# Learning pure quantum states (almost) without regret

arXiv: https://arxiv.org/abs/2406.18370 · HTML: https://mlsanity.com/paper/2406.18370/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 34 references that are in the map.
- First posted on arXiv 26 Jun 2024.

## Builds on (most cited first, 6 of 6)

- [Efficient quantum tomography II](https://mlsanity.com/paper/1612.00034/) (2016)
- [Almost Optimal Algorithms for Linear Stochastic Bandits with Heavy-Tailed Payoffs](https://mlsanity.com/paper/1810.10895/) (2018)
- [Lecture Notes: Selected topics on robust statistical learning theory](https://mlsanity.com/paper/1908.10761/) (2019)
- [When Does Adaptivity Help for Quantum State Learning?](https://mlsanity.com/paper/2206.05265/) (2022)
- [Linear bandits with polylogarithmic minimax regret](https://mlsanity.com/paper/2402.12042/) (2024)
- [Quantum contextual bandits and recommender systems for quantum data](https://mlsanity.com/paper/2301.13524/) (2023)

## Built on by (most cited first, 6 of 6)

- [Variational Quantum Optimization with Continuous Bandits](https://mlsanity.com/paper/2502.04021/) (2025)
- [Quantum state-agnostic work extraction (almost) without dissipation](https://mlsanity.com/paper/2505.09456/) (2025)
- [Online Learning of Pure States is as Hard as Mixed States](https://mlsanity.com/paper/2502.00823/) (2025)
- [Reinforcement learning for quantum processes with memory](https://mlsanity.com/paper/2603.25138/) (2026)
- [Learning Pure Quantum States in Any Dimension (Almost) Without Regret](https://mlsanity.com/paper/2605.09019/) (2026)
- [Quantum Multi-Armed Bandits and Linear Bandits: Lower Bounds and Algorithms](https://mlsanity.com/paper/2608.14319/) (2026)

## Abstract (arXiv)

We initiate the study of sample-optimal quantum state tomography with minimal disturbance to the samples. Can we efficiently learn a precise description of a quantum state through sequential measurements of samples while at the same time making sure that the post-measurement state of the samples is only minimally perturbed? Defining regret as the cumulative disturbance of all samples, the challenge is to find a balance between the most informative sequence of measurements on the one hand and measurements incurring minimal regret on the other. Here we answer this question for qubit states by exhibiting a protocol that for pure states achieves maximal precision while incurring a regret that grows only polylogarithmically with the number of samples, a scaling that we show to be optimal.