Wave-Function Network Description and Kolmogorov Complexity of Quantum Many-Body Systems
Tiago Mendes-Santos, Markus Schmitt, Adriano Angelone, Álex Rodríguez, Pascal Scholl, H. J. Williams, Daniel Barredo, Thierry Lahaye, Antoine Browaeys, Markus Heyl, Marcello Dalmonte
Programmable quantum devices are now able to probe wave functions at unprecedented levels. This is based on the ability to project the many-body state of atom and qubit arrays onto a measurement basis which produces snapshots of the system wave function. Extracting and processing information from such observations remains, however, an open quest. One often resorts to analyzing low-order correlation functions—that is, discarding most of the available information content. Here, we introduce wave-function networks—a mathematical framework to describe wave-function snapshots based on network theory. For many-body systems, these networks can become scale-free—a mathematical structure that has found tremendous success and applications in a broad set of fields, ranging from biology to epidemics to Internet science. We demonstrate the potential of applying these techniques to quantum science by introducing protocols to extract the Kolmogorov complexity corresponding to the output of a quantum simulator and implementing tools for fully scalable cross-platform certification based on similarity tests between networks. We demonstrate the emergence of scale-free networks analyzing experimental data obtained with a Rydberg quantum simulator manipulating up to 100 atoms. Our approach illustrates how, upon crossing a phase transition, the simulator complexity decreases while correlation length increases—a direct signature of buildup of universal behavior in data space. Comparing experiments with numerical simulations, we achieve cross-certification at the wave-function level up to timescales of 4 μs with a confidence level of 90% and determine experimental calibration intervals with unprecedented accuracy. Our framework is generically applicable to the output of quantum computers and simulators with access to the system wave function and requires probing accuracy and repetition rates accessible to most currently available platforms. Published by the American Physical Society 2024
Programmable quantum devices are now able to probe wave functions at unprecedented levels. This is based on the ability to project the many-body state of atom and qubit arrays onto a measurement basis which produces snapshots of the system wave function. Extracting and processing information from such observations remains, however, an open quest. One often resorts to analyzing low-order correlation functions—that is, discarding most of the available information content. Here, we introduce wave-function networks—a mathematical framework to describe wave-function snapshots based on network theory. For many-body systems, these networks can become scale-free—a mathematical structure that has found tremendous success and applications in a broad set of fields, ranging from biology to epidemics to Internet science. We demonstrate the potential of applying these techniques to quantum science by introducing protocols to extract the Kolmogorov complexity corresponding to the output of a quantum simulator and implementing tools for fully scalable cross-platform certification based on similarity tests between networks. We demonstrate the emergence of scale-free networks analyzing experimental data obtained with a Rydberg quantum simulator manipulating up to 100 atoms. Our approach illustrates how, upon crossing a phase transition, the simulator complexity decreases while correlation length increases—a direct signature of buildup of universal behavior in data space. Comparing experiments with numerical simulations, we achieve cross-certification at the wave-function level up to timescales of 4 μs with a confidence level of 90% and determine experimental calibration intervals with unprecedented accuracy. Our framework is generically applicable to the output of quantum computers and simulators with access to the system wave function and requires probing accuracy and repetition rates accessible to most currently available platforms. Published by the American Physical Society 2024
We review methods that allow one to detect and characterize quantum correlations in many-body systems, with a special focus on approaches which are scalable. Namely, those applicable to systems with many degrees of freedom, without requiring a number of measurements or computational resources to analyze the data that scale exponentially with the system size. We begin with introducing the concepts of quantum entanglement, Einstein-Podolsky-Rosen steering, and Bell nonlocality in the bipartite scenario, to then present their multipartite generalization. We review recent progress on characterizing these quantum correlations from partial information on the system state, such as through data-driven methods or witnesses based on low-order moments of collective observables. We then review state-of-the-art experiments that demonstrate the preparation, manipulation and detection of highly-entangled many-body systems. For each platform (e.g. atoms, ions, photons, superconducting circuits) we illustrate the available toolbox for state preparation and measurement, emphasizing the challenges that each system poses. To conclude, we present a list of timely open problems in the field.
In the past decades, rapid advances in the experimental control of quantum systems have opened up unparalleled capabilities of engineering exotic quantum states. Nowadays, we may consider ourselves witnesses of a second quantum revolution as strongly correlated quantum matter is regularly created, on a daily basis, within the most diverse platforms, e.g. arrays of Rydberg atoms, ultra-cold atoms in optical lattices, superconducting qubits, trapped ions, and quantum dots. In this era, dubbed Noisy-Intermediate Scale Quantum (NISQ) era, the effort in pursuing research for realizing quantum technologies with practical purposes, such as quantum computing, simulations, communication and metrology, has greatly accelerated, and we are just starting to experience the immense progress that could be achieved. The remarkable breakthrough we are facing builds upon seminal theoretical and experimental advances in quantum physics. The key achievements in atomic, molecular, and optical (AMO) physics have opened up the possibility of controlling, trapping, and measuring single atoms, one by one, with high accuracy and reliability. In parallel, from a theoretical point of view, the study of quantum entanglement and correlations has bridged AMO physics and quantum information with crucial proposals for the realization of universal quantum computers. In this context, entanglement has emerged as one of the key tools to characterize and to exploit quantum many-body systems for quantum information purposes. We will study quantum entanglement in several scenarios to investigate and probe complex quantum many-body systems. We will consider examples ranging from generic mixed states in equilibrium to out-of-equilibrium dynamics, with and without dissipation, and topologically non-trivial systems. The leitmotif of this work will be how entanglement and correlations can be exploited to characterize the many-body quantum state describing a physical system and how entanglement can be detected in an experimentally efficient manner.
Simulating quantum many-body systems is essential for understanding complex phenomena in condensed matter, chemistry, and materials science. Quantum computing offers a way to overcome the limitations of classical methods by operating directly in exponentially large Hilbert spaces. In particular, classical methods often fail to capture the dynamics of strongly correlated systems, which makes quantum algorithms a promising alternative for exploring these challenging regimes. This thesis uses quantum dynamics simulations to probe the behaviour of many-body systems. We first introduce a spectroscopy-based protocol to extract excitation spectra from the time evolution of observables and analyse its performance on near-term quantum devices. We show that the protocol requires only short evolution times and is resilient to errors, facilitating its implementation on real hardware. Building on this, we simulate fermionic excitation spectra using current quantum devices, introducing resource optimisations - including an approximate free-fermion initial state, a unitary quench without ancillae, and reduced Trotter steps - that significantly lower circuit depth. These improvements enable simulations at system sizes competitive with the state of the art, yielding results that match classical benchmarks. While most studies focus on unitary dynamics of isolated systems, open quantum dynamics is increasingly relevant for describing non-equilibrium processes across biology, materials science, and quantum control. We extend our scope to systems coupled to an environment and governed by Lindbladian evolution. By studying simulation complexity through physically meaningful quantities - such as correlation length and mixing time - we explore whether open-system dynamics can be as tractable as their unitary counterparts. These findings provide insights into the computational complexity landscape of open-system simulation and highlight regimes where quantum methods offer a clear advantage.
Quantum computers and simulators offer unparalleled capabilities of probing quantum many-body states, by obtaining snapshots of the many-body wave function via collective projective measurements. The probability distribution obtained by such snapshots (which are fundamentally limited to a negligible fraction of the Hilbert space) is of fundamental importance to determine the power of quantum computations. However, its relation to many-body collective properties is poorly understood. Here, we develop a theoretical framework to link quantum phases of matter to their snapshots, based on a combination of data complexity and network theory analyses. The first step in our scheme consists of applying Occam's razor principle to quantum sampling: given snapshots of a wave function, we identify a minimal-complexity measurement basis by analyzing the information compressibility of snapshots over different measurement bases. The second step consists of analyzing arbitrary correlations using network theory, building a wave-function network from the minimal-complexity basis data. This approach allows us to stochastically classify the output of quantum computers and simulations, with no assumptions on the underlying dynamics, and in a fully interpretable manner. We apply this method to quantum states of matter in one-dimensional translational invariant systems, where such classification is exhaustive, and where it reveals an interesting interplay between algorithmic and computational complexity for many-body states. Our framework is of immediate experimental relevance, and can be further extended both in terms of more advanced network mathematics, including discrete homology, as well as in terms of applications to physical phenomena, such as time-dependent dynamics and gauge theories.
The simulation of quantum matter with classical hardware plays a central role in the discovery and development of quantum many-body systems, with far-reaching implications in condensed matter physics and quantum technologies. In general, efficient and sophisticated algorithms are required to overcome the severe challenge posed by the exponential scaling of the Hilbert space of quantum systems. In contrast, hardware built with quantum bits of information are inherently capable of efficiently finding solutions of quantum many-body problems. While a universal and scalable quantum computer is still beyond the horizon, recent advances in qubit manufacturing and coherent control of synthetic quantum matter are leading to a new generation of intermediate scale quantum hardware. \n \nThe complexity underlying quantum many-body systems closely resembles the one encountered in many problems in the world of information and technology. In both contexts, the complexity stems from a large number of interacting degrees of freedom. A powerful strategy in the latter scenario is machine learning, a subfield of artificial intelligence where large amounts of data are used to extract relevant features and patterns. In particular, artificial neural networks have been demonstrated to be capable of discovering low-dimensional representations of complex objects from high-dimensional dataset, leading to the profound technological revolution we all witness in our daily life. \n \nIn this Thesis, we envision a new paradigm for scientific discovery in quantum physics. On the one hand, we have the essentially unlimited data generated with the increasing amount of highly controllable quantum hardware. On the other hand, we have a set of powerful algorithms that efficiently capture non-trivial correlations from high-dimensional data. Therefore, we fully embrace this data-driven approach to quantum mechanics, and anticipate new exciting possibilities in the field of quantum many-body physics and quantum information science. We revive a powerful stochastic neural network called a restricted Boltzmann machine, which slowly moved out of fashion after playing a central role in the machine learning revolution of the early 2010s. We introduce a neural-network representation of quantum states based on this generative model. We propose a set of algorithms to reconstruct unknown quantum states from measurement data and numerically demonstrate their potential, with important implications for current experiments. These include the reconstruction of experimentally inaccessible properties, such as entanglement, and diagnostics to determine sources of noise. Furthermore, we introduce a machine learning framework for quantum error correction, where a neural network learns the best decoding strategy directly from data. We expect that the full integration between quantum hardware and artificial intelligence will become the gold standard, and will drive the world into the era of fault-tolerant quantum computing and large-scale quantum simulations.
Programmable quantum devices are now able to probe wave functions at unprecedented levels. This is based on the ability to project the many-body state of atom and qubit arrays onto a measurement basis which produces snapshots of the system wave function. Extracting and processing information from such observations remains, however, an open quest. One often resorts to analyzing low-order correlation functions—that is, discarding most of the available information content. Here, we introduce wave-function networks—a mathematical framework to describe wave-function snapshots based on network theory. For many-body systems, these networks can become scale-free—a mathematical structure that has found tremendous success and applications in a broad set of fields, ranging from biology to epidemics to Internet science. We demonstrate the potential of applying these techniques to quantum science by introducing protocols to extract the Kolmogorov complexity corresponding to the output of a quantum simulator and implementing tools for fully scalable cross-platform certification based on similarity tests between networks. We demonstrate the emergence of scale-free networks analyzing experimental data obtained with a Rydberg quantum simulator manipulating up to 100 atoms. Our approach illustrates how, upon crossing a phase transition, the simulator complexity decreases while correlation length increases—a direct signature of buildup of universal behavior in data space. Comparing experiments with numerical simulations, we achieve cross-certification at the wave-function level up to timescales of 4 μs with a confidence level of 90% and determine experimental calibration intervals with unprecedented accuracy. Our framework is generically applicable to the output of quantum computers and simulators with in situ access to the system wave function and requires probing accuracy and repetition rates accessible to most currently available platforms.