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

• URL: http://arxiv.org/abs/2211.01681v2
• Date: Thu, 27 Apr 2023 03:37:25 GMT
• Title: Matrix Multiplicative Weights Updates in Quantum Zero-Sum Games: Conservation Laws & Recurrence
• Authors: Rahul Jain, Georgios Piliouras and Ryann Sim
• Abstract summary: We focus on learning in quantum zero-sum games under Matrix Multiplicative Weights Update and its continuous analogue, Quantum Replicator Dynamics. Our analysis generalizes previous results in the case of classical games.
• Score: 39.379577980832835
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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