# Principal eigenstate classical shadows

arXiv: https://arxiv.org/abs/2405.13939 · HTML: https://mlsanity.com/paper/2405.13939/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 40 references that are in the map.
- First posted on arXiv 22 May 2024.

## Builds on (most cited first, 8 of 8)

- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Mean estimation and regression under heavy-tailed distributions--a survey](https://mlsanity.com/paper/1906.04280/) (2019)
- [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)
- [Lecture Notes: Selected topics on robust statistical learning theory](https://mlsanity.com/paper/1908.10761/) (2019)
- [Classical Shadows With Noise](https://mlsanity.com/paper/2011.11580/) (2020)
- [Sample-optimal classical shadows for pure states](https://mlsanity.com/paper/2211.11810/) (2022)

## Abstract (arXiv)

Given many copies of an unknown quantum state ρ, we consider the task of learning a classical description of its principal eigenstate. Namely, assuming that ρ has an eigenstate |φrangle with (unknown) eigenvalue λ > 1/2, the goal is to learn a (classical shadows style) classical description of |φrangle which can later be used to estimate expectation values langle φ |O| φ rangle for any O in some class of observables. We consider the sample-complexity setting in which generating a copy of ρ is expensive, but joint measurements on many copies of the state are possible. We present a protocol for this task scaling with the principal eigenvalue λ and show that it is optimal within a space of natural approaches, e.g., applying quantum state purification followed by a single-copy classical shadows scheme. Furthermore, when λ is sufficiently close to 1, the performance of our algorithm is optimal--matching the sample complexity for pure state classical shadows.