arXiv subjects

Giuseppe Carleo

Publications and source records attributed to Giuseppe Carleo.

At least 19 recordsLinked to original sources

arXiv2026

Kibble-Zurek Dynamics in Two-dimensional Frustrated Systems with a Neural Foundation-state Subspace Method

Universal scaling generated when a strongly interacting quantum many-body system is driven across a continuous phase transition provides a dynamical probe of equilibrium criticality. Accessing this regime numerically in two dimensions is challenging because it requires accurate real-time evolution of correlated many-body states over many system sizes and driving rates. We introduce a Neural Foundation-state Subspace (NFS) method for near-adiabatic dynamics. A foundation neural-network quantum state represents the ground-state manifold along the driving path, and a small fidelity-selected subset defines a fixed variational subspace. The many-body Schr\"odinger equation then reduces to the evolution of a few linear coefficients, with projected operators reusable across ramp times. We validate the method on the two-dimensional transverse-field Ising model, recovering the expected Kibble-Zurek scaling and critical exponents in quantitative agreement with ground-state quantum Monte Carlo estimates. Applied to the frustrated square-lattice $J_1$-$J_2$ Heisenberg model up to $16 \times 16$ clusters, our approach provides strong numerical evidence of Kibble-Zurek mechanism across the N\'eel-to-spin-liquid transition at $J_2/J_1=0.49$, yielding ${\nu=1.23(15)}$ and ${\eta=0.409(19)}$ at fixed $z=1$, consistent with static estimates and supporting the proposed continuous critical behavior.

cond-mat.str-el↗

arXiv2026

Unbiased sampling from Boltzmann distributions with noisy energies

Sampling from the Boltzmann distribution is central to computational physics, yet hard when the energy is known only through a stochastic estimate, such as with machine-learned molecular potentials, in variational Monte Carlo, or on quantum computers, because a noisy energy biases the sampled distribution. The penalty method of Ceperley and Dewing corrects this but requires the noise variance and becomes intractable when it is large. We introduce the Poisson product estimator, an unbiased, non-negative estimator of the Boltzmann weight that only needs an upper bound on the energy estimator and remains efficient at high noise. Using it to optimize a variational quantum circuit gradient-free, we recover the $H_3^+$ ground-state energy in a minimal basis and, by sampling rather than following a single trajectory, also map the variational energy landscape.

quant-ph↗

arXiv2026

A Fixed Universal Determinant is Variationally Complete for Continuum Fermions

How many Slater determinants does an accurate variational description of interacting fermions require? Exact expansions in a finite basis need combinatorially many, and state-of-the-art fermionic neural quantum states stack growing numbers of them. We prove that, in the norms that govern variational calculations, at most two are needed, independently of the number of particles and of the target accuracy. A single universal Slater determinant-specified in advance, independent of both the system and the state-multiplied by a smooth bosonic wave function approximates any fermionic wave function in up to three spatial dimensions in the first-order Sobolev norm, which controls the variational energy. Reaching the second-order Sobolev norm-for Coulomb interactions, the domain of the Hamiltonian, which bounds the variance of the local energy at the core of variational Monte Carlo-requires at most one additional fixed determinant, and only in three dimensions. Antisymmetry therefore costs at most two universal determinants and no expressiveness: generalized Slater-Jastrow neural quantum states are variationally complete.

cond-mat.str-el↗

arXiv2026

Variational low-energy subspaces for chemically accurate excited states

Accurate electronic excited states are essential for photochemistry, spectroscopy and non-adiabatic molecular dynamics, but high-level calculations often scale steeply and require prior knowledge of the target state's character or symmetry. Here we show that variational excited-state optimization can be reformulated as an iterated ground-state-like problem for a low-energy subspace of the electronic Hamiltonian. Applying this variational principle to non-orthogonal Slater determinants leads to EXIDOS, an automatic method for excited state calculations controlled only by the number of states and determinants per state. EXIDOS optimizes multiple excited states simultaneously, without explicit orthogonality constraints or imposed spin and point-group symmetries. Benchmarks against FCI and state-of-the-art quantum chemistry methods show chemical accuracy for a multitude of states in N$_2$ and CO, charge-transfer states in HCl, Rydberg states in NH$_3$, double excitations and extended potential-energy curves in C$_2$, and avoided crossings and conical intersections in ethylene. These results establish EXIDOS as a low-scaling, fully variational route to chemically accurate excited states.

physics.chem-ph↗

arXiv2026

Thermalization Dynamics in the Two-Dimensional Hubbard Model with Neural-Network Quantum States

Thermalization in strongly correlated fermionic systems remains a central open problem in quantum many-body physics. In this work, we investigate the real-time dynamics and the approach to thermalization in the two-dimensional Hubbard model, a paradigmatic framework for correlated electrons, relevant to high-temperature superconductivity and ultracold quantum simulation. Focusing on the half-filled square lattice, we monitor the time evolution of the double occupancy following a quench in the on-site interaction $U$, and assess whether its long-time value is captured by a canonical thermal ensemble. We employ time-dependent variational Monte Carlo methods combined with transformer-based Neural-Network Quantum States to accurately describe the nonequilibrium dynamics of fermions, especially for the behavior at long times, thereby accessing regimes that were previously inaccessible to numerical simulations. Our results reveal two dynamical behaviors: for weak to intermediate interactions, the double occupancy rapidly approaches the thermal prediction, consistent with ergodic evolution; beyond a critical interaction $U_{C}$, the dynamics remains distinct from the thermal expectation on the timescales investigated, revealing signatures of a prethermal plateau delaying fast relaxation. These results establish numerical simulation as a powerful tool to probe nonequilibrium quantum phenomena in correlated fermionic matter.

cond-mat.str-el↗

arXiv2026

Scaling Laws for Neural-Network Quantum States

Scaling laws, the power-law relations between loss, architecture size, and compute observed in modern neural networks, offer a quantitative way to characterize the complexity of a learning problem, with the exponent governing the decay of the loss reflecting how rapidly additional resources translate into improved accuracy, and thus how hard the target is to learn. Whether an analogous framework can characterize the complexity of physical problems remains open. We address this question for Neural-Network Quantum States, a leading variational approach for strongly correlated quantum many-body systems. Using transformer wave functions to approximate ground states of the $J_1$-$J_2$ Heisenberg model on triangular and square lattices with up to $20\times 20$ sites, we find that the $V$-score, a measure of accuracy of a variational state, decays as a power law in training compute. Under an appropriate rescaling of compute, results for different system sizes collapse onto a single curve, analogous to scaling collapse in critical phenomena. The resulting power law is, to a good approximation, independent of the number of sites, showing that the transformer Ansatz is size-consistent for the systems considered. The exponent decreases systematically with frustration, identifying it as a quantitative measure of representational difficulty of the ground state and establishing scaling laws as a general framework for benchmarking variational ansätze.

cond-mat.dis-nn↗

arXiv2026

Time-dependent Neural Galerkin Method for Quantum Dynamics

We introduce a classical computational method for quantum dynamics that relies on a global-in-time variational principle. Unlike conventional time-stepping approaches, our scheme computes the entire state trajectory over a finite time window by minimizing a loss function that enforces the Schrödinger's equation. The variational state is parametrized with a Galerkin-inspired ansatz based on a time-dependent linear combination of time-independent Neural Quantum States. This structure is particularly well-suited for exploring long-time dynamics and enables bounding the error with the exact evolution via the global loss function. We showcase the method by simulating global quantum quenches in the paradigmatic Transverse-Field Ising model in both 1D and 2D, uncovering signatures of ergodicity breaking and absence of thermalization in two dimensions. Overall, our method is competitive compared to state-of-the-art time-dependent variational approaches, while unlocking previously inaccessible dynamical regimes of strongly interacting quantum systems.

quant-ph↗

arXiv2026

Neural Wave Functions for High-Pressure Atomic Hydrogen

We leverage the power of neural quantum states to describe the ground state wave function of solid and liquid atomic hydrogen, including both electronic and protonic degrees of freedom. For static protons, the resulting Born-Oppenheimer energies are consistently comparable to or lower than all previous projector Monte Carlo results for systems containing up to $128$ hydrogen atoms. The same level of accuracy is preserved upon inclusion of nuclear quantum effects, thus going beyond the Born-Oppenheimer approximation. In addition, our description overcomes major limitations of current wave functions, notably by avoiding any explicit symmetry assumption on the expected quantum crystal, and sidestepping efficiency issues of imaginary time evolution with disparate mass scales. As a first application, we examine crystal formation in an extremely high-density region up to pressure-induced melting.

cond-mat.str-el↗

arXiv2026

Beyond Variational Bias: Resolving Intertwined Orders in the Hubbard Model

The two-dimensional Hubbard model at finite doping hosts competing or intertwined orders, resulting in conflicting conclusions from different computational approaches regarding its ground state. We show that a key source of such discrepancies is the bias encoded in the variational ansatz. We consider three different Transformer backflow fermionic wave functions based on a Slater determinant, its particle-hole counterpart, and a Pfaffian, initialized without any mean-field pretraining. We show that, despite achieving nearly degenerate, state-of-the-art variational energies, each ansatz converges to a state with qualitatively different spin, charge, and pairing correlations. Upon improving accuracy via symmetry restoration and variance reduction, however, all three converge to the same physical picture: coexisting superconducting and stripe orders. These results demonstrate that variational energy alone is insufficient to identify the ground state in the presence of competing phases, and highlight the importance of tracking how correlation functions evolve as the wave function is systematically improved before drawing physical conclusions.

cond-mat.str-el↗

arXiv2026

A circuit-differentiation framework for Green's functions on quantum computers

We propose a general framework for computing Retarded Green's Functions (RGFs) on quantum computers by recasting their evaluation as a problem of circuit differentiation. Our proposal is based on real-time evolution and specifically designed circuit components, which we refer to as circuit perturbations, acting as a direct representation of the external perturbative force within the quantum circuit in a linear-response setting. The direct mapping between circuit derivatives and the computation of RGFs enables the use of a broad range of differentiation strategies. We provide two such examples, including a class of stochastic estimators which do not require extra qubit connectivity with respect to the underlying time-evolution operations. We demonstrate our approach on interacting spin and fermionic models, showing that accurate dynamical correlations can be obtained even under realistic noise assumptions. Finally, we outline how our proposal can be tied to efficient gradient-estimation techniques relevant for the fault-tolerant regime.

quant-ph↗

arXiv2026

Quantum Finite Temperature Lanczos Method

The computation of thermal properties of quantum many-body systems is a central challenge in our understanding of quantum mechanics. We introduce the Quantum Finite Temperature Lanczos Method (QFTLM), which extends the finite-temperature Lanczos method to quantum computers by combining real-time quantum Krylov methods with efficient preparation of typical states for trace estimation. This approach enables the computation of thermal expectation values while avoiding the exponential scaling inherent to classical exact simulation techniques. Numerical experiments on the transverse-field Ising model show that QFTLM can reproduce thermal observables over a wide temperature range. We further analyze the influence of Krylov dimension, number of trace-estimator states, and Trotter error, and show that suitable regularization is essential for robustness in noisy settings. These results establish QFTLM as a promising framework for finite-temperature quantum simulation.

quant-ph↗

arXiv2026

Variance reduction for forces and pressure in variational Monte Carlo

We present simple and practical strategies to reduce the variance of Monte Carlo estimators. Our focus is on variational Monte Carlo calculations of atomic forces and pressure in electronic systems, although we show that the underlying ideas apply more broadly to other observables, like pair-correlation and angular-distribution functions, and other methods, including molecular dynamics. For Pulay-type contributions, we show that a minor modification based on the Metropolis acceptance ratio softens the power-law divergence of the variance to a logarithmic one, and that inexpensive regularizations can further suppress outliers at the price of a controlled small bias. For Hellmann-Feynman forces, we derive compact variance-reduced estimators for periodic systems that are straightforward to implement in standard Monte Carlo codes. The approach is illustrated for high-pressure metallic hydrogen with more than a hundred atoms described by neural quantum states, including an application to molecular dynamics driven by the improved forces.

cond-mat.str-el↗

arXiv2026

Quantum simulation in the Heisenberg picture via vectorization

We present a general framework for simulating quantum systems in the Heisenberg picture on quantum hardware. Based on the vectorization map, our framework fully exploits the mapping between operators and quantum states, allowing any task defined on Heisenberg operators to be mapped to standard Schrödinger-picture tasks that are naturally accessible via quantum computers and simulators. This yields new or improved protocols for tasks such as operator sampling, the computation of OTOCs/superoperator expectation values and their higher order moments, two-point correlators, and operator stabilizer and entanglement entropies. Our approach is also amenable to implementation, as it inherits the structure and resource requirements of the (forward and time-reversed) Schrödinger-picture quantum simulation problem. We demonstrate this by proposing implementations of our framework for a 2D problem on digital and analog quantum simulators, taking into account device connectivity constraints.

quant-ph↗

arXiv2026

Neural-network quantum states for the nuclear many-body problem

A long-standing goal of nuclear theory is to explain how the structure and dynamics of atomic nuclei and neutron-star matter emerge from the underlying interactions among protons and neutrons. Achieving this goal requires solving the nuclear quantum many-body problem with high accuracy across a wide range of length scales and density regimes. In this review, we discuss how artificial neural network representations of the nuclear many-body wave function have significantly extended the capabilities of continuum quantum Monte Carlo methods. In particular, neural network quantum states enable calculations of larger systems than were previously accessible and provide a flexible framework for capturing phenomena that challenge conventional approaches, including the emergence of nuclear clusters and superfluid phases in dense matter. We highlight recent applications to finite nuclei, infinite nuclear and neutron matter, and dynamical processes relevant to lepton-nucleus and nucleus-nucleus scattering. We also discuss conceptual and methodological connections with condensed matter physics, emphasizing developments in neural network quantum states that bridge strongly correlated systems across disciplines. Together, these developments demonstrate how neural-network methods open new avenues toward unified and accurate descriptions of nuclear structure, matter, and reactions.

nucl-th↗

arXiv2026

Approaching the Thermodynamic Limit with Neural-Network Quantum States

Accessing the thermodynamic-limit properties of strongly correlated quantum matter requires simulations on very large lattices, a regime that remains challenging for numerical methods, especially in frustrated two-dimensional systems. We introduce the Spatial Attention mechanism, a minimal and physically interpretable inductive bias for Neural-Network Quantum States, implemented as a single learned length scale within the Transformer architecture. This bias stabilizes large-scale optimization and enables access to thermodynamic-limit physics through highly accurate simulations on unprecedented system sizes within the Variational Monte Carlo framework. Applied to the spin-$\tfrac12$ triangular-lattice Heisenberg antiferromagnet, our approach achieves state-of-the-art results on clusters of up to $42\times42$ sites. The ability to simulate such large systems allows controlled finite-size scaling of energies and order parameters, enabling the extraction of experimentally relevant quantities such as spin-wave velocities and uniform susceptibilities. In turn, we find extrapolated thermodynamic limit energies systematically better than those obtained with tensor-network approaches such as iPEPS. The resulting magnetization is strongly renormalized, $M_0=0.148(1)$ (about $30\%$ of the classical value), revealing that less accurate variational states systematically overestimate magnetic order. Analysis of the optimized wave function further suggests an intrinsically non-local sign structure, indicating that the sign problem cannot be removed by local basis transformations. We finally demonstrate the generality of the method by obtaining state-of-the-art energies for a $J_1$-$J_2$ Heisenberg model on a $20\times20$ square lattice, outperforming Residual Convolutional Neural Networks.

cond-mat.str-el↗

arXiv2026

Non-stabilizerness of Neural Quantum States

We introduce a methodology to estimate non-stabilizerness or "magic", a key resource for quantum complexity, with Neural Quantum States (NQS). Our framework relies on two schemes based on Monte Carlo sampling to quantify non-stabilizerness via Stabilizer Rényi Entropy (SRE) in arbitrary variational wave functions. When combined with NQS, this approach is effective for systems with strong correlations and in dimensions larger than one, unlike Tensor Network methods. Firstly, we study the magic content in an ensemble of random NQS, demonstrating that neural network parametrizations of the wave function capture finite non-stabilizerness besides large entanglement. Secondly, we investigate the non-stabilizerness in the ground state of the $J_1$-$J_2$ Heisenberg model. In 1D, we find that the SRE vanishes at the Majumdar-Ghosh point $J_2 = J_1/2$, consistent with a stabilizer ground state. In 2D, a dip in the SRE is observed near maximum frustration around $J_2/J_1 \approx 0.6$, suggesting a Valence Bond Solid between the two antiferromagnetic phases.

quant-ph↗

arXiv2025

Foundation Neural-Networks Quantum States as a Unified Ansatz for Multiple Hamiltonians

Foundation models are highly versatile neural-network architectures capable of processing different data types, such as text and images, and generalizing across various tasks like classification and generation. Inspired by this success, we propose Foundation Neural-Network Quantum States (FNQS) as an integrated paradigm for studying quantum many-body systems. FNQS leverage key principles of foundation models to define variational wave functions based on a single, versatile architecture that processes multimodal inputs, including spin configurations and Hamiltonian physical couplings. Unlike specialized architectures tailored for individual Hamiltonians, FNQS can generalize to physical Hamiltonians beyond those encountered during training, offering a unified framework adaptable to various quantum systems and tasks. FNQS enable the efficient estimation of quantities that are traditionally challenging or computationally intensive to calculate using conventional methods, particularly disorder-averaged observables. Furthermore, the fidelity susceptibility can be easily obtained to uncover quantum phase transitions without prior knowledge of order parameters. These pretrained models can be efficiently fine-tuned for specific quantum systems. The architectures trained in this paper are publicly available at https://huggingface.co/nqs-models, along with examples for implementing these neural networks in NetKet.

quant-ph↗

arXiv2025

Grassmann Variational Monte Carlo with neural wave functions

Excited states play a central role in determining the physical properties of quantum matter, yet their accurate computation in many-body systems remains a formidable challenge for numerical methods. While neural quantum states have delivered outstanding results for ground-state problems, extending their applicability to excited states has faced limitations, including instability in dense spectra and reliance on symmetry constraints or penalty-based formulations. In this work, we rigorously formalize the framework introduced by Pfau et al.~\cite{pfau2024accurate} in terms of Grassmann geometry of the Hilbert space. This allows us to generalize the Stochastic Reconfiguration method for the simultaneous optimization of multiple variational wave functions, and to introduce the multidimensional versions of operator variances and overlaps. We validate our approach on the Heisenberg quantum spin model on the square lattice, achieving highly accurate energies and physical observables for a large number of excited states.

quant-ph↗