Abstract We explore the interplay of quantum computing and machine learning to advance experimental protocols for observing measurement-induced phase transitions (MIPTs) in quantum devices. In particular, we focus on trapped ion monitored circuits and apply the cross entropy benchmark recently introduced by Li et al (2023 Phys. Rev. Lett. 130 220404), which can mitigate the post-selection problem. By doing so, we reduce the number of projective measurements—the sample complexity—required per random circuit realization, which is a critical limiting resource in real devices. Since these projective measurement outcomes form a classical probability distribution, they are suitable for learning with a standard machine learning generative model. In this paper, we use a recurrent neural network to learn a representation of the measurement record for a native trapped-ion MIPT, and show that using this generative model can substantially reduce the number of measurements required to accurately estimate the cross entropy. This illustrates the potential of combining quantum computing and machine learning to overcome practical challenges in realizing quantum experiments.
Abstract We explore the interplay of quantum computing and machine learning to advance experimental protocols for observing measurement-induced phase transitions (MIPTs) in quantum devices. In particular, we focus on trapped ion monitored circuits and apply the cross entropy benchmark recently introduced by Li et al (2023 Phys. Rev. Lett. 130 220404), which can mitigate the post-selection problem. By doing so, we reduce the number of projective measurements—the sample complexity—required per random circuit realization, which is a critical limiting resource in real devices. Since these projective measurement outcomes form a classical probability distribution, they are suitable for learning with a standard machine learning generative model. In this paper, we use a recurrent neural network to learn a representation of the measurement record for a native trapped-ion MIPT, and show that using this generative model can substantially reduce the number of measurements required to accurately estimate the cross entropy. This illustrates the potential of combining quantum computing and machine learning to overcome practical challenges in realizing quantum experiments.
Open quantum systems have been shown to host a plethora of exotic dynamical phases. Measurement-induced entanglement phase transitions in monitored quantum systems are a striking example of this phenomena. However, naive realizations of such phase transitions requires an exponential number of repetitions of the experiment which is practically unfeasible on large systems. Recently, it has been proposed that these phase transitions can be probed locally via entangling reference qubits and studying their purification dynamics. In this work, we leverage modern machine learning tools to devise a neural network decoder to determine the state of the reference qubits conditioned on the measurement outcomes. We show that the entanglement phase transition manifests itself as a stark change in the learnability of the decoder function. We study the complexity and scalability of this approach in both Clifford and Haar random circuits and discuss how it can be utilized to detect entanglement phase transitions in generic experiments.
The rise of programmable quantum devices has motivated the exploration of circuit models which could realize novel physics. A promising candidate is a class of hybrid circuits, where entangling unitary dynamics compete with disentangling measurements. Novel phase transitions between different entanglement regimes have been identified in their dynamical states, with universal properties hinting at unexplored critical phenomena. Trapped-ion hardware is a leading contender for the experimental realization of such physics, which requires not only traditional two-qubit entangling gates, but also a constant rate of local measurements accurately addressed throughout the circuit. Recent progress in engineering high-precision optical addressing of individual ions makes preparing a constant rate of measurements throughout a unitary circuit feasible. Using tensor network simulations, we show that the resulting class of hybrid circuits, prepared with native gates, exhibits a volume-law to area-law transition in the entanglement entropy. This displays universal hallmarks of a measurement-induced phase transition. Our simulations are able to characterize the critical exponents using circuit sizes with tens of qubits and thousands of gates. We argue that this transition should be robust against additional sources of experimental noise expected in modern trapped-ion hardware and will rather be limited by statistical requirements on postselection. Our work highlights the powerful role that tensor network simulations can play in advancing the theoretical and experimental frontiers of critical phenomena.
Monitored quantum circuits host a rich variety of exotic non-equilibrium phases. Among the most representative examples are measurement-induced phase transitions between distinct area-law entangled states. However, because these transitions are characterized by specific entanglement quantities such as mutual information or topological entanglement entropy that are nonlinear functionals of the density matrix, their experimental observation requires multiple identical quantum trajectories via post-selection, which becomes exponentially unfeasible for large systems. Here, we leverage modern machine learning tools to address this challenge. We devise a neural network architecture combining a convolutional neural network with an attention mechanism, and use raw measurement outcomes directly as input to classify trivial, long-range entangled, and symmetry-protected topological phases. We show that the system's relaxation to a steady-state phase manifests as a sharp convergence in the classifier's accuracy, entirely bypassing the need for quantum state reconstruction. We systematically study the performance of our network as a function of sample size, input data, spatial and temporal constraints, and system size scalability. Our results demonstrate that this approach is robust and post-selection free, offering a practical pathway for experimentally probing measurement-induced phases.
Machine Learning (ML) has been widely applied across numerous domains due to its ability to automatically identify informative patterns from data for various tasks. The availability of large-scale data and advanced computational power enables the development of sophisticated models and training strategies, leading to state-of-the-art performance, but it also introduces substantial challenges. Quantum Computing (QC), which exploits quantum mechanisms for computation, has attracted growing attention and significant global investment as it may address these challenges. Consequently, Quantum Machine Learning (QML), the integration of these two fields, has received increasing interest, with a notable rise in related studies in recent years. We are motivated to review these existing contributions regarding quantum circuit-based learning models for classical data analysis and highlight the identified potentials and challenges of this technique. Specifically, we focus not only on QML models, both kernel-based and neural network-based, but also on recent explorations of their integration with classical machine learning layers within hybrid frameworks. Moreover, we examine both theoretical analysis and empirical findings to better understand their capabilities, and we also discuss the efforts on noise-resilient and hardware-efficient QML that could enhance its practicality under current hardware limitations. In addition, we cover several emerging paradigms for advanced quantum circuit design and highlight the adaptability of QML across representative application domains. This study aims to provide an overview of the contributions made to bridge quantum computing and machine learning, offering insights and guidance to support its future development and pave the way for broader adoption in the coming years.
Measurement-induced entanglement (MIE) captures how local measurements generate long-range quantum correlations and drive dynamical phase transitions in many-body systems. Yet estimating MIE experimentally remains challenging: direct evaluation requires extensive post-selection over measurement outcomes, raising the question of whether MIE is accessible with only polynomial resources. We address this challenge by reframing MIE detection as a data-driven learning problem that assumes no prior knowledge of state preparation. Using measurement records alone, we train a neural network in a self-supervised manner to predict the uncertainty metric for MIE--the gap between upper and lower bounds of the average post-measurement bipartite entanglement. Applied to random circuits with one-dimensional all-to-all connectivity, our method reveals a learnability transition with increasing circuit depth: below a threshold the MIE can be effectively learned with resources that grow only polynomially with system size, whereas above it the required resources grow exponentially. This computational phase transition coincides with the breakdown of efficient classical simulation of the underlying quantum state. We further observe signatures of this transition on current noisy quantum devices. These results highlight the power of data-driven approaches for learning MIE and delineate the practical limits of its classical learnability.
The combination of quantum and classical computational resources towards more effective algorithms is one of the most promising research directions in computer science. In such a hybrid framework, existing quantum computers can be used to their fullest extent and for practical applications. Generative modeling is one of the applications that could benefit the most, either by speeding up the underlying sampling methods or by unlocking more general models. In this work, we design a number of hybrid generative models and validate them on real hardware and datasets. The quantum-assisted Boltzmann machine is trained to generate realistic artificial images on quantum annealers. Several challenges in state-of-the-art annealers shall be overcome before one can assess their actual performance. We attack some of the most pressing challenges such as the sparse qubit-to-qubit connectivity, the unknown effective-temperature, and the noise on the control parameters. In order to handle datasets of realistic size and complexity, we include latent variables and obtain a more general model called the quantum-assisted Helmholtz machine. In the context of gate-based computers, the quantum circuit Born machine is trained to encode a target probability distribution in the wavefunction of a set of qubits. We implement this model on a trapped ion computer using low-depth circuits and native gates. We use the generative modeling performance on the canonical Bars-and-Stripes dataset to design a benchmark for hybrid systems. It is reasonable to expect that quantum data, i.e., datasets of wavefunctions, will become available in the future. We derive a quantum generative adversarial network that works with quantum data. Here, two circuits are optimized in tandem: one tries to generate suitable quantum states, the other tries to distinguish between target and generated states.