# Adaptive Online Learning of Quantum States

arXiv: https://arxiv.org/abs/2206.00220 · HTML: https://mlsanity.com/paper/2206.00220/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 57 references that are in the map.
- First posted on arXiv 1 Jun 2022.

## Builds on (most cited first, 11 of 11)

- [Introduction to Online Convex Optimization](https://mlsanity.com/paper/1909.05207/) (2019)
- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Adaptive Online Learning in Dynamic Environments](https://mlsanity.com/paper/1810.10815/) (2018)
- [Strongly Adaptive Online Learning](https://mlsanity.com/paper/1502.07073/) (2015)
- [Efficient quantum tomography II](https://mlsanity.com/paper/1612.00034/) (2016)
- [Improved Strongly Adaptive Online Learning using Coin Betting](https://mlsanity.com/paper/1610.04578/) (2016)
- [Dynamical Models and Tracking Regret in Online Convex Programming](https://mlsanity.com/paper/1301.1254/) (2013)
- [Online Learning of Quantum States](https://mlsanity.com/paper/1802.09025/) (2018)
- [Pushing the Efficiency-Regret Pareto Frontier for Online Learning of Portfolios and Quantum States](https://mlsanity.com/paper/2202.02765/) (2022)
- [Learning Temporal Quantum Tomography](https://mlsanity.com/paper/2103.13973/) (2021)
- [More Practical and Adaptive Algorithms for Online Quantum State Learning](https://mlsanity.com/paper/2006.01013/) (2020)

## Built on by (most cited first, 12 of 12)

- [A survey on the complexity of learning quantum states](https://mlsanity.com/paper/2305.20069/) (2023)
- [Learning Quantum Processes and Hamiltonians via the Pauli Transfer Matrix](https://mlsanity.com/paper/2212.04471/) (2022)
- [ShadowNet for Data-Centric Quantum System Learning](https://mlsanity.com/paper/2308.11290/) (2023)
- [Online Self-Concordant and Relatively Smooth Minimization, With Applications to Online Portfolio Selection and Learning Quantum States](https://mlsanity.com/paper/2210.00997/) (2022)
- [Logarithmic-Regret Quantum Learning Algorithms for Zero-Sum Games](https://mlsanity.com/paper/2304.14197/) (2023)
- [Online learning of a panoply of quantum objects](https://mlsanity.com/paper/2406.04245/) (2024)
- [Online Learning Quantum States with the Logarithmic Loss via VB-FTRL](https://mlsanity.com/paper/2311.04237/) (2023)
- [Online Learning of Pure States is as Hard as Mixed States](https://mlsanity.com/paper/2502.00823/) (2025)
- [Instance-Optimal Matrix Multiplicative Weight Update and Its Quantum Applications](https://mlsanity.com/paper/2509.08911/) (2025)
- [Quantum Algorithm for Sparse Online Learning with Truncated Gradient Descent](https://mlsanity.com/paper/2411.03925/) (2024)
- [Differential equation quantum solvers: engineering measurements to reduce cost](https://mlsanity.com/paper/2503.22656/) (2025)
- [Guiding Agents of Quantum Games to Equilibrium using Matrix Exponential Fixed-Point Iteration](https://mlsanity.com/paper/2609.21944/) (2026)

## Abstract (arXiv)

The problem of efficient quantum state learning, also called shadow tomography, aims to comprehend an unknown d-dimensional quantum state through POVMs. Yet, these states are rarely static; they evolve due to factors such as measurements, environmental noise, or inherent Hamiltonian state transitions. This paper leverages techniques from adaptive online learning to keep pace with such state changes. The key metrics considered for learning in these mutable environments are enhanced notions of regret, specifically adaptive and dynamic regret. We present adaptive and dynamic regret bounds for online shadow tomography, which are polynomial in the number of qubits and sublinear in the number of measurements. To support our theoretical findings, we include numerical experiments that validate our proposed models.