# Learning in quantum games

arXiv: https://arxiv.org/abs/2302.02333 · HTML: https://mlsanity.com/paper/2302.02333/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 52 references that are in the map.
- First posted on arXiv 5 Feb 2023.

## Builds on (most cited first, 9 of 9)

- [Introduction to Multi-Armed Bandits](https://mlsanity.com/paper/1904.07272/) (2019)
- [Quantum generative adversarial networks](https://mlsanity.com/paper/1804.08641/) (2018)
- [Cycles in adversarial regularized learning](https://mlsanity.com/paper/1709.02738/) (2017)
- [Tsallis-INF: An Optimal Algorithm for Stochastic and Adversarial Bandits](https://mlsanity.com/paper/1807.07623/) (2018)
- [Learning in games with continuous action sets and unknown payoff functions](https://mlsanity.com/paper/1608.07310/) (2016)
- [No-regret learning and mixed Nash equilibria: They do not mix](https://mlsanity.com/paper/2010.09514/) (2020)
- [Online Learning of Quantum States](https://mlsanity.com/paper/1802.09025/) (2018)
- [Quantum Wasserstein Generative Adversarial Networks](https://mlsanity.com/paper/1911.00111/) (2019)
- [A continuous-time approach to online optimization](https://mlsanity.com/paper/1401.6956/) (2014)

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

- [Hamiltonian of polymatrix zero-sum games](https://mlsanity.com/paper/2505.12609/) (2025)
- [Near-Optimal Quantum Algorithms for Computing (Coarse) Correlated Equilibria of General-Sum Games](https://mlsanity.com/paper/2510.16782/) (2025)
- [Guiding Agents of Quantum Games to Equilibrium using Matrix Exponential Fixed-Point Iteration](https://mlsanity.com/paper/2609.21944/) (2026)
- [Average-and Last-Iterate Lower Bounds for Optimistic Matrix Mirror-Prox in Quantum Zero-Sum Games](https://mlsanity.com/paper/2609.38835/) (2026)

## Abstract (arXiv)

In this paper, we introduce a class of learning dynamics for general quantum games, that we call "follow the quantum regularized leader" (FTQL), in reference to the classical "follow the regularized leader" (FTRL) template for learning in finite games. We show that the induced quantum state dynamics decompose into (i) a classical, commutative component which governs the dynamics of the system's eigenvalues in a way analogous to the evolution of mixed strategies under FTRL; and (ii) a non-commutative component for the system's eigenvectors which has no classical counterpart. Despite the complications that this non-classical component entails, we find that the FTQL dynamics incur no more than constant regret in all quantum games. Moreover, adjusting classical notions of stability to account for the nonlinear geometry of the state space of quantum games, we show that only pure quantum equilibria can be stable and attracting under FTQL while, as a partial converse, pure equilibria that satisfy a certain "variational stability" condition are always attracting. Finally, we show that the FTQL dynamics are Poincar'e recurrent in quantum min-max games, extending in this way a very recent result for the quantum replicator dynamics.