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arXiv:2509.24569

Bandits roaming Hilbert space

Josep Lumbreras

Published Sep 29, 2025• 0 citations
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Abstract

This thesis studies the exploration and exploitation trade-off in online learning of properties of quantum states using multi-armed bandits. Given streaming access to an unknown quantum state, in each round we select an observable from a set of actions to maximize its expectation value. Using past information, we refine actions to minimize regret; the cumulative gap between current reward and the maximum possible. We derive information-theoretic lower bounds and optimal strategies with matching upper bounds, showing regret typically scales as the square root of rounds. As an application, we reframe quantum state tomography to both learn the state efficiently and minimize measurement disturbance. For pure states and continuous actions, we achieve polylogarithmic regret using a sample-optimal algorithm based on a weighted online least squares estimator. The algorithm relies on the optimistic principle and controls the eigenvalues of the design matrix. We also apply our framework to quantum recommender systems and thermodynamic work extraction from unknown states. In this last setting, our results demonstrate an exponential advantage in work dissipation over tomography-based protocols.

DOI
10.48550/arxiv.2509.24569

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This thesis studies the exploration and exploitation trade-off in online learning of properties of quantum states using multi-armed bandits. Given streaming access to an unknown quantum state, in each round we select an observable from a set of actions to maximize its expectation value. Using past information, we refine actions to minimize regret; the cumulative gap between current reward and the maximum possible. We derive information-theoretic lower bounds and optimal strategies with matching upper bounds, showing regret typically scales as the square root of rounds. As an application, we reframe quantum state tomography to both learn the state efficiently and minimize measurement disturbance. For pure states and continuous actions, we achieve polylogarithmic regret using a sample-optimal algorithm based on a weighted online least squares estimator. The algorithm relies on the optimistic principle and controls the eigenvalues of the design matrix. We also apply our framework to quantum recommender systems and thermodynamic work extraction from unknown states. In this last setting, our results demonstrate an exponential advantage in work dissipation over tomography-based protocols.
We study the problem of online learning of quantum states with logarithmic loss, which is a quantum generalization of online portfolio selection and has applications to quantum state tomography, the task of estimating an unknown quantum state from measurement outcomes. We first establish a regret lower bound, showing that the regret rates of Exponential Weights and the algorithm of Tseng et al. (2025) have optimal polynomial dependence on the dimension. Our analysis leads to two further results. First, our lower-bound proof leads to a quantum state tomography algorithm that is sample optimal up to polylogarithmic factors in the sample size. Second, we develop an algorithm for constructing confidence sequences for the mean of a matrix-valued random variable. A numerical experiment shows that the latter algorithm is competitive with existing methods.
We initiate the study of tradeoffs between exploration and exploitation in online learning of properties of quantum states. Given sequential oracle access to an unknown quantum state, in each round, we are tasked to choose an observable from a set of actions aiming to maximize its expectation value on the state (the reward). Information gained about the unknown state from previous rounds can be used to gradually improve the choice of action, thus reducing the gap between the reward and the maximal reward attainable with the given action set (the regret). We provide various information-theoretic lower bounds on the cumulative regret that an optimal learner must incur, and show that it scales at least as the square root of the number of rounds played. We also investigate the dependence of the cumulative regret on the number of available actions and the dimension of the underlying space. Moreover, we exhibit strategies that are optimal for bandits with a finite number of arms and general mixed states.
In reinforcement learning, an agent interacts sequentially with an environment to maximize a reward, receiving only partial, probabilistic feedback. This creates a fundamental exploration-exploitation trade-off: the agent must explore to learn the hidden dynamics while exploiting this knowledge to maximize its target objective. While extensively studied classically, applying this framework to quantum systems requires dealing with hidden quantum states that evolve via unknown dynamics. We formalize this problem via a framework where the environment maintains a hidden quantum memory evolving via unknown quantum channels, and the agent intervenes sequentially using quantum instruments. For this setting, we adapt an optimistic maximum-likelihood estimation algorithm. We extend the analysis to continuous action spaces, allowing us to model general positive operator-valued measures (POVMs). By controlling the propagation of estimation errors through quantum channels and instruments, we prove that the cumulative regret of our strategy scales as O~(K)\widetilde{\mathcal{O}}(\sqrt{K})O(K​) over KKK episodes. Furthermore, via a reduction to the multi-armed quantum bandit problem, we establish information-theoretic lower bounds demonstrating that this sublinear scaling is strictly optimal up to polylogarithmic factors. As a physical application, we consider state-agnostic work extraction. When extracting free energy from a sequence of non-i.i.d. quantum states correlated by a hidden memory, any lack of knowledge about the source leads to thermodynamic dissipation. In our setting, the mathematical regret exactly quantifies this cumulative dissipation. Using our adaptive algorithm, the agent uses past energy outcomes to improve its extraction protocol on the fly, achieving sublinear cumulative dissipation, and, consequently, an asymptotically zero dissipation rate.
We study quantum multi-armed bandits (QMAB) and quantum linear bandits (QLB), where the learner queries each arm or action through a quantum reward oracle or its inverse. Prior work gives algorithms over horizon TTT with regret O(Klog⁡T)O(K\log T)O(KlogT) for QMAB with KKK arms and O(d2polylog⁡T)O(d^2\operatorname{polylog} T)O(d2polylogT) for ddd-dimensional QLB. This leaves open the optimal dependence on KKK and TTT and whether the dependence on ddd can be further improved. In this work, we prove the first tight minimax regret bound of Θ(Klog⁡(1+T/K))Θ(K\log(1+T/K))Θ(Klog(1+T/K)) for QMAB and the first lower bound of Ω(dlog⁡(1+T/d))Ω(d\log(1+T/d))Ω(dlog(1+T/d)) for finite-action QLB, ruling out regret independent of TTT. Our lower bounds rely on a high-confidence single-arm quantum testing lower bound for distinguishing a fixed reward mean from an interval of alternatives. A bandit-to-testing reduction then lifts it to the QMAB lower bound, while a linear embedding gives the finite-action QLB lower bound. The matching QMAB upper bound is obtained using a tail bound for the Quantum Monte Carlo (QMC) estimator. For finite-action QLB, we propose a phased elimination algorithm that combines a low-bias low-variance quantum mean estimator with a small-support GGG-optimal design through a query allocation matched to the design weights. When the action set has size poly⁡(d)\operatorname{poly}(d)poly(d), its regret is nearly linear in ddd and matches our lower bound up to polylogarithmic factors.
We initiate the study of sample-optimal quantum state tomography with minimal disturbance to the samples. Can we efficiently learn a precise description of a quantum state through sequential measurements of samples while at the same time ensuring that the post-measurement states are only minimally perturbed? We quantify the accumulated cost of the protocol over T samples by an additive regret objective, which controls the cumulative expected post-measurement infidelity of the consumed copies. The challenge is to balance informative measurements with measurements that keep this accumulated cost small. Here, we answer this question for all pure states by exhibiting an adaptive protocol whose expected regret is O(d3log⁡2T)O({d}^{3}{\log }^{2}T)O(d3log2T) , while the infidelity of the online estimate after t samples scales as O~(1/t)\widetilde{O}(1/t)O(1/t) . We also prove a logarithmic lower bound, showing that for fixed dimension, the dependence on the number of samples is optimal up to one logarithmic factor.
We extend quantum state tomography with minimal cumulative disturbance, first investigated in [arXiv:2406.18370], to arbitrary finite-dimensional pure states. A learner sequentially receives fresh copies of an unknown pure state, chooses a rank-one projector for each copy using the previous outcomes, and performs the corresponding two-outcome projective measurement. The goal is to learn the state while keeping the chosen projectors close to the unknown state in order to minimize disturbance. The qubit solution relies on the special geometry of the Bloch sphere and does not extend directly to qudits, where pure states form a curved manifold. We show that this obstruction can be overcome by working locally on the pure-state manifold. The algorithm proceeds in epochs. In each epoch, it fixes a current estimate, measures pairs of nearby rank-one projectors obtained by moving in opposite tangent directions, and takes differences of the corresponding outcomes. This gives an exact linear observation of the tangent component of the error. The resulting local linear models are combined with a robust variance-adaptive estimator and a hot-start regularization that transfers precision across epochs. For every unknown pure state in dimension ddd, after TTT measured copies, our protocol achieves cumulative regret O(d3log⁡2T)\mathcal{O}(d^3\log^2 T)O(d3log2T), and at each intermediate time t≤Tt\leq Tt≤T its current estimate has online infidelity O(d3log⁡(T)/t)\mathcal{O}(d^3\log(T)/t)O(d3log(T)/t). Hence, pure-state tomography with essentially no cumulative disturbance is not a peculiarity of qubits but a geometric phenomenon that persists for qudits.