# Matrix Multiplicative Weights Updates in Quantum Zero-Sum Games: Conservation Laws & Recurrence

arXiv: https://arxiv.org/abs/2211.01681 · HTML: https://mlsanity.com/paper/2211.01681/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Nov 2022.

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

- [Quantum thermodynamics and semi-definite optimization](https://mlsanity.com/paper/2505.04514/) (2025)
- [Configurable Mirror Descent: Towards a Unification of Decision Making](https://mlsanity.com/paper/2405.11746/) (2024)
- [Hamiltonian of polymatrix zero-sum games](https://mlsanity.com/paper/2505.12609/) (2025)
- [Optimistic Online Learning in Symmetric Cone Games](https://mlsanity.com/paper/2504.03592/) (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)

## Abstract (arXiv)

Recent advances in quantum computing and in particular, the introduction of quantum GANs, have led to increased interest in quantum zero-sum game theory, extending the scope of learning algorithms for classical games into the quantum realm. In this paper, we focus on learning in quantum zero-sum games under Matrix Multiplicative Weights Update (a generalization of the multiplicative weights update method) and its continuous analogue, Quantum Replicator Dynamics. When each player selects their state according to quantum replicator dynamics, we show that the system exhibits conservation laws in a quantum-information theoretic sense. Moreover, we show that the system exhibits Poincare recurrence, meaning that almost all orbits return arbitrarily close to their initial conditions infinitely often. Our analysis generalizes previous results in the case of classical games.