We show that combining randomized measurement protocols with importance sampling allows for characterizing entanglement in significantly larger quantum systems and in a more efficient way than in previous work. A drastic reduction of statistical errors is obtained using classical techniques of machine learning and tensor networks using partial information on the quantum state. In current experimental settings of engineered many-body quantum systems this significantly increases the (sub-)system sizes for which entanglement can be measured. In particular, we show an exponential reduction of the required number of measurements to estimate the purity of product states and GHZ states.
We show that combining randomized measurement protocols with importance sampling allows for characterizing entanglement in significantly larger quantum systems and in a more efficient way than in previous work. A drastic reduction of statistical errors is obtained using classical techniques of machine learning and tensor networks using partial information on the quantum state. In current experimental settings of engineered many-body quantum systems this significantly increases the (sub-)system sizes for which entanglement can be measured. In particular, we show an exponential reduction of the required number of measurements to estimate the purity of product states and GHZ states.
There have been recently significant progress in quantum technologies to realize physically diverse quantum computation and simulation platforms.The aim of this thesis is to develop theoretical approaches based on measurement protocols to experimentally characterize entanglement generated in these state-of-art quantum technological platforms. In the past few years, protocols belonging to the "randomized measurements toolbox" (RM toolbox) have shown to have the ability to measure bipartite entanglement (entanglement between two parts of a quantum system). These protocols have been implemented in several experimental platforms consisting of quantum simulators and quantum computers beyond the regime of full state tomography.The aim of this manuscript is principally dedicated to the development of new approaches based on the RM toolbox to provide experimental groups with advanced tools and methods to probe new fundamental quantities of interest related to entanglement prepared in large quantum platforms.The first objective of this thesis is to provide a new protocol to optimize the existing RM protocol by reducing the statistical errors in the estimation of the quantities. This is done with two approaches: (i) via importance sampling of local random unitaries and (ii) by implementing common randomized measurements. These techniques automatically enable us to implement the RM toolbox to estimate properties associated to entanglement on larger experimental platforms. Then, a central part of this work proposes an alternate effective protocol to measure a fundamental quantity in quantum metrology known as the "quantum Fisher information" that enables us to characterize the multipartite entanglement content of the underlying quantum state. This novel protocol is based on estimating the QFI via the RM toolbox, as a converging series of lower bound polynomials of the density matrix.Additionally, the manuscript also focuses on solving a practical bottleneck of the current RM toolbox caused by the over-burdening classical cost to post-process experimental RM data to evaluate unbiased estimations of relevant quantities.This motivates the introduction of the "batch shadow formalism" that circumvents this problem and provides an efficient data treatment technique with rigorous performance guarantees to get estimates of quantities accessible from the experimental RM data. Lastly, owing to these developments of the RM toolbox, the manuscript reports estimation of some quantities that were accessed for the first time by experimental implementation of this toolbox on two different architectures of quantum platform consisting of superconducting qubits and trapped ions.
Abstract We present a systematic comparative analysis of machine learning and traditional approaches for quantum entanglement characterization in bi-partite ququart systems. Quantifying entanglement traditionally requires full quantum state tomography, necessitating Nmub2D2Ncop measurements, where Nmub is the number of mutually unbiased bases, D is the subsystem dimensionality, and Ncop is the number of identical copies. For ququart systems with D=4, this translates to hundreds of distinct measurement settings, each requiring multiple copies (Ncop typically in thousands)—resulting in hundreds of thousands of total measurements in practice. Our research demonstrates that neural networks achieve comparable or superior accuracy with orders of magnitude fewer measurement settings. We evaluate three architectures—Multi-Layer Perceptron (MLP), Convolutional Neural Network (CNN), and Transformer—against Maximum Likelihood Estimation (MLE) and Bayesian methods across measurement counts ranging from 10 to 400. Neural approaches achieve up to 1650× faster computation times compared to traditional methods while maintaining competitive accuracy. At 100 measurements, the Transformer achieves Mean Squared Error (MSE) of 1.49×10−1, while MLE yields 2.04×10−1—over 10× higher error—despite taking 178 times longer. All neural methods in our study show error reduction scaling as approximately 1/N with increased measurements. While this observed scaling might be influenced by our specific architectures, it provides important practical guidance for experimental design. The Transformer architecture demonstrates exceptional sample efficiency, achieving with 100 measurements what traditional methods require 250-400 measurements to accomplish—a significant advantage for resource-constrained quantum experiments. This research provides a viable pathway for real-time entanglement characterization in higher-dimensional quantum systems where traditional methods become computationally prohibitive.
We introduce a method for extracting meaningful entanglement measures of tensor network states in general dimensions.Current methods require the explicit reconstruction of the density matrix, which is highly demanding, or the contraction of replicas, which requires an effort exponential in the number of replicas and which is costly in terms of memory.In contrast, our method requires the stochastic sampling of matrix elements of the classically represented reduced states with respect to random states drawn from simple product probability measures constituting frames.Even though not corresponding to physical operations, such matrix elements are straightforward to calculate for tensor network states, and their moments provide the Rényi entropies and negativities as well as their symmetry-resolved components.We test our method on the one-dimensional critical XX chain and the two-dimensional toric code in a checkerboard geometry.Although the cost is exponential in the subsystem size, it is sufficiently moderate so that-in contrast with other approaches-accurate results can be obtained on a personal computer for relatively large subsystem sizes.
Entanglement not only plays a crucial role in quantum technologies, but is key to our understanding of quantum correlations in many-body systems. However, in an experiment, the only way of measuring entanglement in a generic mixed state is through reconstructive quantum tomography, requiring an exponential number of measurements in the system size. Here, we propose a machine-learning-assisted scheme to measure the entanglement between arbitrary subsystems of size N_{A} and N_{B}, with O(N_{A}+N_{B}) measurements, and without any prior knowledge of the state. The method exploits a neural network to learn the unknown, nonlinear function relating certain measurable moments and the logarithmic negativity. Our procedure will allow entanglement measurements in a wide variety of systems, including strongly interacting many-body systems in both equilibrium and nonequilibrium regimes.
A defining feature of quantum many-body systems is the exponential scaling of the Hilbert space with the number of degrees of freedom. This exponential complexity naïvely renders a complete state characterization, for instance via the complete set of bipartite Renyi entropies for all disjoint regions, a challenging task. Recently, a compact way of storing subregions' purities by encoding them as amplitudes of a fictitious quantum wave function, known as entanglement feature, was proposed. Notably, the entanglement feature can be a simple object even for highly entangled quantum states. However the complexity and practical usage of the entanglement feature for general quantum states has not been explored. In this work, we demonstrate that the entanglement feature can be efficiently learned using only a polynomial amount of samples in the number of degrees of freedom through the so-called tensor cross interpolation (TCI) algorithm, assuming it is expressible as a finite bond dimension MPS. We benchmark this learning process on Haar and random MPS states, confirming analytic expectations. Applying the TCI algorithm to quantum eigenstates of various one dimensional quantum systems, we identify cases where eigenstates have entanglement feature learnable with TCI. We conclude with possible applications of the learned entanglement feature, such as quantifying the distance between different entanglement patterns and finding the optimal one-dimensional ordering of physical indices in a given state, highlighting the potential utility of the proposed purity interpolation method.
Entanglement is a key feature of many-body quantum systems. Measuring the entropy of different partitions of a quantum system provides a way to probe its entanglement structure. Here, we present and experimentally demonstrate a protocol for measuring the second-order Rényi entropy based on statistical correlations between randomized measurements. Our experiments, carried out with a trapped-ion quantum simulator with partition sizes of up to 10 qubits, prove the overall coherent character of the system dynamics and reveal the growth of entanglement between its parts, in both the absence and presence of disorder. Our protocol represents a universal tool for probing and characterizing engineered quantum systems in the laboratory, which is applicable to arbitrary quantum states of up to several tens of qubits.