# Predicting quantum channels over general product distributions

arXiv: https://arxiv.org/abs/2409.03684 · HTML: https://mlsanity.com/paper/2409.03684/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 29 references that are in the map.
- First posted on arXiv 5 Sep 2024.

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

- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Quantum advantage in learning from experiments](https://mlsanity.com/paper/2112.00778/) (2021)
- [Information-theoretic bounds on quantum advantage in machine learning](https://mlsanity.com/paper/2101.02464/) (2021)
- [Learning to predict arbitrary quantum processes](https://mlsanity.com/paper/2210.14894/) (2022)
- [Learning shallow quantum circuits](https://mlsanity.com/paper/2401.10095/) (2024)
- [Dynamical simulation via quantum machine learning with provable generalization](https://mlsanity.com/paper/2204.10269/) (2022)
- [Learning low-degree functions from a logarithmic number of random queries](https://mlsanity.com/paper/2109.10162/) (2021)
- [Learning low-degree quantum objects](https://mlsanity.com/paper/2405.10933/) (2024)

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

- [Artificial intelligence for representing and characterizing quantum systems](https://mlsanity.com/paper/2509.04923/) (2025)
- [Efficient learning of bosonic Gaussian unitaries](https://mlsanity.com/paper/2510.05531/) (2025)

## Abstract (arXiv)

We investigate the problem of predicting the output behavior of unknown quantum channels. Given query access to an n-qubit channel E and an observable O, we aim to learn the mapping beginequation* ρ mapsto Tr(O E[ρ]) endequation* to within a small error for most ρ sampled from a distribution D. Previously, Huang, Chen, and Preskill proved a surprising result that even if E is arbitrary, this task can be solved in time roughly nO(log(1/ε)), where ε is the target prediction error. However, their guarantee applied only to input distributions D invariant under all single-qubit Clifford gates, and their algorithm fails for important cases such as general product distributions over product states ρ. In this work, we propose a new approach that achieves accurate prediction over essentially any product distribution D, provided it is not "classical" in which case there is a trivial exponential lower bound. Our method employs a "biased Pauli analysis," analogous to classical biased Fourier analysis. Implementing this approach requires overcoming several challenges unique to the quantum setting, including the lack of a basis with appropriate orthogonality properties. The techniques we develop to address these issues may have broader applications in quantum information.