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.
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.
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.
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 purpose of this research work is to learn the quantum states in an ideal environments analytically, computationally, and graphically. The analysis starts with the learning of quantum states in identity channels with the help of the Regularized Follow the Leader (RFTL) algorithm. Our machine will try to learn the states based on the previous information, which is called the online learning model. The objective of this problem is to minimize regret by utilizing a learning algorithm that successively anticipates quantum states through observed measurements and losses. We have to produce many copies of quantum state ρ to perform analysis on them, which indicates the use of the shadow tomography approach in an ideal situation. Our goal is to learn the shadow of the state ρ by using a series of measurement operators that have two outcomes in nature. Aaronson et al. [1] developed an online setting for a non-realizable case, where the maximum possible loss is O( √ Tn) for the best possible state up to T−measurements . It is noteworthy that this outcome is an extension of the Aaronson PAC-like findings [2]. GANIT J. Bangladesh Math. Soc. 43.1 (2023) 65-74
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.
In Online Shadow Tomography, we are given copies of an unknown d-dimensional quantum state ρ, an adversary (adaptively) proposes a sequence of bounded observables A(1),…,A(m), and after each A(t) is given we must estimate Tr(A(t)ρ) to within ±ε. This is the direct quantum generalization of the classical problem of Adaptive Data Analysis. Prior results for online Shadow Tomography were suboptimal in all three parameters m,d,ε, lagging behind the best known and classical rates, for which there is some evidence of optimality. In this work, we finally close this gap, giving a pair of algorithms matching the classical rates. Our first algorithm is the first to achieve o(log2m)-dependence together with poly(log(d)/ε); moreover, it improves all three exponents even in the Offline Shadow Tomography setting. Our second algorithm is known to be optimal among bounds independent of d, and improves the best prior result by a mlogm factor.
The key to our proof is a new framework for quantifying post-measurement damage, based on the quantum Efron-Stein decomposition.
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.