# Classical Shadows With Noise

arXiv: https://arxiv.org/abs/2011.11580 · HTML: https://mlsanity.com/paper/2011.11580/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Nov 2020.

## Built on by (most cited first, 14 of 14)

- [Learning to predict arbitrary quantum processes](https://mlsanity.com/paper/2210.14894/) (2022)
- [A Primer on Quantum Machine Learning](https://mlsanity.com/paper/2511.15969/) (2025)
- [Demonstration of Robust and Efficient Quantum Property Learning with Shallow Shadows](https://mlsanity.com/paper/2402.17911/) (2024)
- [Hardware-efficient learning of quantum many-body states](https://mlsanity.com/paper/2212.06084/) (2022)
- [Hamiltonian-Driven Shadow Tomography of Quantum States](https://mlsanity.com/paper/2102.10132/) (2021)
- [Scalable bayesian shadow tomography for quantum property estimation with set transformers](https://mlsanity.com/paper/2509.18674/) (2025)
- [Quantum-Inspired Self-Attention in a Large Language Model](https://mlsanity.com/paper/2603.03318/) (2026)
- [A quantum tug of war between randomness and symmetries on homogeneous spaces](https://mlsanity.com/paper/2309.05253/) (2023)
- [Noise-tolerant learnability of shallow quantum circuits from statistics and the cost of quantum pseudorandomness](https://mlsanity.com/paper/2405.12085/) (2024)
- [Principal eigenstate classical shadows](https://mlsanity.com/paper/2405.13939/) (2024)
- [Predicting adaptively chosen observables in quantum systems](https://mlsanity.com/paper/2410.15501/) (2024)
- [Classical Shadows with Improved Median-of-Means Estimation](https://mlsanity.com/paper/2412.03381/) (2024)
- [Privacy-Utility Tradeoffs in Quantum Information Processing](https://mlsanity.com/paper/2602.10510/) (2026)
- [Symmetry Discovery in Quantum Learning: Observable-Level and Task-Level Inference from Finite Measurements](https://mlsanity.com/paper/2610.00157/) (2026)

## Abstract (arXiv)

The classical shadows protocol, recently introduced by Huang, Kueng, and Preskill [Nat. Phys. 16, 1050 (2020)], is a quantum-classical protocol to estimate properties of an unknown quantum state. Unlike full quantum state tomography, the protocol can be implemented on near-term quantum hardware and requires few quantum measurements to make many predictions with a high success probability. In this paper, we study the effects of noise on the classical shadows protocol. In particular, we consider the scenario in which the quantum circuits involved in the protocol are subject to various known noise channels and derive an analytical upper bound for the sample complexity in terms of a shadow seminorm for both local and global noise. Additionally, by modifying the classical post-processing step of the noiseless protocol, we define a new estimator that remains unbiased in the presence of noise. As applications, we show that our results can be used to prove rigorous sample complexity upper bounds in the cases of depolarizing noise and amplitude damping.