Quantum Safe Stochastic Linear Bandits

We initiate the study of safe stochastic linear bandits under quantum feedback models. The learner must maximize an unknown linear reward while satisfying unknown linear constraints at every interaction. Under coherent unitary access to the joint reward-and-constraint distribution, we develop QRS-COLTS, a staged quantum analog of constrained linear Thompson sampling. With high probability, QRS-COLTS plays only feasible actions and achieves regret polylogarithmic in the quantum query budget; an isotropic multivariate estimator reduces the vector-feedback cost from linear to square-root dependence on the number of constraints, up to logarithmic factors. We also formulate a stronger query-weighted coherent-exploration model with a safe-action bomb flag. In this model, the learner may query action superpositions, the bomb provides exact but dangerous safe-set access, and regret is charged by the action weights in the query state. We present a quantum algorithm, BCO-QLinTS, that uses bomb-certified quantum convex optimization to select safe actions directly, thereby avoiding the need to estimate the constraint matrix. Its query-weighted regret is polylogarithmic in the horizon, with a bomb-safe optimization overhead that depends only polynomially on the number of constraints.