Machine Learning for Quantum Matter

Techniques of machine learning and AI are having an ever-increasing impact across quantum technologies, quantum many-body systems, and quantum foundations. In this talk, I will start by giving an overview of our use of reinforcement learning applied to quantum feedback control. I will focus in particular on a recently introduced model-based approach, feedback GRAPE, and the promising results we obtained for logical-state stabilization in the Gottesman-Kitaev-Preskill code using non-Markovian feedback. I will then discuss a new analog quantum machine learning approach that enables the efficient physics-based training of arbitrary quantum simulator platforms: Quantum Equilibrium Propagation. Finally, I will highlight the use of automatic differentiation and numerical optimization in the construction of local hidden variable theories. The technique I present enabled us to obtain new thresholds for the locality-to-nonlocality transition of Werner states and other classes of states. It also allows us to construct local hidden variable models for quantum many body states.

Present-day quantum computers and simulators provide unique capabilities to control and observe quantum matter at the level of individual constituents. In particular, this has made it possible to take snapshots of many-body wave functions through projective measurements. These developments have sparked recent intensive efforts to theoretically characterise physical properties from finite sets of such snapshots. In this talk, I will present a framework based on network theory that links quantum phases of matter to the data properties of their snapshots. First, we identify a minimal-complexity basis by quantifying correlations through the intrinsic dimension of datasets in different bases. Next, we examine the network structure of the data, leveraging a construction commonly used in data science methods such as data clustering and topological data analysis. I will then discuss how, for one-dimensional systems, this framework enables a complete classification of quantum phases of matter. Specifically, paramagnetic and symmetry-broken phases exhibit low-complexity network descriptions, whereas gapless and topological phases feature high-complexity networks. The latter can be further distinguished by their specific network structures, which are either scale-free or random. Finally, we validate this classification using tensor network simulations of prototypical spin chains.

Randomness is a powerful tool to characterize, simulate, and benchmark quantum many-body systems. We show that, when quantum simulation (both analogue or digital) is subject to noise, the most efficient way of estimating physical observable is available when the system is scrambled. Motivated by the potential of early fault-tolerant quantum computers to perform superlogarithmic depth quantum circuits, we further introduce the notion of symmetric Clifford group which can applied to realize a symmetry-constrained twirling. We further show numerical results for symmetric twirling that accelerates the realization of global depolarizing noise.

The efficient preparation of quantum many-body systems on quantum or classical devices is a major challenge, as the corresponding states typically exhibit an exponentially large number of amplitudes. A popular approach to managing this complexity is using exponential ansätze, such as the Coupled-Cluster method, that capture the essential structural features of the corresponding wavefunction. However, these ansätze are often system-dependent, making it challenging to generalize findings across different physical systems. Here, we develop a strategy to learn the parameters of the universal, exact exponential ansatz for quantum many-body physics. Our approach generalizes the contracted Schrödinger equation for electronic systems, which, quite remarkably, allows the ansatz to retain the same degrees of freedom as the original many-body Hamiltonian. Inspired by architectures used for operator learning, particularly in the context of driven-dissipative quantum mechanics, the learning process leverages these advancements to achieve efficient state preparation for arbitrary quantum many-body systems. We illustrate the power of this approach by training neural networks to learn the parameters of challenging quantum systems, including molecules interacting with light and mixtures of bosonic quantum systems forming supersolids, highlighting its superior performance over traditional methods, such as the (polaritonic) coupled-cluster approach.

Ultracold quantum gases can benefit from modern machine learning tools in various ways, e.g., in the data evaluation or in the read-out process. As an example of the former, I will present our results on identifying phase transitions from experimental quantum-gas data enhanced via supervised and unsupervised machine learning. As an example of the latter, I will explain our recently introduced phase microscope and discuss possibilities for improving it via machine learning.

Qubit decoherence in solid-state devices arises from temporally correlated noise, limiting control fidelity. Bayesian tracking estimates fluctuating Hamiltonian parameters but requires balancing accuracy and operational overhead. We frame adaptive estimation as a sequential decision problem and propose a reinforcement learning (RL) approach that optimally allocates measurements based on learned noise dynamics. The RL agent selects probing times and updates uncertainty to mitigate noise drift, improving efficiency. Evaluated on simulated 1/f noise, our approach enhances estimation-accuracy trade-offs and enables FPGA-compatible real-time control.

In the ongoing race towards experimental implementations of quantum error correction (QEC), finding ways to automatically discover codes and encoding strategies tailored to the qubit hardware platform is emerging as a critical problem. In this work, we use reinforcement learning (RL) to automate the discovery of QEC codes, along with their hardware-efficient encoding circuits. Our approach adapts these codes to diverse noise models and qubit connectivity constraints, making them highly tailored to specific quantum hardware platforms of interest. I will also show more recent work in progress where we train generative models on the synthetic dataset of QEC codes discovered with the RL strategy.

Classical machine learning (ML) provides a potentially powerful approach to solving challenging quantum many-body problems in physics and chemistry. However, the advantages of ML over traditional methods have not been firmly established. In this work, we prove that classical ML algorithms **can** efficiently learn to predict important properties of a quantum many-body system. In particular, ML can provably predict ground-state properties of gapped Hamiltonians after learning from other Hamiltonians in the same quantum phase of matter. Our proof technique combines signal processing with quantum many-body physics and also builds upon the recently developed framework of classical shadows. I will try to convey the ideas and also present numerical experiments that confirm our theoretical findings. This colloquium talk is based on joint work with Hsin-Yuan (Robert) Huang, Giacomo Torlai, Victor Albert and John Preskill, see [Huang et al., Provably efficient machine learning for quantum many-body problems, Science 2022]

The parquet equations are a set of field-theoretical integral equations for the self-energy and two-particle vertex functions of an interacting many-body system, that have to be solved self-consistently. One of the numerical challenges in solving them is that the vertex functions require huge amounts of memory when stored using conventional discretization grids. The quantics tensor train (QTT) representation of multivariate functions has already been around for a decade or so, but it was only recently applied to quantum field theory. This representation, based on length/energy scale separation, leads to significant dimensional reduction of the parquet equations problem, removing memory bottlenecks. The computational cost becomes logarithmic in grid size and depends strongly only on the maximum bond dimension, which is small enough so that the overall computational cost is significantly reduced. In my talk I will present the results of the application of quantics tensor trains (QTTs) and tensor cross interpolation (TCI) to the solution of a full set of parquet equations for a simple model, where exact solution is known, the Hubbard atom, and for the single impurity Anderson model. The two-particle vertices in these models are functions dependent on three frequencies, but not on momenta of the scattering particles. I will show that the steps needed to evaluate the equations (Bethe–Salpeter equation, parquet equation and Schwinger–Dyson equation) can be decomposed into basic operations on the QTT-TCI (QTCI) compressed objects. The repeated application of these operations does not lead to a loss of accuracy beyond a specified tolerance and the iterative scheme converges even for numerically demanding parameters. The results show that this approach allows for an exponential increase of the number of grid points included in the calculations leading to an exponentially improving computational error for a linear increase in computational cost.

There have been some exciting advances in the solutions to correlated quantum many-body problems in recent years with the progress and synergies of machine learning, quantum computation and tensor networks. However, as we push to higher accuracy in demanding problems, we have to ensure that the field does not become disconnected from questions that take place on the long timescales of atomic (rather than electronic) motion. This is a huge challenge, because propagating atoms in time along with the explicit electronic structure entails a prohibitive number of electronic structure calculations at high accuracy. We will present our work to straddle this gap, with a simple yet surprisingly effective interpolation scheme, allowing for efficient inference of the correlated many-body electronic wave function through chemical space, and coupling this to the explicit motion of the atoms undergoing reactive chemistry. This has enabled the advances in high-accuracy electronic structure to be coupled to real molecular dynamics with unprecedented accuracy, allowing for the first time molecules to move on the exact potential energy surface as provided by state-of-the-art tensor network and machine learning inspired many-electron states.

Large language models, like transformers, have recently demonstrated immense powers in text and image generation. This success is driven by the ability to capture long-range correlations between elements in a sequence. The same feature makes the transformer a powerful wavefunction ansatz that addresses the challenge of describing correlations in simulations of qubit systems. In this talk I consider two-dimensional Rydberg atom arrays to demonstrate that transformers reach higher accuracies than conventional recurrent neural networks for variational ground state searches. I introduce and discuss model-inspired modifications of transformers that can significantly accelerate the simulations, allowing for the consideration of large-scale quantum many-body systems.

I will give an overview of recent progress on fermionic neural quantum states, with a focus on `hidden fermions' determinantal states and applications.

I present a variational approach to simulating the dynamics of open quantum systems using artificial neural networks. The parameters of a compressed representation of a probability distribution representing the quantum state are adapted dynamically according to the Lindblad master equation by employing a time-dependent variational principle. As an example, we apply this method to the dissipative quantum Heisenberg model in one and two dimensions. Moreover we extend this approach to the domain of continuous variables and use normalizing flows to variationally solve diffusive classical dynamics, as well as driven dissipative dynamics of multimode bosonic systems.

Neural-network wave-functions have been proven to serve as flexible variational ansatz for the ground state search of many-body Hamiltonians. At the same time, they are notoriously hard to optimize, in particular in the presence of frustration. The interplay between trainability and expressivity of the ansatz is further affected by computational limitations that pose constraints on the number of parameters and/or the number of samples. We investigate this broad question in a systematic approach, by comparing the optimization landscape of some of the most common neural-network architectures (RBMs, RNNs and GCNNs). In particular, we analyze fundamental similarities and differences to more standard machine learning tasks and discuss the notion of overparametrization in the context of neural-network quantum states.

We will discuss the usefulness of artificial neural networks in solving quantum many-body problems at zero temperature and finite temperatures.

Neural quantum states have emerged as a novel promising numerical method to solve the quantum many-body problem both in and out of equilibrium. In this talk I will highlight the recent progress in particular concerning correlated quantum matter in two spatial dimensions both for the equilibrium ground-state as well nonequilibrium real-time dynamics problem. For the calculations of ground states I will introduce the minimum-step stochastic reconfiguration method that reduces the optimization complexity by orders of magnitude. I will show that with this method we can now accurately train on unprecedentedly deep neural quantum states with millions of parameters allowing us to obtain the lowest variational energies as compared to existing numerical results for frustrated quantum magnets. Further, I will highlight the recent results on solving the real-time dynamics of correlated quantum matter using neural quantum states, which has allowed us to verify for instance for the first time the quantum Kibble-Zurek mechanism for interacting quantum many-body systems in two spatial dimensions.

The application of the novel neural quantum state method in fermionic systems has been based on multiple design choices, including neural Jastrow factor, neural backflow, and hidden fermion determinant state (HFDS). Here, we propose a natural generalization of HFDS named hidden fermion pfaffian state (HFPS). In HFPS, there are visible and hidden orbitals respectively given by mean-field pfaffians, and the neural network is employed to encode their coupling. We show that the HFPS architecture matches the convolutional neural network architecture and the pfaffian local update technique. We apply HFPS to the study of the Hubbard model on the square lattice, showing better accuracy compared with existing neural network methods.

Recent advances in quantum simulation are focused on combining matter and light to engineer new types of interactions, typically characterized by long-range effects, requiring the development of advanced numerical simulation techniques. For instance, ordered arrays of atoms placed at distances smaller than the wavelength of light display photo-mediated long-range interactions and a peculiar correlated emission. The main features observed when starting from a highly excited initial state are a superradiant burst at short times, followed by a non-trivial subradiant critical regime with a slow power-law relaxation. By integrating out the photonic degrees of freedom, the dynamics are effectively described by a Lindblad equation with long-range interactions and dissipation. To simulate these dynamics, we employ a recently proposed numerical approach that combines a positive operator-valued measure (POVM) description of the density matrix—approximated by a neural network—with a time-dependent variational principle (TDVP) to project the evolution of the state onto the neural network manifold. We explore upscaling to larger system sizes as a complementary tool to standard tensor network techniques, especially for long-range interactions and two-dimensional setups. From a more physical perspective, by applying a time-dependent Generalized Gibbs ensemble Ansatz, we uncover the role of (approximate) integrability at long times, which leads to the observed polynomial decay.

Machine learning is rapidly proving indispensable in tuning and characterising quantum devices. By facilitating the exploration of complex high-dimensional parameter spaces, these algorithms not only allow for the identification of optimal operational conditions but also surpass human experts in the characterisation of different operational regimes. I will present the first fully autonomous tuning of a spin qubit. This is a major advancement for scaling semiconductor quantum technologies and understanding variability in nominally identical devices.

The importance of machine learning (ML) in quantum physics has been rising, especially in studies of quantum phases of matter or finding ground states of interacting Hamiltonians. The more widespread its use, the more urgent the critical questions of how to extract meaningful physical insights from ML. This talk will explore the differences between explaining a trained model’s behavior (post-hoc explainability) and designing machine learning models with interpretable parts from the ground up. Using two case studies from our research - neural networks applied to the Su-Schrieffer-Heeger (SSH) model and the TetrisCNN model tailored to detecting phase transitions and their order parameters in spin systems - we will show the limitations of post-hoc explainability and the advantages of interpretable architectures. We argue that the most insightful interpretable models for non-tabular data are largely task-dependent, and we share our recipe for their design.

Over the years our group has developed a machine learning algorithm suitable for phase classification. The framework builds on a support vector machine based on tensorial kernels (TKSVM). It has the additional advantage that it can be made quasi-unsupervised in combination with graph theory. This is leveraged, in combination with the strong interpretability properties of TKSVM, to detect (novel) phases of matter in a typical research setting consisting of rather few data of noisy quality. I will demonstrate our approach for two examples: First, I will show how many iterations between TKSVM and human interpretation allowed us to elucidate the ordering phase of a classical rank-2 U(1) spin liquid, found at very low temperature on a breathing pyrochlore lattice, where previous Monte Carlo simulations failed to converge. The resulting structure features a 32-site unit cell with hybridized nematic order, spin-plane selection, an order-by-disorder mechanism, and the emergence of a subdimensional symmetry. Monte Carlo simulations initiated with the correct structure are then shown to converge nicely. Second, we have extended the algorithm so that it becomes suitable for the post-processing of quantum data. In particular, experimental data on trapped ions from the University of Innsbruck have been successfully classified, discriminating between a trivial paramagnetic phase and a symmetry protected phase characterized by string order. The precise form of the string correlators could be inferred, and interpreted. This was demonstrated for the cluster Hamiltonian with spin-1/2 qubits, as well as for Haldane order with spin-1 particles, implemented with both qubits and qutrits. Our hybrid approach paves the way to classify data of future quantum experiments with many qubits in regimes where no classical algorithms are feasible.

In the past years, machine learning applications in the field of quantum physics and quantum technologies have surged along various directions [1]. In this presentation we discuss how reinforcement learning provides a flexible framework for the optimal control of out-of-equilibrium open quantum systems. We discuss applications in quantum thermodynamics, ranging from the optimization of the performance of quantum thermal machines [2,3,4], including quantum measurements and feedback control [5], to the charging power of a Dicke quantum battery exhibiting a collective speedup of the charging power [6]. Novel control strategies are discovered that outperform previous proposals made in literature, and provide physical insights into the design of optimal control strategies. REFERENCES: [1] M. Krenn, J. Landgraf, T. Foesel, and F. Marquardt, Phys. Rev. A 107, 010101 (2023). [2] P.A. Erdman and F. Noé, NPJ Quantum Inf. 8, 1 (2022). [3] P.A. Erdman and F. Noé, PNAS Nexus 2, pgad248 (2023). [4] P. A. Erdman, A. Rolandi, P. Abiuso, M. Perarnau-Llobet and F. Noé, Phys. Rev. Res. 5, L022017 (2023). [5] P.A. Erdman, R. Czupryniak, B. Bhandari, A.N. Jordan, F. Noé, J. Eisert, and G. Guarnieri, arXiv:2408.15328 (2024). [6] P.A. Erdman, G. M. Andolina, V. Giovannetti, and F. Noé, arXiv:2212.12397, accepted in Phys. Rev. Lett. (2024).

Neural Quantum Sates (NQS) constitute a variational wave function ansatz, that can provably efficiently represent even highly entangled quantum many-body states. Beyond their representative power, NQS inherit the speed of modern neural networks (NN) and equally profit from the enormous development that NNs have recently received. In this work we show that NQS can efficiently find the ground state of quantum impurity models with large baths, allowing us to compute high quality real-frequency, zero-temperature Green's functions by means of a Krylov-like method. We demonstrate the capability of this approach and its potential as dynamical mean-field theory (DMFT) solver at the example of the Bethe lattice and other benchmarks.

Simulating quantum many-body systems on near-term quantum hardware is challenging due to noise, their limited size, and the need for discretization. In this talk, I will present two hybrid quantum-classical ansatze that combine quantum resources with classical methods to simulate both ground-state and dynamical properties of quantum systems. The first approach introduces a discretization-free, variational ansatz for continuous-space Hamiltonians, optimized via variational Monte Carlo, and applied to systems such as the quantum rotor model and the two-dimensional electron gas. By systematically increasing circuit parameters, we achieve improved simulation accuracy and demonstrate an advantage of the hybrid ansatz over other purely classical approaches. The second method addresses the challenge of simulating quantum many-body dynamics of discrete systems. Here, we evolve an initial state on a quantum computer according to a simplified Hamiltonian. A classical model corrects for omitted terms, mitigates errors from Trotterization, and captures additional degrees of freedom without increasing the qubit requirements on the quantum device. To showcase these capabilities, we apply our technique to spin and impurity systems.

Quantum dynamics compilation is an important task for improving quantum simulation efficiency: It aims to synthesize multi-qubit target dynamics into a circuit consisting of as few elementary gates as possible. Compared to deterministic methods such as Trotterization, variational quantum compilation (VQC) methods employ variational optimization to reduce gate costs while maintaining high accuracy. In this work, we explore the potential of a VQC scheme by making use of out-of-distribution generalization results in quantum machine learning (QML): By learning the action of a given many-body dynamics on a small data set of product states, we can obtain a unitary circuit that generalizes to highly entangled states such as the Haar random states. The efficiency in training allows us to use tensor network methods to compress such time-evolved product states by exploiting their low entanglement features. Our approach exceeds state-of-the-art compilation results in both system size and accuracy in one dimension (1D). For the first time, we extend VQC to systems on two-dimensional (2D) strips with a quasi-1D treatment, demonstrating a significant resource advantage over standard Trotterization methods, highlighting the method's promise for advancing quantum simulation tasks on near-term quantum processors.

Quantum-classical hybrid dynamics is crucial for accurately simulating complex systems where both quantum and classical behaviors need to be considered. However, coupling between classical and quantum degrees of freedom and the exponential growth of the Hilbert space present significant challenges. Current machine learning approaches for predicting such dynamics, while promising, remain unknown in their error bounds, sample complexity, and generalizability. In this work, we establish a generic theoretical framework for analyzing quantum-classical adiabatic dynamics with learning algorithms. Based on quantum information theory, we develop a provably efficient adiabatic learning (PEAL) algorithm with logarithmic system size sampling complexity and favorable time scaling properties. We benchmark PEAL on the Holstein model, and demonstrate its accuracy in predicting single-path dynamics and ensemble dynamics observables as well as transfer learning over a family of Hamiltonians. Our framework and algorithm open up new avenues for reliable and efficient learning of quantum-classical dynamics.

In recent years, neural quantum states (NQS) have established themselves as a state-of-the-art method for tackling paradigmatic quantum many-body problems (J1-J2 model, we are looking at you). However, as the Anna Karenina principle of quantum many-body problems suggests, “every problem is hard in its own way," and ab initio quantum chemistry is a perfect example. While the potential of NQS to accurately represent molecular ground states was widely acknowledged, scaling to larger system sizes and reaching good accuracy has posed significant challenges. The two key bottlenecks are: 1. Molecular wave functions are peaked, necessitating large batch sizes to obtain meaningful observable estimates. 2. The Hamiltonian involves too many terms (quartic in the number of spin-orbitals), making local energy evaluation prohibitively expensive. In this talk, I will present our approach, which tackles both issues in a unified manner—and show how the first "curse" might, in fact, be a "blessing" when addressing the second. Specifically, I will introduce a novel autoregressive sampling procedure that efficiently generates the required number of unique samples, along with a GPU-friendly algorithm for local energy calculations, designed to work in a compressed format. Additionally, I will highlight the importance of code optimization, as reducing the computation time can be as crucial as developing new ansatzes. Our results demonstrate how tasks that initially took 2 days are now completed in 10 minutes, allowing us to study systems with up to 118 qubits and 3 million terms, outperforming conventional quantum chemistry methods in certain cases.

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)

Neural-network quantum states are becoming one of the most powerful method to simulate quantum many-body systems. We present a simple framework in which a single architecture is trained to approximate simultaneously the ground state of multiple systems. Performing a single simulation, we get an accurate description of the entire phase diagram of quantum spin models. This approach yields comparable accuracy to training separate networks from scratch at each point on the phase diagram, yet demands only the computational cost of a single simulation. Moreover, our findings demonstrate that the network exhibits remarkable generalization capabilities across unseen models.