipe for efficient agnostic tomography and magic estimation](https://mlsanity.com/paper/2408.06967/) (2024)
- [Agnostic Tomography of Stabilizer Product States](https://mlsanity.com/paper/2404.03813/) (2024)
- [Tolerant Testing of Stabilizer States with Mixed State Inputs](https://mlsanity.com/paper/2411.08765/) (2024)
- [Polynomial-time tolerant testing stabilizer states](https://mlsanity.com/paper/2408.06289/) (2024)

## Built on by (most cited first, 2 of 2)

- [Efficient learning of bosonic Gaussian unitaries](https://mlsanity.com/paper/2510.05531/) (2025)
- [Parameterised graph theory for tensor networks: entanglement rerouting, structural simplification, and agnostic tomography](https://mlsanity.com/paper/2609.04165/) (2026)

## Abstract (arXiv)

Recent work has shown that one can efficiently learn fermionic Gaussian unitaries, also commonly known as nearest-neighbor matchcircuits or non-interacting fermionic unitaries. However, one could ask a similar question about unitaries that are near Gaussian: for example, unitaries prepared with a small number of non-Gaussian circuit elements. These operators find significance in quantum chemistry and many-body physics, yet no algorithm exists to learn them. We give the first such result by devising an algorithm which makes queries to an n-mode fermionic unitary U prepared by at most O(t) non-Gaussian gates and returns a circuit approximating U to diamond distance varepsilon in time textrmpoly(n,2t,1/varepsilon). This resolves a central open question of Mele and Herasymenko under the strongest distance metric. In fact, our algorithm is much more general: we define a property of unitary Gaussianity known as unitary Gaussian dimension and show that our algorithm can learn n-mode unitaries of Gaussian dimension at least 2n - O(t) in time textrmpoly(n,2t,1/varepsilon). Indeed, this class subsumes unitaries prepared by at most O(t) non-Gaussian gates but also includes several unitaries that require up to 2O(t) non-Gaussian gates to construct. In addition, we give a textrmpoly(n,1/varepsilon)-time algorithm to distinguish whether an n-mode unitary is of Gaussian dimension at least k or varepsilon-far from all such unitaries in Frobenius distance, promised that one is the case. Along the way, we prove structural results about near-Gaussian fermionic unitaries that are likely to be of independent interest.