# Fermionic tomography and learning

arXiv: https://arxiv.org/abs/2207.14787 · HTML: https://mlsanity.com/paper/2207.14787/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Jul 2022.

## Built on by (most cited first, 3 of 3)

- [Learning to predict arbitrary quantum processes](https://mlsanity.com/paper/2210.14894/) (2022)
- [A survey on the complexity of learning quantum states](https://mlsanity.com/paper/2305.20069/) (2023)
- [A Primer on Quantum Machine Learning](https://mlsanity.com/paper/2511.15969/) (2025)

## Abstract (arXiv)

Shadow tomography via classical shadows is a state-of-the-art approach for estimating properties of a quantum state. We present a simplified, combinatorial analysis of a recently proposed instantiation of this approach based on the ensemble of unitaries that are both fermionic Gaussian and Clifford. Using this analysis, we derive a corrected expression for the variance of the estimator. We then show how this leads to efficient estimation protocols for the fidelity with a pure fermionic Gaussian state (provably) and for an X-like operator of the form (|mathbf 0ranglelanglepsi| + h.c.) (via numerical evidence). We also construct much smaller ensembles of measurement bases that yield the exact same quantum channel, which may help with compilation. We use these tools to show that an n-electron, m-mode Slater determinant can be learned to within ε fidelity given O(n2 m7 log(m / δ) / ε2) samples of the Slater determinant.