Machine Learning for Quantum Matter

Finite-temperature effects play an important role in the design and optimization of quantum devices, as decoherence and noise often originate from thermal fluctuations. At finite temperatures, quantum systems are described by a statistical ensemble of states rather than a single pure state. Simulating such thermal states requires constructing the thermal density matrix, which suffers from significant computational challenges due to the exponential growth of the Hilbert space with system size. So far, purification methods(thermofield) in the context of MPS and Minimally Entangled Typical Thermal States (METTS) approach have been developed in the context of tensor networks. In this work, we propose using neural quantum states (NQS), leveraging the expressivity and scalability of transformer-based architectures to address the challenges of thermal equilibrium density matrix representation.

Quantum computers and simulators, in the noisy intermediate-scale quantum era, offer a unique ability to control and probe individual quantum states at the many-body level. This capability allows for wave function snapshots via collective projective measurements, which are key to understanding the power of quantum computations. However, the connection between these snapshots and many-body collective properties remains poorly understood. We develop a theoretical framework using network theory to link quantum phases of matter to their snapshots. First, we identify a minimal-complexity basis by analyzing the information compressibility of snapshots. Then, we build a wave-function network to study correlations in this basis. This approach reveals a stochastic classification of quantum states in one dimension: low-complexity networks correspond to paramagnetic and symmetry-broken phases, while high-complexity networks characterize conformal and topological phases. The latter can be distinguished by their network structures - which are either scale-free or Erdos-Renyi. We corroborate this classification using extensive tensor network numerical experiments, complemented with state of the art network theory analysis, that also point to an interesting interplay between algorithmic and computational complexity for many-body states. The classification is of immediate experimental relevance, and draws a clear connection between probability distribution sampling and physical properties that is uncovered by network theory.

We study nonequilibrium quantum dynamics of spin chains by employing principal component analysis on data sets of wave function snapshots and examine how information propagates within these data sets. The quantities we employ are derived from the spectrum of the sample second moment matrix, built directly from data sets. Our investigations on several interacting spin chains featuring distinct spin or energy transport reveal that the growth of data information spreading follows the same dynamical exponents as that of the underlying quantum transport of spin or energy. Specifically, our approach enables an easy, data-driven, and, importantly, interpretable diagnostic to track energy transport with a limited number of samples, which is usually challenging without any assumption on the Hamiltonian form. These observations are obtained at a modest finite-size and evolution time, which aligns with experimental and numerical constraints. Our framework directly applies to experimental quantum simulator data sets of dynamics in higher-dimensional systems, where classical simulation methods usually face significant limitations and apply equally to both near- and far-from-equilibrium quenches.

I report progress in our variational monte carlo study of the SU(N) Fermi-Hubbard model, with a focus on understanding the ground state phase diagram and finite temperature behaviour of the Hubbard model involving more than 2 flavours of Fermion. We utilise novel neural network quantum state ansatz to improve the expressibility of the variational wavefunction, and compare with state of the art methods where they exist. This is a famously hard system to solve, given the presence of Fermionic sign problems in QMC approaches, and the exponential growth of the Hilbert space with N. Low temperature physics is to be probed using a finite temperature variational principle, to obtain observables that can be compared with and provide prediction for recent experiments on ultra-cold atoms.

Quantum impurity models are computationally expensive, especially in non-equilibrium settings. In equilibrium, it is possible to deterministically map interactions of the Anderson impurity model (AIM) onto a Resonant Level Model (RLM) with customized degrees of freedom [1]. Out of equilibrium, such mapping does not extend automatically to a generalized time-evolving setting. To overcome this, we use machine learning techniques to find optimal parameters for an auxiliary chain representation (ACR) that reproduces the dynamics of the impurity occupation number following a quench. We show that this ACR provides accurate approximations of the AIM non-equilibrium behavior, offering a computationally efficient way to study quantum impurity dynamics. An interesting feature of our approach is that, it allows us to understand interacting non-equilibrium behaviour through a non-interacting lens. We can show for example that the optimized ACR is Anderson localised but with a particular structure that allows it to account for the for the complex dynamics of the interacting system. [1] Sen S., Wong P., Mitchell A., Phys. Rev. B 102, 081110(R)

Simulating large, strongly interacting fermionic systems remains a major challenge for existing numerical methods. In this work, we present, for the first time, the application of neural quantum states - specifically, hidden fermion determinant states (HFDS) - to simulate the strongly interacting limit of the Fermi-Hubbard model, namely the t−J model, across the entire doping regime. We demonstrate that HFDS achieve energies competitive with matrix product states (MPS) on lattices as large as 8×8 sites while using several orders of magnitude fewer parameters, suggesting the potential for efficient application to even larger system sizes. This remarkable efficiency enables us to probe low-energy physics across the full doping range, providing new insights into the competition between kinetic and magnetic interactions and the nature of emergent quasiparticles. Starting from the low-doping regime, where magnetic polarons dominate the low energy physics, we track their evolution with increasing doping through analyses of spin and polaron correlation functions. Our findings demonstrate the potential of determinant-based neural quantum states with inherent fermionic sign structure, opening the way for simulating large-scale fermionic systems at any particle filling.

In this research, we investigate the dynamics of entanglement in Cliord circuits by employing a reinforcement learning (RL) algorithm in competition with a random agent. The RL agent is designed to strategically place gates that decrease entanglement, while the random agent aims to increase entanglement. This interaction between the two agents results in an entanglement transition, the nature of which is induced by the level of information accessible by the RL agent. By systematically varying the information provided to the RL agent, we analyze its impact on the transition characteristics. Our findings provide new insights into the interplay between entanglement manipulation and information constraints, shedding light on the fundamental mechanisms governing quantum circuit dynamics.

NeuralQuantumStates.jl is a Julia package under development to facilitate the training of neural quantum states (NQS) by variational Monte Carlo (VMC). The package aims to provide an efficient and extensible environment for the simulation of closed many-body quantum systems by exploiting the power of neural networks and modern computational resources. Inspired by established Python libraries such as NetKet and jVMC, NeuralQuantumStates.jl focuses on providing a machine learning toolbox for quantum many-body systems in a Julia-based environment. The package's primary modules will include a variety of functionalities, including a `Lattices` module for generating Bravais lattices, a `Networks` module for constructing artificial neural networks via Flux.jl, and a `VarStates` module for defining variational quantum states. Other modules in the works include `Hilberts`, for defining Hilbert spaces, and `Operators`, for implementing arbitrary quantum operators on computational bases. In addition, `Samplers` will enable variational state sampling using Markov chain Monte Carlo (MCMC) techniques, while `Handlers` will support state optimization using gradient-based and imaginary time evolution methods. The package will also integrate support for distributed and parallel computing via MPI.jl, and GPU acceleration via CUDA.jl, AMDGPU.jl, and Metal.jl. Although still in the early stages of development, NeuralQuantumStates.jl aims to provide a versatile and high-performance toolkit for the quantum research community. Future efforts will focus on improving computational efficiency across CPU and GPU architectures, stabilizing the package's API, and expanding the range of neural network architectures and optimizations supported within the package. [1] https://github.com/cevenkadir/NeuralQuantumStates.jl

Neural quantum state (NQS) ans\"atze have shown promise in variational Monte Carlo algorithms by their theoretical capability of representing any quantum state. However, the reason behind the practical improvement in their performance with an increase in the number of parameters is not fully understood. In this work, we systematically study the efficiency of a shallow neural network to represent the ground states in different phases of the spin-1 bilinear-biquadratic chain, as the number of parameters increases. We train our ansatz by a supervised learning procedure, minimizing the infidelity w.r.t. the exact ground state. We observe that the accuracy of our ansatz improves with the network width in most cases, and eventually saturates. We demonstrate that this can be explained by looking at the spectrum of the quantum geometric tensor (QGT), particularly its rank. By introducing an appropriate indicator, we establish that the QGT rank provides a useful diagnostic for the practical representation power of an NQS ansatz."

We investigate numerical algorithms for the compression of quantum circuits into shallower ones, better suited for simulations on near-term quantum devices. We apply our compression scheme to a time periodic random circuit, which can exhibit many-body localization (MBL). As MBL has been observed in finite Hamiltonian models as a crossover of different regimes, it is still an open question, to what extent it persists at longer timescales or in larger systems. We explore circuit compressibility of a possible scalable probe of localization. Furthermore, compressed Floquet random circuits might open a practical route to observe dynamical signatures of localization in digital quantum simulations on near-term quantum processors.

We introduce a neural network-based variational approach for studying dense hydrogen systems, which exhibit a range of intriguing physical phenomena such as liquid-liquid transitions, solidification, and metallization. Accurately predicting the phase diagram and understanding the underlying physical mechanisms in these systems has long been a challenge for computational methods. Our method combines deep generative models with fermionic electron wave function neural networks. The joint optimization of the considered neural networks allows for accurate calculation of the variational energy and free energy at many-body level, enabling the exploration of both classical and quantum solid hydrogen systems.

Neural Quantum States (NQS) have shown to be a reliable and efficient method for numerically simulating the ground states of two-dimensional quantum systems. Of particular interest for current research are fractional quantum Hall models and lattice gauge theories, both of which present significant challenges for state-of-the-art numerics. In this study, we demonstrate that NQS are capable of effectively simulating such complex systems. We focus on evaluating the strengths and weaknesses of this Ansatz from a physical perspective, providing deeper insights into the potential difficulties encountered during optimization.

Novel imaging techniques have repeatedly revolutionized science, medicine, and industry. While working with quantum probe particles such as photons or electrons, traditional techniques rely on intensity-based measurements that collapse the wavefunction of the probe particle, effectively ignoring information about the sample. Here, we explore the possibilities that arise when we allow the transfer of the wavefunction to a quantum system that we can manipulate. This may allow measurements with higher sensitivity beyond the standard quantum limit and the imaging of novel quantities of interest. A promising way to design measurement protocols is model-aware reinforcement learning, where an agent learns an optimal adaptive strategy that decides the next measurement based on previous outcomes.

Shannon and Rényi mutual information in quantum amny-body systems provides us with information about their underlying universal behaviour [4, 1, 3]. Experimentally extracting Shannon and Rényi entropies, which are basis-dependent quantities, requires estimating the probability distribution corresponding to all configurations of all spins in the measurement basis. However, as the system size grows, calculating this probability distribution from samples becomes challenging. Using Unsupervised Generative Models based on Matrix Product States [2] and recurrent neural networks, we can approximate the probability distribution of measurement outcomes in several bases, offering a strategy to use experimental measurements including a wide array of randomized measurement techniques to infer the universal behavior of large quantum many-body states. [1] F. C. Alcaraz and M. A. Rajabpour. Universal behavior of the shannon mutual information of critical quantum chains. Phys. Rev. Lett., 111:017201, Jul 2013. [2] Z.-Y. Han, J. Wang, H. Fan, L. Wang, and P. Zhang. Unsupervised generative modeling using matrix product states. Phys. Rev. X, 8:031012, Jul 2018. [3] G. Misguich, V. Pasquier, and M. Oshikawa. Finite-size scaling of the Shannon-Rényi entropy in two-dimensional systems with spontaneously broken continuous symmetry. Physical Review B, 95(19):195161, May 2017. [4] J.-M. Stéphan. Shannon and rényi mutual information in quantum critical spin chains. Phys. Rev. B, 90:045424, Jul 2014.

Finding the stationary states of a quantum many-body system is a task of pivotal importance for understanding and predicting crucial properties of quantum many-body systems, including complex molecules and materials. The variational principle has played a central role in motivating the development of several hybrid quantum-classical algorithms, the most popular of which is the variational quantum eigensolver (VQE). Despite extensive investigation into their resilience and efficiency, VQE-based approaches continue to require measurement counts that scale exponentially with the Hamiltonian's connectivity. Although quantum phase estimation (QPE) approaches have been proposed for estimating energies, they have not been widely adapted for variational state preparation. In this work, we propose a QPE-based variational algorithm that complements the existing methods for estimating the ground and excited states of quantum many-body systems. Our variational objective requires measuring only one Pauli observable, significantly reducing the measurement costs associated with current approaches. The method relies on executing a controlled unitary time evolution over a very short time through a Suzuki-Trotter expansion. Hence, optimally trading off the measurement count advantage with circuits deeper than the current methods but still substantially shallower than other QPE variants. We demonstrate the utility of our algorithm by starting from a VQE estimate and optimizing the parameters to recover stationary states. While the measurement counts with VQE scale as O(n^k) for spin chains with k-body terms and O(n^4) for molecular Hamiltonians, our approach maintains a significantly lower scaling across all cases. Finally, we provide preliminary estimates for the effects of hardware noise on our error scalings. This work presents novel findings that pave the way toward next-to-near-term fault-tolerant algorithms and their potential adaptations to NISQ devices.

Among the two primary families of variational algorithms for approximating Schrödinger dynamics—time-dependent Variational Monte Carlo (tVMC) and projected tVMC (p-tVMC)—the former is traditionally favored for its computational efficiency. However, recent findings demonstrate that tVMC suffers from systematic statistical bias and exponential sample complexity, particularly in cases where the wave function exhibits vanishing amplitudes, as commonly encountered in dissipative and continuously monitored systems. In contrast, the p-tVMC method provides a robust alternative, free from these limitations, albeit at a higher computational cost. In this talk, I will present a rigorous formalization of the p-tVMC framework, focusing on advancements in two critical areas: stochastic infidelity minimization and the discretization of unitary evolution. I will discuss improvements in stochastic infidelity minimization through second-order optimization techniques, emphasizing the stability of various Monte Carlo estimators and the development of adaptive regularization methods that eliminate the need for manual hyperparameter tuning. Additionally, I will introduce high-order integration schemes specifically designed for p-tVMC, leveraging Taylor expansions, Padé approximants, and Trotter splitting to enhance both accuracy and scalability. These results demonstrate the potential of p-tVMC to achieve state-of-the-art performance in simulating unitary dynamics in large quantum systems.

To transcend current-day noisy intermediate-scale quantum (NIQS) platforms to fault-tolerant quantum computing architectures, scalable and highly efficient quantum error correction schemes are moving into technological focus. One approach to overcome the scaling limitation of combinatorial decoders (such as minimum-weight perfect matching, MWPM) is to implement hierarchical decoders that employ machine-learning assisted pattern recognition techniques [1]. In this talk, we will present adaptations of this approach to rotated surface codes and simulations of both, compact codes (with code distance d=3,5,7) and large-scale codes with 10 to 100 thousands of qubits. We will discuss our results in context of recent experimental realizations of rotated surface code in transmon qubit [2] and ion-based qubit [3] systems. [1] Kai Meinerz et al., Scalable Neural Decoder for Topological Surface Codes, University of Cologne (2021), arXiv:2101.07285 [quant-ph] [2] Google Quantum AI et al., Quantum error correction below the surface code threshold, Google Quantum AI (2024), arXiv:2408.13687 [quant-ph] [3] Ben W. Reichardt et al. Demonstration of quantum computation and error correction with a tesseract code, Microsoft Azure Quantum / Quantinuum (2024), arXiv:2409.04628 [quant-ph]

With the increasing complexity of quantum computers, standard methods for characterizing quantum states based on direct measurements and classical post-processing have become impractical due to large measurement costs. Quantum neural networks can directly process quantum states to identify underlying characteristics with reduced measurement efforts, but they require deep quantum circuits that cannot be implemented on existing noisy intermediate-scale quantum devices. To overcome this challenge, we introduce hybrid quantum-classical neural networks that consist of a short-depth parametrized quantum circuit, measurement and a feedforward neural network. Using supervised learning, we train these hybrid neural networks to detect a topological phase of the surface code perturbed by an external field, and to distinguish between two symmetry-protected topological phases of a generalized cluster-Ising model. The parametrized quantum circuit performs a nonlocal transformation of the measurement basis that is trained to variationally maximize the statistical distance between data obtained by measuring quantum states. This significantly reduces the sample complexity required to recognize the topological phases using the feedforward neural network. Furthermore, we employ unsupervised learning by confusion to identify the boundaries between these phases and we show that the hybrid neural networks approximate fidelity susceptibility, which is known to detect a wide range of quantum phase transitions but is hard to measure directly. These hybrid neural networks feature short quantum circuits that can be readily implemented on existing quantum computers and they, thus, open the way for the efficient characterization of quantum states.

Time evolution in several classes of quantum devices is generated through the application of quantum gates. Resetting is a critical technological feature in these systems allowing for mid-circuit measurement and complete or partial qubit reset. The possibility of realising discrete-time reset dynamics on quantum computers makes it important to investigate the steady-state properties of such dynamics. Here, we explore the behaviour of generic discrete-time unitary dynamics interspersed by random reset events. For Poissonian resets, we compute the stationary state of the process and demonstrate, by taking a weak-reset limit, the existence of "resonances" in the quantum gates, allowing for the emergence of steady state density matrices which are not diagonal in the eigenbasis of the generator of the unitary gate. Such resonances are a genuine discrete-time feature and impact on quantum and classical correlations even beyond the weak-reset limit. Furthermore, we consider non-Poissonian reset processes and explore conditions for the existence of a steady state. We show that, when the reset probability vanishes sufficiently rapidly with time, the system does not approach a steady state. Our results highlight key differences between continuous-time and discrete-time stochastic resetting and may be useful to engineer states with controllable correlations on existing devices.

Neural networks have been shown to be effective variational ansätze for the representation of ground state wavefunctions of strongly correlated Hamiltonians, including states that exhibit volume-law entanglement in systems of interacting spins and fermions (https://arxiv.org/abs/2309.11534). However, little is known about the precise dependence of these entanglement features on the nature of the variational parameter space of these models. Here, we investigate the entanglement properties of two autoregressive neural quantum state (NQS) architectures initialized at random, namely Recurrent Neural Networks (RNNs) and Transformers. We uncover regimes of varying average entanglement entropy that are initialization-dependent. We also uncover insights about other properties of these random NQS such as the effective trainability of the models, as well as quantities important to the study of phenomena such as many-body localization. We conclude by testing the effect of various wavefunction initializations on the convergence time of ground state optimizations at and away from criticality for various Hamiltonians of interest. Our work builds on the work presented in Phys. Rev. B 106, 115138 (2022) that investigated the entanglement features of random restricted Boltzmann machines.

We propose a method to post-process a variational simulation of quantum dynamics to improve its accuracy at a negligible cost. Our method consists in constructing the variational state of linear combinations of the states of this dynamics, and evolving this new variational state from the initial time. The time-dependent variational principle applied on this linear variational state gives rise to an effective Schrödinger equation that drives the dynamics in the constructed subspace, and where the effective Hamiltonian needs only be sampled once, leading to an inexpensive time evolution. Numerical experiments on the 2-dimensional transverse field Ising model suggest that this method is very efficient at mitigating integration error, such that it could serve as an efficient way of increasing the length of time steps, while preserving the precision of the resulting dynamics.

A one-dimensional Kitaev model implemented in a quantum dot system coupled to superconductors can support Majorana zero modes at the ends of the chain. Recent experiments [1] have shown that even a two-dot system can host “Poor Man’s Majorana” modes in a specific regime of the Hamiltonian, known as the sweet spot [2]. In this regime, the strengths of elastic co-tunneling (ECT) and crossed Andreev reflection (CAR) are equal, which is crucial for the appearance of Poor Man’s Majoranas. Previous studies have demonstrated that a generative machine learning model can predict the underlying Hamiltonian parameters based on experimental data [3]. In this work, we introduce an automated tuning algorithm that leverages a convolutional neural network to infer the Hamiltonian state, allowing the quantum dot system to reach the sweet spot regime autonomously [4]. By combining theoretical insights, machine learning, and experimental techniques, we lay the groundwork for efficient, automated tuning of longer Kitaev chains, with promising applications in quantum information and computing. [1] Tom Dvir, et al. "Realization of a minimal Kitaev chain in coupled quantum dots." Nature 614.7948 (2023): 445-450. [2] Martin Leijnse, and Karsten Flensberg. "Parity qubits and poor man's Majorana bound states in double quantum dots." Physical Review B 86.13 (2012): 134528. [3] Rouven Koch, David van Driel, Alberto Bordin, Jose L. Lado, and Eliska Greplova. “Adversarial Hamiltonian learning of quantum dots in a minimal Kitaev chain.” Phys. Rev. Applied 20, 044081 (2023). [4] David van Driel, et al. "Cross-Platform Autonomous Control of Minimal Kitaev Chains." arXiv preprint arXiv:2405.04596 (2024).

Quantum coherence is a valuable measure worth exploring. So far, research of this quantity was limited to only a few spins because of the exponential scaling required for both analytical and numerical computation. In this work we present a method that enables numerical computation of quantum coherence for larger systems by incorporating Tensor Cross Interpolation algorithm. We focus our investigation on topologically frustrated models and comparison with analytical results near the classical point. Our findings can be contrasted with other entropic quantum resources such as Entanglement Entropy and Stabilizer 2-Rényi Entropy which suggest a more general picture with regards to phenomenology of topological frustration in the thermodynamic limit.

The properties of interfaces are key to understand the physics of matter. However, the study of quantum interface dynamics has remained an outstanding challenge. Here, we use large-scale Tree Tensor Network simulations to identify the dynamical signature of an interface roughening transition within the ferromagnetic phase of the 2D quantum Ising model. For initial domain wall profiles we find extended prethermal plateaus for smooth interfaces, whereas above the roughening transition the domain wall decays quickly. Our results can be readily explored experimentally in Rydberg atomic systems.

Quantum algorithms require repeated estimation of expectation values using consistent control procedures. However, fluctuations in control Hamiltonian parameters, caused by slowly varying environments, often lead to dephasing linked to uncertainties in these parameters. Nevertheless, temporal correlations in such fluctuations present an opportunity for feedback-based control strategies. One approach is real-time tracking of Hamiltonian parameters, which originally achieved a three-orders-of-magnitude enhancement in phase gate performance [0]. More recently, in collaboration with experiments, we demonstrated noise-driven coherent qubit oscillations using real-time Hamiltonian estimation on an FPGA [1]. Physics-informed [2] and greedy [3] approaches further improved performance by incorporating noise modeling and interleaving estimation and execution phases. To address faster and more complex environments, we developed a reinforcement learning strategy that optimizes the trade-off between estimation time and algorithm fidelity. By interacting with a simulated environment, the agent learns to exploit temporal noise correlations through experience, without requiring prior knowledge of the noise model. Analysis of the learned policy reveals adaptive resource allocation between estimation and operation shots, as well as novel estimation strategies tailored to specific noise spectra. These findings pave the way for FPGA-based implementations of adaptive control algorithms trained directly on experimental signals. [0] M. Shulman, S. Harvey, J. Nichol et al., Nat Commun 5, 5156 (2014) [1] F. Berritta, T. Rasmussen, J. A. Krzywda, et al., Nature Communications, 15(1), 1676 (2024) [2] F. Berritta, J. A. Krzywda et al., Physical Review Applied, 22(1), 014033. (2024) [3] J. Benestad, J. A. Krzywda et al., SciPost Physics, 17(1), 014. (2024)

Simulating strongly interacting fermionic systems is crucial for understanding correlated phases like unconventional superconductivity, yet it remainsa challenge for numerical and experimental methods in many cases. In this presentation, I will discuss how neural network representations of quantum states and quantum simulation platforms can be combined to gain insights into the physics of (doped) quantum magnets. First, I will talk about a hybrid training scheme, where the neural quantum states are initialized based on experimental measurements from different measurement bases. Then, I will discuss fermionic neural quantum states, in particular hidden fermion determinant states (HFDS). I will show results on the strongly interacting limit of the Fermi-Hubbard model across the entire doping regime. The HFDS achieve energies competitive with matrix product states (MPS) on lattices as large as 8×8 sites while using several orders of magnitude fewer parameters, suggesting the potential for efficient application to even larger system sizes. This remarkable efficiency enables us to probe low-energy physics across the full doping range, providing new insights into the competition between kinetic and magnetic interactions and the nature of emergent quasiparticles. Starting from the low-doping regime, where magnetic polarons dominate the low energy physics, we track their evolution with increasing doping through analyses of spin and polaron correlation functions and compare them with experimental measurements. Our findings demonstrate the potential of determinant-based neural quantum states with inherent fermionic sign structure, opening the way for simulating large-scale fermionic systems at any particle filling.

I would like to present an overview of how tensor networks have offered theoretical insights into Neural Networks for sequence modelling, leading to our work 'A Tensor Decomposition Perspective on Second-order RNNs' (abstract below). I will then present new results analyzing the impact of depth on the expressivity of Recurrent Neural Networks. "Second-order Recurrent Neural Networks (2RNNs) extend RNNs by leveraging second-order interactions for sequence modelling. These models are provably more expressive than their first-order counterparts and have connections to well-studied models from formal language theory. However, their large parameter tensor makes computations intractable. To circumvent this issue, one approach known as MIRNN consists in limiting the type of interactions used by the model. Another is to leverage tensor decomposition to diminish the parameter count. In this work, we study the model resulting from parameterizing 2RNNs using the CP decomposition, which we call CPRNN. Intuitively, the rank of the decomposition should reduce expressivity. We analyze how rank and hidden size affect model capacity and show the relationships between RNNs, 2RNNs, MIRNNs, and CPRNNs based on these parameters. We support these results empirically with experiments on the Penn Treebank dataset which demonstrate that, with a fixed parameter budget, CPRNNs outperforms RNNs, 2RNNs, and MIRNNs with the right choice of rank and hidden size."

Fracton models host unconventional topological orders in three and higher dimensions and provide promising candidates for quantum memory platforms. Understanding their robustness against quantum fluctuations is an important task but also poses great challenges due to the lack of efficient numerical tools. In this work, we establish neural quantum states (NQS) as new tools to study phase transitions in these models. Exact and efficient parametrizations are derived for three prototypical fracton codes --- the checkerboard and X-cube model, as well as Haah's code --- both in terms of a restricted Boltzmann machine (RBM) and a correlation-enhanced RBM. We then adapt the correlation-enhanced RBM architecture to a perturbed checkerboard model and reveal its strong first-order phase transition between the fracton phase and a trivial field-polarizing phase. To this end, we simulate this highly entangled system on lattices of up to 512 qubits with high accuracy, representing a cutting-edge application of variational neural-network methods. Our work demonstrates the remarkable potential of NQS in studying complicated three-dimensional problems and highlights physics-oriented constructions of NQS architectures.

The rational in silico design of chemical compounds requires a deep understanding of both the structure–property and property–property relationships that exist across chemical compound space (CCS), as well as efficient methodologies for defining an inverse property-to-structure mapping. This presentation will discuss these relationships in the CCS sector spanned by small [Sci. Data 8, 43 (2021)] and large [Sci. Data 11, 742 (2024)] drug-like molecules, highlighting the existence of the “freedom of design” principle [Chem. Sci. 14, 10702 (2023)]. The insights gained are subsequently leveraged to design molecules with desired properties. To this end, we first developed a variational autoencoder (VAE) approach and demonstrated that CCS can be parameterized using a finite set of quantum-mechanical (QM) properties [Nat. Commun. 15, 6061 (2024)]. We showcased the capabilities of this method by conditionally generating de novo molecular structures, interpolating transition paths for chemical reactions, and providing insightful insights into property–structure relationships. Then, a diffusion generative model on both small and large drug-like molecules was trained to examine the limits of scalability and chemical diversity when targeting diverse sets of QM properties. We expect our work will contribute to the development of advanced generative frameworks that enhance the in silico design and identification of molecules for specific chemical processes.

Variational quantum calculations have offered a controllable and powerful framework for capturing many-body quantum physics for decades. The recent introduction of expressive neural network quantum states has made it possible to accurately represent complex ground-state wavefunctions across the phase diagram for many Hamiltonians of interest. In this work, we introduce generalized variational representations of low-lying excited states, a key ingredient in the characterization of phases of matter and in the modeling of spectral properties. Exploiting continuity in Hamiltonian parameters, we adiabatically connect spectra of diagonalizable unperturbed Hamiltonians with fully interacting states at finite coupling, allowing us to tractably connect ground and excited states at different points of the phase diagram. Tests performed on spin Hamiltonians show an accurate representation of excited states and gaps closing at criticality. These results are achieved using a formalism of adiabatic gauge potentials (AGPs) which we use to derive a set of general update equations that can be seamlessly integrated with any variational calculation.

Machine-learning-based variational Monte Carlo simulations are a promising approach for targeting quantum many body ground states, especially in two-dimensions and in cases where the ground state is known to have a non-trivial sign structure. While many state-of-the-art variational energies have been reached with these methods for finite-size systems, little work has been done to use these results to extract information about the target systems in the thermodynamic limit. Here, we employ recurrent neural networks (RNNs) as our variational ansatzë, because the recurrent nature of the architecture allows one to iteratively retrain the same network for progressively larger physical systems. This transfer learning technique, which has been termed ``iterative retraining’’, allows us to study lattices with L up to 36 without beginning optimization from scratch for each system size. More specifically, we study Heisenberg antiferromagnets on square and triangular lattices with open and periodic boundary conditions and we perform finite-size scaling studies of our variational energies and of the spin structure factor. We first study the square lattice and show that our results are in good agreement with existing literature. For the triangular lattice, it is more difficult to carefully benchmark our results; however, we find a non-vanishing extrapolation of the structure factor, which is indicative of the expected long-range order of the target ground state. These results demonstrate how the unique ability of RNN wave functions to generalize in physical system size allows us to study the physics of the target system in the thermodynamic limit.

Neural network quantum states have emerged in the past few years as a leading numerical method for studying the ground states of frustrated many-body spin Hamiltonians. Recently, promising results were obtained with patched visual transformers (ViTs), originally developed for image analysis but which otherwise lack a clear physical motivation. Using numerical experiments, we aim to clarify the success of these architectures, highlighting the necessary ingredients to well-approximate ground states. First, by using convolutional layers we show that it is sufficient to encode correlations in a local, hierarchical manner. Second, we show that the role of patching is most significant for enforcing translational symmetries. For lattices with a multi-site unit cell, it allows them to be enforced at reduced computational cost. This furthers our empirical understanding of how neural networks are able to accurately approximate ground states, and informs how we can tackle challenging ground state problems, for example in frustrated magnetism.

Proposals for quantum advantage rely on proven computational complexity separations between quantum and classical methods. Problems with this known separation provide an excellent investigative platform for neural quantum state approaches, as the associated complexity guarantees offer insights into the expected scaling of computational resources. This work examines the limitations of various neural quantum state methods in preparing 2D cluster states, a quantum simulation task with known unfavourable classical complexity. By analyzing the failure modes of these classical methods, we aim to deepen the understanding of their limitations and identify potential pathways for improving their performance.

Neural networks have shown to be a powerful tool to represent ground state of quantum many-body systems, including for fermionic systems. In this work, we introduce a framework for embedding lattice symmetries in Neural Slater-Backflow-Jastrow wavefunction ansatzes, and demonstrate how our model allows us to target the ground state and low-lying excited states. To capture the Hamiltonian symmetries, we introduce group-equivariant backflow transformations. We study the low-energy excitation spectrum of the t-V model on a square lattice away from half-filling, and find that our symmetry-aware backflow significantly improves the ground-state energies, and yields accurate low-lying excited states for up to 10 × 10 lattices.

Floquet engineering enables access to interesting states and phases of matter by modifying the properties of otherwise trivial systems through high-frequency periodic driving. However, simulating the long-time dynamics of two-dimensional systems in this regime is computationally challenging, due to the exponential scaling of the Hilbert space and limitations on the timesteps imposed by the drive frequency. We address these challenges by using a neural network quantum state ansatz that directly incorporates the structure induced by the periodic drive. This approach relies on the formulation of Floquet dynamics in Sambe space to encode the fast drive dynamics into the ansatz, allowing the variational optimization to focus on the slower, relevant dynamics of the effective Floquet Hamiltonian.

Periodic driving has become crucial in quantum simulation and quantum sensing, and has led to a new paradigm of non-equilibrium order in condensed matter. Our present-day understanding of periodically driven systems is based on the Floquet theorem, which guarantees the existence of a rotating reference frame in which a time-independent, so-called, Floquet Hamiltonian describes the system. Despite significant advancements, numerous open questions persist. For instance, computing the Floquet Hamiltonian is notoriously hard, and the absence of a variational principle limits approximation techniques. Moreover, the quasi-energies of the Floquet Hamiltonian are ambiguous, precluding the unique definition of a Floquet ground state. In this talk, I introduce a geometric formulation of Floquet theory. We establish a duality between periodically driven systems and transitionless counterdiabatic driving, which paves the way for formulating a variational principle for the Floquet Hamiltonian. Moreover, we can identify quasienergy folding as a consequence of an incomplete gauge-fixing $U(1) \mapsto \mathbb{Z}$. This allows us to introduce a unique gauge-fixing based on inherently gauge-invariant quantities, which decomposes the dynamics into a purely geometric and a purely dynamical evolution. The dynamical average-energy operator provides an unambiguous sorting of the quasienergy spectrum, as I will demonstrate on an exactly solvable two-level system, and a non-integrable kicked Ising chain. Moreover, I will show that the geometric contribution accounts for inherently nonequilibrium effects—like the $\pi$-quasienergy gap in discrete time crystals or anomalous edge modes in anomalous Floquet topological insulators.

The Fermi-Hubbard model is a cornerstone of condensed matter physics, describing strongly correlated electrons and phenomena like Mott insulators, antiferromagnetism, and unconventional superconductivity. Solving this model is challenging, especially in strongly correlated and frustrated regimes. We develop Neural Quantum State (NQS) architectures that respect the translational symmetry of the lattice, enabling scalable and efficient wavefunction evaluations. By leveraging convolutional networks for translational invariance and backflow transformations for fermionic antisymmetry, our framework can reach competitive variational energies. Our results demonstrate the critical role of symmetry in reducing the variational search space, significantly improving the accuracy and efficiency of NQS for the Fermi-Hubbard model.

Magic, or non-stabilizerness, has recently emerged as a crucial measure of quantum complexity and classical simulability. In this work, we introduce a novel framework for quantifying magic in quantum many-body systems based on Neural Quantum States (NQS). Our approach targets systems in high-dimension and with large entanglement, pushing the limits of existing computational paradigms. After benchmarking against established techniques based on Matrix Product States, we apply our method to the paradigmatic J1-J2 Heisenberg model on the square lattice, whose study has proven to be challenging for traditional numerical schemes. We compute the magic diagram of this 2D system, revealing a sharp dip of the non-stabilizerness in the regime of maximum frustration, in conjunction with the putative phase transition of the model.

Autoregressive neural network quantum states are a powerful class of generative variational ansätze which allow efficient and exact sampling. Within the Variational Monte Carlo framework, we show how physically motivated variational bases can influence the convergence and latent feature space of transformers, giving rise to a more interpretable and computationally efficient representation of the ground state of a non-local many-body Hamiltonian containing a metal-insulator phase transition.

Variational techniques have long been at the heart of atomic, solid-state, and many-body physics. They have recently extended to quantum and classical machine learning, providing a basis for representing quantum states via neural networks. These methods generally aim to minimize the energy of a given ansätz, though open questions remain about the expressivity of quantum and classical variational ansätze. The connection between variational techniques and quantum computing, through variational quantum algorithms, offers opportunities to explore the quantum complexity of classical methods. We demonstrate how the concept of non-stabilizerness, or magic, can create a bridge between quantum information and variational techniques and we show that energy accuracy is a necessary but not always sufficient condition for accuracy in non-stabilizerness. Through systematic benchmarking of neural network quantum states, matrix product states, and variational quantum methods, we show that while classical techniques are more accurate in non-stabilizerness, not accounting for the symmetries of the system can have a severe impact on this accuracy. Our findings form a basis for a universal expressivity characterization of both quantum and classical variational methods.

The search for the optimal variational wave function underpins much of the contemporary efforts to combat the quantum many-body problem with a classical computer. Although recent advances based on Neural Network Quantum States (NNQS) have shown to be extremely potent, they suffer from bad navigability during the optimization procedures, rendering practical use limited. In this work, we introduce a new generation of variational wave functions, that combine the ideas stemming from machine learning with those from the tensor network community. In doing so we introduce a "perceptrain", a generalization of the perceptron, which retains many useful properties of tensor networks, such as compressibility and the possibility of local optimization. We carefully study the expressivity of different variational ansätze and show that indeed such wave functions lead to higher expressivity, while navigability is preserved due to the possibility of gradually increasing the expressivity of the model. We present optimization techniques useful for such states and demonstrate the usefulness of Green Function Monte Carlo (GFMC), which gives us access to exact solutions and exact overlap of the variational wave function with the ground state.

Thinking in a classical phase space picture, a many-body ground state should be localized around the minimum of the classical mean-field energy landscape with stable integrable features. But here, we investigate many-body ground states on chaotic features, as the phase space picture is actually fragile if we increase the system size and keep the quantum scale (the effective Plank constant $\hbar_{\rm eff}$) fixed. With the new degrees of freedom, we disturb the energy landscape in the classical limit more and more such that classical chaos is present even for low energies. We show this phenomenon, called 'chaotic melting', is indeed happening in the Bose-Hubbard system with disorder. By using neural quantum states we can push quantum calculations for ground states to large systems and find signatures of chaos at the ground state. An intriguing application for these large systems is that the Bose-Hubbard Hamiltonian with disorder is an effective model for transmon arrays which are a prime candidate for quantum computer hardware. Therefore we also gain access to quantum states describing a possible quantum computer with chaotic features.

Simulating the dynamics of many-body quantum systems is a cornerstone of modern theoretical physics. Traditional methods for time evolution often rely on step-by-step integration schemes, which can struggle with time integration schemes, or approximations of the time derivative of the wavefunction. In this work, we discuss a novel approach using a time-dependent neural quantum state (t-NQS) framework. Our method leverages a transformer architecture to represent the quantum state across all time steps, treating time as an explicit input parameter. This paradigm enables global optimization of the dynamics via automatic differentiation, bypassing the need for sequential time integration. We demonstrate the effectiveness of t-NQS on the transverse field Ising model, achieving accurate results and extrapolation for quench dynamics in two-dimensional systems. Our results highlight the ability of t-NQS to efficiently capture spatial and temporal correlations, offering a scalable and versatile tool for exploring quantum dynamics in high-dimensional settings.

Characterizing multipartite quantum systems is crucial for quantum computing and many-body physics. The problem, however, becomes challenging when the system size is large and the properties of interest involve correlations among a large number of particles. Here we introduce a neural network model that can predict various quantum properties of many-body quantum states with constant correlation length, using only measurement data from a small number of neighboring sites. The model is based on the technique of multi-task learning, which we show to offer several advantages over traditional single-task approaches. Through numerical experiments, we show that multi-task learning can be applied to sufficiently regular states to predict global properties, like string order parameters, from the observation of short-range correlations, and to distinguish between quantum phases that cannot be distinguished by single-task networks. Remarkably, our model appears to be able to transfer information learnt from lower dimensional quantum systems to higher dimensional ones, and to make accurate predictions for Hamiltonians that were not seen in the training

I will discuss a recently proposed variational ground state quantum adiabatic theorem, which establishes that, under specific conditions, adiabatic dynamics restricted to a variational manifold can closely track the instantaneous variational ground state. The focus will be on low-entanglement variational manifolds and target Hamiltonians with classical ground states. Remarkably, I will show that even when the exact quantum annealing path involves highly entangled intermediate states, the variational evolution can successfully converge to the target ground state. To support this, I will present illustrative examples consistent with the theoretical predictions. Additionally, I will explore potential limitations and extensions involving the adiabatic gauge potential and Matrix Product State (MPS)-catalyst Hamiltonians.