Sample-optimal learning of quantum states using gentle measurements

Gentle measurements of quantum states do not entirely collapse the initial state. Instead, they provide a post-measurement state at a prescribed trace distance $\alpha$ from the initial state together with a random variable used for quantum learning of the initial state. We introduce here the class of $\alpha-$locally-gentle measurements ($\alpha-$LGM) on a finite dimensional quantum system which are product measurements on product states and prove a strong quantum Data-Processing Inequality (qDPI) on this class using an improved relation between gentleness and quantum differential privacy. We further show a gentle quantum Neyman-Pearson lemma which implies that our qDPI is asymptotically optimal (for small $\alpha$). This inequality is employed to show that the necessary number of quantum states for prescribed accuracy $\epsilon$ is of order $1/(\epsilon^2 \alpha^2)$ for both quantum tomography and quantum state certification. Finally, we propose an $\alpha-$LGM called quantum Label Switch that attains these bounds. It is a general implementable method to turn any two-outcome measurement into an $\alpha-$LGM.

QubitLens: An Interactive Learning Tool for Quantum State Tomography

Quantum state tomography is a fundamental task in quantum computing, involving the reconstruction of an unknown quantum state from measurement outcomes. Although essential, it is typically introduced at the graduate level due to its reliance on advanced concepts such as the density matrix formalism, tensor product structures, and partial trace operations. This complexity often creates a barrier for students and early learners. In this work, we introduce QubitLens, an interactive visualization tool designed to make quantum state tomography more accessible and intuitive. QubitLens leverages maximum likelihood estimation (MLE), a classical statistical method, to estimate pure quantum states from projective measurement outcomes in the X, Y, and Z bases. The tool emphasizes conceptual clarity through visual representations, including Bloch sphere plots of true and reconstructed qubit states, bar charts comparing parameter estimates, and fidelity gauges that quantify reconstruction accuracy. QubitLens offers a hands-on approach to learning quantum tomography without requiring deep prior knowledge of density matrices or optimization theory. The tool supports both single- and multi-qubit systems and is intended to bridge the gap between theory and practice in quantum computing education.

Quantum state-agnostic work extraction (almost) without dissipation

We investigate work extraction protocols designed to transfer the maximum possible energy to a battery using sequential access to $N$ copies of an unknown pure qubit state. The core challenge is designing interactions to optimally balance two competing goals: charging of the battery optimally using the qubit in hand, and acquiring more information by qubit to improve energy harvesting in subsequent rounds. Here, we leverage exploration-exploitation trade-off in reinforcement learning to develop adaptive strategies achieving energy dissipation that scales only poly-logarithmically in $N$. This represents an exponential improvement over current protocols based on full state tomography.

Online Learning of Pure States is as Hard as Mixed States

Quantum state tomography, the task of learning an unknown quantum state, is a fundamental problem in quantum information. In standard settings, the complexity of this problem depends significantly on the type of quantum state that one is trying to learn, with pure states being substantially easier to learn than general mixed states. A natural question is whether this separation holds for any quantum state learning setting. In this work, we consider the online learning framework and prove the surprising result that learning pure states in this setting is as hard as learning mixed states. More specifically, we show that both classes share almost the same sequential fat-shattering dimension, leading to identical regret scaling under the $L_1$-loss. We also generalize previous results on full quantum state tomography in the online setting to learning only partially the density matrix, using smooth analysis.

Optimal Allocation of Pauli Measurements for Low-rank Quantum State Tomography

The process of reconstructing quantum states from experimental measurements, accomplished through quantum state tomography (QST), plays a crucial role in verifying and benchmarking quantum devices. A key challenge of QST is to find out how the accuracy of the reconstruction depends on the number of state copies used in the measurements. When multiple measurement settings are used, the total number of state copies is determined by multiplying the number of measurement settings with the number of repeated measurements for each setting. Due to statistical noise intrinsic to quantum measurements, a large number of repeated measurements is often used in practice. However, recent studies have shown that even with single-sample measurements--where only one measurement sample is obtained for each measurement setting--high accuracy QST can still be achieved with a sufficiently large number of different measurement settings. In this paper, we establish a theoretical understanding of the trade-off between the number of measurement settings and the number of repeated measurements per setting in QST. Our focus is primarily on low-rank density matrix recovery using Pauli measurements. We delve into the global landscape underlying the low-rank QST problem and demonstrate that the joint consideration of measurement settings and repeated measurements ensures a bounded recovery error for all second-order critical points, to which optimization algorithms tend to converge. This finding suggests the advantage of minimizing the number of repeated measurements per setting when the total number of state copies is held fixed. Additionally, we prove that the Wirtinger gradient descent algorithm can converge to the region of second-order critical points with a linear convergence rate. We have also performed numerical experiments to support our theoretical findings.

Active Learning with Variational Quantum Circuits for Quantum Process Tomography

Quantum process tomography (QPT), used for reconstruction of an unknown quantum process from measurement data, is a fundamental tool for the diagnostic and full characterization of quantum systems. It relies on querying a set of quantum states as input to the quantum process. Previous works commonly use a straightforward strategy to select a set of quantum states randomly, overlooking differences in informativeness among quantum states. Since querying the quantum system requires multiple experiments that can be prohibitively costly, it is always the case that there are not enough quantum states for high-quality reconstruction. In this paper, we propose a general framework for active learning (AL) to adaptively select a set of informative quantum states that improves the reconstruction most efficiently. In particular, we introduce a learning framework that leverages the widely-used variational quantum circuits (VQCs) to perform the QPT task and integrate our AL algorithms into the query step. We design and evaluate three various types of AL algorithms: committee-based, uncertainty-based, and diversity-based, each exhibiting distinct advantages in terms of performance and computational cost. Additionally, we provide a guideline for selecting algorithms suitable for different scenarios. Numerical results demonstrate that our algorithms achieve significantly improved reconstruction compared to the baseline method that selects a set of quantum states randomly. Moreover, these results suggest that active learning based approaches are applicable to other complicated learning tasks in large-scale quantum information processing.

Sample-Optimal Quantum State Tomography for Structured Quantum States in One Dimension

Quantum state tomography (QST) remains the gold standard for benchmarking and verifying quantum devices. A recent study has proved that, with Haar random projective measurements, only a $O(n^3)$ number of state copies is required to guarantee bounded recovery error of an matrix product operator (MPO) state of qubits $n$. While this result provides a formal evidence that quantum states with an efficient classical representation can be reconstructed with an efficient number of state copies, the number of state copies required is still significantly larger than the number of independent parameters in the classical representation. In this paper, we attempt to narrow this gap and study whether the number of state copies can saturate the information theoretic bound (i.e., $O(n)$, the number of parameters in the MPOs) using physical quantum measurements. We answer this question affirmatively by using a class of Informationally Complete Positive Operator-Valued Measures (IC-POVMs), including symmetric IC-POVMs (SIC-POVMs) and spherical $t$-designs. For SIC-POVMs and (approximate) spherical 2-designs, we show that the number of state copies to guarantee bounded recovery error of an MPO state with a constrained least-squares estimator depends on the probability distribution of the MPO under the POVM but scales only linearly with $n$ when the distribution is approximately uniform. For spherical $t$-designs with $t\ge3$, we prove that only a number of state copies proportional to the number of independent parameters in the MPO is needed for a guaranteed recovery of any state represented by an MPO. Moreover, we propose a projected gradient descent (PGD) algorithm to solve the constrained least-squares problem and show that it can efficiently find an estimate with bounded recovery error when appropriately initialized.

Agnostic Process Tomography

Characterizing a quantum system by learning its state or evolution is a fundamental problem in quantum physics and learning theory with a myriad of applications. Recently, as a new approach to this problem, the task of agnostic state tomography was defined, in which one aims to approximate an arbitrary quantum state by a simpler one in a given class. Generalizing this notion to quantum processes, we initiate the study of agnostic process tomography: given query access to an unknown quantum channel $\Phi$ and a known concept class $\mathcal{C}$ of channels, output a quantum channel that approximates $\Phi$ as well as any channel in the concept class $\mathcal{C}$, up to some error. In this work, we propose several natural applications for this new task in quantum machine learning, quantum metrology, classical simulation, and error mitigation. In addition, we give efficient agnostic process tomography algorithms for a wide variety of concept classes, including Pauli strings, Pauli channels, quantum junta channels, low-degree channels, and a class of channels produced by $\mathsf{QAC}^0$ circuits. The main technical tool we use is Pauli spectrum analysis of operators and superoperators. We also prove that, using ancilla qubits, any agnostic state tomography algorithm can be extended to one solving agnostic process tomography for a compatible concept class of unitaries, immediately giving us efficient agnostic learning algorithms for Clifford circuits, Clifford circuits with few T gates, and circuits consisting of a tensor product of single-qubit gates. Together, our results provide insight into the conditions and new algorithms necessary to extend the learnability of a concept class from the standard tomographic setting to the agnostic one.

Universal Quantum Tomography With Deep Neural Networks

Quantum state tomography is a crucial technique for characterizing the state of a quantum system, which is essential for many applications in quantum technologies. In recent years, there has been growing interest in leveraging neural networks to enhance the efficiency and accuracy of quantum state tomography. Still, many of them did not include mixed quantum state, since pure states are arguably less common in practical situations. In this research paper, we present two neural networks based approach for both pure and mixed quantum state tomography: Restricted Feature Based Neural Network and Mixed States Conditional Generative Adversarial Network, evaluate its effectiveness in comparison to existing neural based methods. We demonstrate that our proposed methods can achieve state-of-the-art results in reconstructing mixed quantum states from experimental data. Our work highlights the potential of neural networks in revolutionizing quantum state tomography and facilitating the development of quantum technologies.

Learning pure quantum states (almost) without regret

We initiate the study of quantum state tomography with minimal regret. A learner has sequential oracle access to an unknown pure quantum state, and in each round selects a pure probe state. Regret is incurred if the unknown state is measured orthogonal to this probe, and the learner's goal is to minimise the expected cumulative regret over $T$ rounds. The challenge is to find a balance between the most informative measurements and measurements incurring minimal regret. We show that the cumulative regret scales as $\Theta(\operatorname{polylog} T)$ using a new tomography algorithm based on a median of means least squares estimator. This algorithm employs measurements biased towards the unknown state and produces online estimates that are optimal (up to logarithmic terms) in the number of observed samples.