Among recent insights into learning quantum states, online learning and shadow tomography procedures are notable for their ability to accurately predict expectation values even of adaptively chosen observables. In contrast to the state case, quantum process learning tasks with a similarly adaptive nature have received little attention. In this work, we investigate online learning tasks for quantum processes. Whereas online learning is infeasible for general quantum channels, we show that channels of bounded gate complexity as well as Pauli channels can be online learned in the regret and mistake-bounded models of online learning. In fact, we can online learn probabilistic mixtures of any exponentially large set of known channels. We also provide a provably sample-efficient shadow tomography procedure for Pauli channels. Our results extend beyond quantum channels to non-Markovian multi-time processes, with favorable regret and mistake bounds, as well as a shadow tomography procedure. We complement our online learning upper bounds with mistake as well as computational lower bounds. On the technical side, we make use of the multiplicative weights update algorithm, classical adaptive data analysis, and Bell sampling, as well as tools from the theory of quantum combs for multi-time quantum processes. Our work initiates a study of online learning for classes of quantum channels and, more generally, non-Markovian quantum processes. Given the importance of online learning for state shadow tomography, this may serve as a step towards quantum channel variants of adaptive shadow tomography.
Among recent insights into learning quantum states, online learning and shadow tomography procedures are notable for their ability to accurately predict expectation values even of adaptively chosen observables. In contrast to the state case, quantum process learning tasks with a similarly adaptive nature have received little attention. In this work, we investigate online learning tasks for quantum processes. Whereas online learning is infeasible for general quantum channels, we show that channels of bounded gate complexity as well as Pauli channels can be online learned in the regret and mistake-bounded models of online learning. In fact, we can online learn probabilistic mixtures of any exponentially large set of known channels. We also provide a provably sample-efficient shadow tomography procedure for Pauli channels. Our results extend beyond quantum channels to non-Markovian multi-time processes, with favorable regret and mistake bounds, as well as a shadow tomography procedure. We complement our online learning upper bounds with mistake as well as computational lower bounds. On the technical side, we make use of the multiplicative weights update algorithm, classical adaptive data analysis, and Bell sampling, as well as tools from the theory of quantum combs for multi-time quantum processes. Our work initiates a study of online learning for classes of quantum channels and, more generally, non-Markovian quantum processes. Given the importance of online learning for state shadow tomography, this may serve as a step towards quantum channel variants of adaptive shadow tomography.
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.
Online learning is a framework for interfacing with environments whose structure is revealed through interactions; quantum information theory is a means of representing data and its processing consistent with modern understanding of physics. Their union is symbiotic---sequential processing is light on precious quantum resources, and quantum algorithms can surpass known bounds of classical algorithms---their combination being an area of independent interest. This thesis aims to exemplify the above point via five case studies on tomography, change detection, and function evaluation. In the process of deriving efficient tomography procedures, prior work has shown that quantum states can be learned in online adversarial environments. We extend this notion to subsets of positive semidefinite operators, including general quantum objects such as states, measurements, channels, strategies, co-strategies, and Gram matrices. We tailor a regularized follow-the-leader algorithm and achieve sublinear regret. As a byproduct, we show a generalization of Pinsker's inequality that accounts for differences in trace, its proof method more broadly applicable to general divergences. We further extend online learnability to objects in arbitrary physical theories. We derive mappings between Euclidean Jordan algebras and generalized probabilistic theories, using methods for the former to show that states in the latter are learnable. Specifically, we develop a projective version of the symmetric cone multiplicative weights update algorithm that achieves sublinear regret when learning states of generalized probabilistic theories. We consider the task of detecting changes in sequences of unknown quantum states with a constraint of no false positives, developing efficient and optimal online quantum algorithms. These include online algorithms with small amounts of quantum memory based on the swap test, and optimal algorithms projecting onto the symmetric subspace. We find circuit constructions for these algorithms, analyze their complexity, and show their equivalence under certain memory limitations. The proposed algorithms are applicable to quality control and anomaly detection in quantum devices, and may be of independent interest for identity testing and optimal cloning. We elaborate on the quality control aspect of changepoint methods by developing procedures for aiding the calibration of near-term quantum devices. These consist of algorithms for certification and changepoint detection in Hamiltonian dynamics, which enable continuous monitoring and trigger recalibration procedures. Using a cumulative sum procedure, we attain an asymptotically-optimal scaling of the average times to detection versus false positive, dependent primarily on Hamiltonian norm bounds. Lastly, we quantify the effect of memory limits on Boolean function evaluation of quantum sequences---a task applicable to distributed sensing and quantum reading of classical memories, among others. We show that minimum-error function evaluation in the memoryless regime is equivalent to the so-called pretty good measurement and hence shares its performance guarantee. Additionally, we characterize the precise set of functions on which this strategy performs optimally---affine functions. The above studies reveal insights about learnability as a universal property and the tradeoffs between performance and resource requirements in online settings, informing the choice of quantum settings best-suited for online learning. We conclude by summarizing these takeaways, along with listing remaining open questions.
Learning about physical systems from quantum-enhanced experiments can outperform learning from experiments in which only classical memory and processing are available. Whereas quantum advantages have been established for state learning, quantum process learning is less understood. We establish an exponential quantum advantage for learning an unknown n -qubit quantum process N . We show that a quantum memory allows to efficiently solve the following tasks: (a) learning the Pauli transfer matrix (PTM) of an arbitrary N , (b) predicting expectation values of Pauli-sparse observables measured on the output of an arbitrary N upon input of a Pauli-sparse state, and (c) predicting expectation values of arbitrary observables measured on the output of an unknown N with sparse PTM upon input of an arbitrary state. With quantum memory, these tasks can be solved using linearly-in- n many copies of the Choi state of N . In contrast, any learner without quantum memory requires exponentially-in- n many queries, even when using adaptively designed experiments. In proving this separation, we extend existing shadow tomography bounds from states to channels. Moreover, we combine PTM learning with polynomial interpolation to learn arbitrary Hamiltonians from short-time dynamics. Our results highlight the power of quantum-enhanced experiments for learning highly complex quantum dynamics.
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.
The problem of efficient quantum state learning, also called shadow tomography, aims to comprehend an unknown d -dimensional quantum state through POVMs. Yet, these states are rarely static; they evolve due to factors such as measurements, environmental noise, or inherent Hamiltonian state transitions. This paper leverages techniques from adaptive online learning to keep pace with such state changes.The key metrics considered for learning in these mutable environments are enhanced notions of regret, specifically adaptive and dynamic regret. We present adaptive and dynamic regret bounds for online shadow tomography, which are polynomial in the number of qubits and sublinear in the number of measurements. To support our theoretical findings, we include numerical experiments that validate our proposed models.
Shadow tomography for quantum states provides a sample efficient approach for predicting the properties of quantum systems when the properties are restricted to expectation values of 2-outcome POVMs. However, these shadow tomography procedures yield poor bounds if there are more than 2 outcomes per measurement. In this paper, we consider a general problem of learning properties from unknown quantum states: given an unknown d-dimensional quantum state ρ and M unknown quantum measurements M1,...,MM with K≥2 outcomes, estimating the probability distribution for applying Mi on ρ to within total variation distance ε. Compared to the special case when K=2, we need to learn unknown distributions instead of values. We develop an online shadow tomography procedure that solves this problem with high success probability requiring O~(Klog2Mlogd/ε4) copies of ρ. We further prove an information-theoretic lower bound that at least Ω(min{d2,K+logM}/ε2) copies of ρ are required to solve this problem with high success probability. Our shadow tomography procedure requires sample complexity with only logarithmic dependence on M and d and is sample-optimal for the dependence on K.