Such systems exhibit a variety of electronic phases, including metallic, insulating, and superconducting states [1], and represent a versatile design space for technological applications in electronics, quantum computing, and sensing. Designing new correlated materials with targeted properties therefore depends on the ability to solve the many-body electronic Hamiltonian, a computationally demanding task [2]. Quantum embedding (QE) methods provide a robust framework for overcoming these challenges [3-5]. The common strategy underlying these methods is to describe each fragment of interacting orbitals through an effective model, where its complex environment is replaced by a simpler, entangled quantum bath, designed to approximate its influence [3, 6]. The link between this effective model and the original system is established through self-consistency conditions, and different embedding schemes are defined by their choice of which physical property to match [3]. Dynamic mean-field theory (DMFT) [7-14], for instance, uses frequency-dependent one-body Green's functions, whereas density matrix embedding theory (DMET) [5, 15-19] typically uses one-and two-body density matrices. The recently developed gGA [20-26] is a powerful variational method that generalizes the standard Gutzwiller Approximation (GA) [27-34] by systematically extending its variational space with auxiliary "ghost" fermionic degrees of freedom. This approach yields results in remarkable agreement with DMFT but at a much lower computational cost, as it requires calculating only the ground state of a finite-size impurity model, whereas DMFT requires the full spectra from an impurity model that corresponds to an infinite bath. Successful applications of gGA include accurate modelling of the Anderson lattice systems [21], excitonic phenomena [20], non-equilibrium systems [35] and altermagnetic systems [36], along with extensions that achieve charge self-consistency with density functional theory (DFT) [22], demonstrating its versatility and practical utility in real-material contexts.

We propose a hybrid variational framework that enhances Neural Quantum States (NQS) with a Normalising Flow-based sampler to improve the expressivity and trainability of quantum many-body wavefunctions. Our approach decouples the sampling task from the variational ansatz by learning a continuous flow model that targets a discretised, amplitude-supported subspace of the Hilbert space. This overcomes limitations of Markov Chain Monte Carlo (MCMC) and autoregressive methods, especially in regimes with long-range correlations and volume-law entanglement. Applied to the transverse-field Ising model with both short- and long-range interactions, our method achieves comparable ground state energy errors with state-of-the-art matrix product states and lower energies than autoregressive NQS. For systems up to 50 spins, we demonstrate high accuracy and robust convergence across a wide range of coupling strengths, including regimes where competing methods fail. Our results showcase the utility of flow-assisted sampling as a scalable tool for quantum simulation and offer a new approach toward learning expressive quantum states in high-dimensional Hilbert spaces.

A naive classical representation of an n-qubit state requires specifying exponentially many amplitudes in the computational basis. Past works have demonstrated that classical neural networks can succinctly express these amplitudes for many physically relevant states, leading to computationally powerful representations known as neural quantum states. What underpins the efficacy of such representations? We show that conditional correlations present in the measurement distribution of quantum states control the performance of their neural representations. Such conditional correlations are basis dependent, arise due to measurement-induced entanglement, and reveal features not accessible through conventional few-body correlations often examined in studies of phases of matter. By combining theoretical and numerical analysis, we demonstrate how the state's entanglement and sign structure, along with the choice of measurement basis, give rise to distinct patterns of short- or long-range conditional correlations. Our findings provide a rigorous framework for exploring the expressive power of neural quantum states.

A central problem of quantum physics, be it fundamental quantum physics or applications for quantum technology, is the ground state problem. It can be defined as finding a state vector |Ψ that minimises the expected value of the Hamiltonian Ĥ that represents the energetic interactions between the different parts that make up a quantum physical system. It is well-known that the difficulty of solving the ground state problem for a physical system arises from the exponential growth of the Hilbert space with respect to the number of the system components and their dimension. Therefore, techniques such as exact diagonalisation of Ĥ quickly render insufficient to find the ground state, and other approximate methods have to be used. Interestingly, other central problems of quantum physics such as finding the evolution of a quantum system can be cast into the ground state problem, as demonstrated by the Feynman-Kitaev formalism [24]. An immediate implication of using this formalism is that the computational tools historically developed for solving the ground state problem can be used to find the dynamics of a physical system. Broadly speaking, the Feynman-Kitaev formalism appends a clock as an auxilliary subsystem of the main physical system, i.e. the Hilbert space H of the whole system is H = P C, where P is the Hilbert space of the main physical system and C is the Hilbert space of the clock.

One of the promising applications of early quantum computers is the simulation of quantum systems. Variational methods for near-term quantum computers, such as the variational quantum eigensolver (VQE), are a promising approach to finding ground states of quantum systems relevant in physics, chemistry, and materials science. These approaches, however, are constrained by the effects of noise as well as the limited quantum resources of near-term quantum hardware, motivating the need for quantum error mitigation techniques to reduce the effects of noise. Here we introduce $\textit{neural error mitigation}$, a novel method that uses neural networks to improve estimates of ground states and ground-state observables obtained using VQE on near-term quantum computers. To demonstrate our method's versatility, we apply neural error mitigation to finding the ground states of H$_2$ and LiH molecular Hamiltonians, as well as the lattice Schwinger model. Our results show that neural error mitigation improves the numerical and experimental VQE computation to yield low-energy errors, low infidelities, and accurate estimations of more-complex observables like order parameters and entanglement entropy, without requiring additional quantum resources. Additionally, neural error mitigation is agnostic to both the quantum hardware and the particular noise channel, making it a versatile tool for quantum simulation. Applying quantum many-body machine learning techniques to error mitigation, our method is a promising strategy for extending the reach of near-term quantum computers to solve complex quantum simulation problems.

One of the most challenging problems in modern theoretical physics is the so-called many-body problem. Typical many-body systems are composed of a large number of strongly interacting particles. Few such systems are amenable to exact mathematical treatment and numerical techniques are needed to make progress. However, since the resources required to specify a generic many-body quantum state depend exponentially on the number of particles in the system (more precisely, on the number of degrees of freedom), even today's best supercomputers lack sufficient power to exactly encode such states (they can handle only relatively small systems, with less than 45 particles). As we shall see, recent applications of machine learning techniques (artificial neural networks in particular) have been shown to provide highly efficient representations of such complex states, making their overwhelming complexity computationally tractable.