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 \(O({d}^{3}{\log }^{2}T)\) , while the infidelity of the online estimate after t samples scales as \(\widetilde{O}(1/t)\) . 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 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({d}^{3}{\log }^{2}T)) , while the infidelity of the online estimate after t samples scales as (\widetilde{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 d, after T measured copies, our protocol achieves cumulative regret O(d3log2T), and at each intermediate time t≤T its current estimate has online infidelity 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.
How many samples of a quantum state are required to learn a complete description of it? As we will see in this thesis, the fine-grained answer depends on the measurements available to the learner, but in general it is at least Ω(d^2/ϵ^2) where d is the dimension of the state and ϵ the trace distance accuracy. Optimal algorithms for this task -- known as quantum state tomography -- make use of powerful, yet highly impractical entangled measurements, where some joint measurement is performed on all copies of the state. What can be accomplished without such measurements, where one must perform measurements on individual copies of the states? \n \nIn Chapter 2 we show a relationship between the recently proposed quantum online learning framework and quantum state tomography. Specifically, we show that tomography can be accomplished using online learning algorithms in a black-box manner and O(d^4/ϵ^4) two-outcome measurements on separate copies of the state. The interpretation of this approach is that the experimentalist uses informative measurements to teach the learner by helping it make "mistakes" on measurements as early as possible. \n \nWe move on to proving lower bounds on tomography in Chapter 3. First, we review a known lower bound for entangled measurements as well as a Ω(d^3/ϵ^2) lower bound in the setting where non-entangled measurements are made non-adaptively, both due to Ref.[18]. We then derive a novel bound of Ω(d^4/ϵ^2) samples when the learner is further restricted to observing a constant number of outcomes (e.g., two-outcome measurements). This implies that the folklore "Pauli tomography" algorithm is optimal in this setting. \n \nUnderstanding the power of adaptive measurements, where measurement choices can depend on previous outcomes, is currently an open problem. In Chapter 4 we present two scenarios in which adapting on previous outcomes makes no difference to the number of samples required. In the first, the learner is limited to adapting on at most o(d^2/ϵ^2) of the previous outcomes. In the second, measurements are drawn from some set of at most exp(O(d)) measurements. In particular, this second lower bound implies that adaptivity makes no difference in the regime of efficiently implementable measurements, in the context of quantum computing. \n \nFinally, we apply the above technique to the problems of classical shadows and shadow tomography to obtain similar lower bounds. Here, one is interested only in determining the expectations of some fixed set of observables. We once again find that, for the worst-case input of observables, adaptivity makes no difference to the sample complexity when considering efficient, non-entangled measurements. As a corollary, we find a straightforward algorithm for shadow tomography is optimal in this setting.
Quantum state tomography is fundamental to quantum information processing but becomes infeasible at scale due to the exponential growth of the state space. Shadow tomography alleviates this challenge by focusing on predicting measurement outcomes rather than reconstructing the full state. Its online variant models adaptive and potentially adversarial measurement scenarios, where a learner sequentially predicts outcomes while competing with the best fixed quantum state in hindsight.
We show that exploiting additional structure in the measurements leads to significantly stronger regret guarantees. In particular, under the assumption that the adversarial measurements have bounded Frobenius norm, we analyze online mirror descent and derive optimal regret bounds that depend on intrinsic structural properties, such as rank or sparsity in the standard basis, rather than the dimension of the measurement operators. As a complementary result, we also show that, even in the setting where adversarial measurements are known to be sparse in the Pauli basis commonly used in variational quantum eigensolvers and near-term quantum error mitigation, the underlying regret bound for learning quantum states is the same as that obtained in the generic setting, where the adversarial measurements are not known to possess any particular structure.
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.
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.
Unlike in finite dimensions, quantum information in continuous-variable systems has the peculiar feature that without imposing physical constraints, the sample complexity of state tomography can be unbounded. Remarkably, this is even the case for state-of-the-art protocols for learning Gaussian states, which have finite-dimensional descriptions: the best known rates scale with loglogE, where E is the energy of the system. We prove this is not an artifact of existing analyses, but a fundamental limitation of the measurements used. We show: (1) Any protocol that uses Gaussian measurements, even entangled or adaptively chosen ones, must incur a loglogE dependence. This answers an open question posed by a number of previous works. (2) There is a smooth tradeoff between the number of rounds of adaptivity and the energy dependence, and we give a matching protocol achieving this interpolated rate. (3) With highly entangled, non-Gaussian measurements, one can learn n-mode pure Gaussian states with O(n2/ε2) samples, independent of E. This answers an open question posed by Chen et al. (4) A simple protocol based on the single-copy canonical phase POVM of Holevo and Helstrom learns single-mode pure Gaussian states with O(1/ε2) samples, again independent of E.
Our results clarify the role of energy in bosonic state tomography and shed new light on the intriguing interplay between adaptivity, entanglement, and magic in quantum learning.