# Sample-optimal classical shadows for pure states

arXiv: https://arxiv.org/abs/2211.11810 · HTML: https://mlsanity.com/paper/2211.11810/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 31 references that are in the map.
- First posted on arXiv 21 Nov 2022.

## Builds on (most cited first, 5 of 5)

- [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)

## Built on by (most cited first, 4 of 4)

- [Improved classical shadows from local symmetries in the Schur basis](https://mlsanity.com/paper/2405.09525/) (2024)
- [Optimal high-precision shadow estimation](https://mlsanity.com/paper/2407.13874/) (2024)
- [Principal eigenstate classical shadows](https://mlsanity.com/paper/2405.13939/) (2024)
- [Classical Shadows with Improved Median-of-Means Estimation](https://mlsanity.com/paper/2412.03381/) (2024)

## Abstract (arXiv)

We consider the classical shadows task for pure states in the setting of both joint and independent measurements. The task is to measure few copies of an unknown pure state ρ in order to learn a classical description which suffices to later estimate expectation values of observables. Specifically, the goal is to approximate Tr(O ρ) for any Hermitian observable O to within additive error ε provided Tr(O2)≤ B and lVert O rVert = 1. Our main result applies to the joint measurement setting, where we show tildeTheta(√Bε-1 + ε-2) samples of ρ are necessary and sufficient to succeed with high probability. The upper bound is a quadratic improvement on the previous best sample complexity known for this problem. For the lower bound, we see that the bottleneck is not how fast we can learn the state but rather how much any classical description of ρ can be compressed for observable estimation. In the independent measurement setting, we show that mathcal O(√Bd ε-1 + ε-2) samples suffice. Notably, this implies that the random Clifford measurements algorithm of Huang, Kueng, and Preskill, which is sample-optimal for mixed states, is not optimal for pure states. Interestingly, our result also uses the same random Clifford measurements but employs a different estimator.