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arXiv:2606.03109

Game, Set, Quantum: Parameterized Quantum Circuit for Correlated Equilibrium in Bayesian Games

Param Pathak, Vidhi Oad, Nouhaila Innan, Adarsh Ganesan, Muhammad Shafique

quant-phSubmitted Jun 2, 2026• Announced Jun 3, 2026• 0 citations

v1 · Jun 2, 2026 v2 · Aug 21, 2026

Abstract

Strategic decision-making among many agents under incomplete information is central to economics, security, and multi-agent artificial intelligence (AI). Computing equilibria in such settings is challenging because the joint type-action space grows exponentially with the number of players. In binary-type, binary-action Bayesian games, an explicit representation over type-action profiles requires O(22n) entries, making direct linear-programming (LP) formulations memory intensive at moderate player counts. We propose a hybrid quantum-classical framework for approximating Bayes correlated equilibrium using a parameterized quantum circuit (PQC). The PQC represents the conditional strategy distribution with O(nL) trainable parameters, where n is the number of players and L is the circuit depth; for the largest setting studied here, n = 10 and L = 2, this corresponds to 60 trainable angles. The circuit is trained by gradient-based regret minimization with a negative entropy regularizer and a curriculum schedule over player counts. On a poker-style Bayesian game with two to ten players, the proposed solver achieves lower mean clipped regret than MCCFR across all tested player counts and lower regret than DCFR up to eight players, while DCFR performs best at ten players. These results show that compact PQC parameterizations can provide a viable variational representation for approximate equilibrium computation, while highlighting the roles of ansatz expressivity, optimization strategy, and classical simulation cost.

Comments: 19 pages, 10 figures

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Computing Nash equilibria of zero-sum games in classical and quantum settings is extensively studied. For general-sum games, computing Nash equilibria is PPAD-hard and the computing of a more general concept called correlated equilibria has been widely explored in game theory. In this paper, we initiate the study of quantum algorithms for computing

\varepsilon

-approximate correlated equilibria (CE) and coarse correlated equilibria (CCE) in multi-player normal-form games. Our approach utilizes quantum improvements to the multi-scale Multiplicative Weight Update (MWU) method for CE calculations, achieving a query complexity of

\tilde{O}(m\sqrt{n})

for fixed

\varepsilon

. For CCE, we extend techniques from quantum algorithms for zero-sum games to multi-player settings, achieving query complexity

\tilde{O}(m\sqrt{n}/\varepsilon^{2.5})

. Both algorithms demonstrate a near-optimal scaling in the number of players

m

and actions

n

, as confirmed by our quantum query lower bounds.

Parameterized quantum circuits (PQC, aka, variational quantum circuits) are among the proposals for a computational advantage over classical computation of near-term (not fault tolerant) digital quantum computers. PQCs have to be "trained" -- i.e., the expectation value function has to be maximized over the space of parameters. This paper deals with the number of samples (or "runs" of the quantum computer) which are required to train the PQC, and approaches it from an information theoretic viewpoint. The main take-away is a disparity in the large amount of information contained in a single exact evaluation of the expectation value, vs the exponentially small amount contained in the random sample obtained from a single run of the quantum circuit.

As quantum processors advance, the emergence of large-scale decentralized systems involving interacting quantum-enabled agents is on the horizon. Recent research efforts have explored quantum versions of Nash and correlated equilibria as solution concepts of strategic quantum interactions, but these approaches did not directly connect to decentralized adaptive setups where agents possess limited information. This paper delves into the dynamics of quantum-enabled agents within decentralized systems that employ no-regret algorithms to update their behaviors over time. Specifically, we investigate two-player quantum zero-sum games and polymatrix quantum zero-sum games, showing that no-regret algorithms converge to separable quantum Nash equilibria in time-average. In the case of general multi-player quantum games, our work leads to a novel solution concept, that of the separable quantum coarse correlated equilibria (QCCE), as the convergent outcome of the time-averaged behavior no-regret algorithms, offering a natural solution concept for decentralized quantum systems. Finally, we show that computing QCCEs can be formulated as a semidefinite program and establish the existence of entangled (i.e., non-separable) QCCEs, which are unlearnable via the current paradigm of no-regret learning.