Three-dimensional representation of the many-body quantum state

This chapter describes the usage of ML to study condensed matter. ML is shown to accelerate the sampling of spin glass configurations in computer simulations by up to five orders of magnitude. On the analysis side, ML is very efficient for evaluating the output of simulations of topological order transitions. The latter are characterized by the temperature-driven unbinding of vortex/antivortex pairs. Such transitions may be detected by analyzing configurations using neural networks implemented with the aim of learning vortices. Incorporated in the heart of computer simulations, ML is used to study quantum many-body systems. Here, the mathematical object describing the quantum-mechanical state of a system – the many-body wave function – is implemented in a neural-network representation. Away from pure theory, ML allows the quantum state of a system to be reconstructed based on experimental measurements. Here, the experimental data are used to train a neural network via unsupervised learning to output the state as a neural-network quantum state. As a more technological application, ML is efficient in predicting new materials with desired properties. Neural networks or other ML techniques enable the prediction of thermodynamically stable compounds with a desired structure by searching over the entire compound space.

Quantum many-body scar states are special eigenstates of nonintegrable models with distinctive entanglement features that give rise to infinitely long-lived coherent dynamics under quantum quenches from certain initial states. We elaborate on a construction of quantum many-body scar states in which they emerge from Einstein-Podolsky-Rosen states in systems with two layers, wherein the two layers are maximally entangled. We apply this construction to spin systems as well as systems of itinerant fermions and bosons and demonstrate how symmetries can be harnessed to enhance its versatility. We show that several well-known examples of quantum many-body scars, including the tower of states in the spin-1 $XY$ model and the $\ensuremath{\eta}$-pairing states in the Fermi-Hubbard model, can be understood within this formalism. We also demonstrate how an infinite tower of many-body scar states can emerge in bilayer Bose-Hubbard models with charge conservation.

<title>Abstract</title> Learning many-body quantum states and quantum phase transitions remains a major challenge in quantum many-body physics. Classical machine learning methods offer certain advantages in addressing these difficulties. In this work, we propose a novel framework that bypasses the need to measure physical observables by directly learning the parameters of parameterized quantum circuits. By integrating the attention mechanism from large language models (LLMs) with a variational autoencoder (VAE), we efficiently capture hidden correlations within the circuit parameters. These correlations allow us to extract information about quantum phase transitions in an unsupervised manner. Moreover, our VAE acts as a classical representation of parameterized quantum circuits and the corresponding many-body quantum states, enabling the efficient generation of quantum states associated with specific phases. We apply our framework to a variety of quantum systems and demonstrate its broad applicability, with particularly strong performance in identifying topological quantum phase transitions.

The state space of quantum system is a Hilbert space with complex inner product. The two states quantum system is called a qubit. Its quantum state gives different geometric representations. The density matrix of a quantum state can describe all the information of a quantum system. In particular, the representation of quantum pure states by means of extended complex plane is given. On this basis, combined with the matrix representation of the quantum states of the composite system, the criteria for quantum entanglement and separability are studied. Finally, the measurement method of quantum pure state entanglement is given.

Two types of dynamic quantum state secret sharing based on tensor networks states

Tracking the time evolution of a quantum state allows one to verify the thermalization rate or the propagation speed of correlations in generic quantum systems. Inspired by the energy-time uncertainty principle, bounds have been demonstrated on the maximal speed at which a quantum state can change, resulting in immediate and practical tasks. Based on a programmable superconducting quantum processor, we test the dynamics of various emulated quantum mechanical systems encompassing single- and many-body states. We show that one can test the known quantum speed limits and that modifying a single Hamiltonian parameter allows the observation of the crossover of the different bounds on the dynamics. We also unveil the observation of minimal quantum speed limits in addition to more common maximal ones, i.e., the lowest rate of change of a unitarily evolved quantum state. Our results show a comprehensive experimental characterization of quantum speed limits and enhance the understanding for their subsequent study in engineered non-unitary conditions.

Representation of quantum states by statistical ensembles on the quantum phase space in the Hamiltonian form of quantum mechanics is analyzed. Various mathematical properties and some physical interpretations of the equivalence classes of ensembles representing a mixed quantum state in the Hamiltonian formulation are examined. In particular, non-uniqueness of the quantum phase space probability density associated with the quantum mixed state, Liouville dynamics of the probability densities and the possibility to represent the reduced states of bipartite systems by marginal distributions are discussed in detail. These considerations are used to study ensembles of hybrid quantum-classical systems. In particular, nonlinear evolution of a single hybrid system in a pure state and unequal evolutions of initially equivalent ensembles are discussed in the context of coupled hybrid systems.

Numerical modeling of fermionic many-body quantum systems presents similar challenges across various research domains, necessitating universal tools, including state-of-the-art machine learning techniques. Here, we introduce SOLAX, a Python library designed to compute and analyze fermionic quantum systems using the formalism of second quantization. SOLAX provides a modular framework for constructing and manipulating basis sets, quantum states, and operators, facilitating the simulation of electronic structures and determining many-body quantum states in finite-size Hilbert spaces. The library integrates machine learning capabilities to mitigate the exponential growth of Hilbert space dimensions in large quantum clusters. The core low-level functionalities are implemented using the recently developed Python library JAX. Demonstrated through its application to the Single Impurity Anderson Model, SOLAX offers a flexible and powerful tool for researchers addressing the challenges of many-body quantum systems across a broad spectrum of fields, including atomic physics, quantum chemistry, and condensed matter physics.

The Landauer cost for erasing information demands that information about a physical system be included in the total entropy, as proposed by Zurek [Nature 341, 119 (1989); Phys. Rev. A 40, 4731 (1989)]. A consequence is that most system states---either classical phase-space distributions or quantum pure states---have total entropy much larger than thermal equilibrium. If total entropy is to be a useful concept, this must imply that work can be extracted in going from equilibrium to a typical system state. The work comes from randomization of a ``memory'' that holds a record of the system state.

With rapid progress across platforms for quantum systems, the problem of many-body quantum state reconstruction for noisy quantum states becomes an important challenge. There has been a growing interest in approaching the problem of quantum state reconstruction using generative neural network models. Here we propose the ‘attention-based quantum tomography’ (AQT), a quantum state reconstruction using an attention mechanism-based generative network that learns the mixed state density matrix of a noisy quantum state. AQT is based on the model proposed in ‘Attention is all you need’ by Vaswani et al (2017 NIPS) that is designed to learn long-range correlations in natural language sentences and thereby outperform previous natural language processing (NLP) models. We demonstrate not only that AQT outperforms earlier neural-network-based quantum state reconstruction on identical tasks but that AQT can accurately reconstruct the density matrix associated with a noisy quantum state experimentally realized in an IBMQ quantum computer. We speculate the success of the AQT stems from its ability to model quantum entanglement across the entire quantum system much as the attention model for NLP captures the correlations among words in a sentence.

Measurements and feedback have emerged as powerful resources for creating many-body quantum states. However, a detailed understanding has been restricted to fixed-point representatives of phases of matter. Here, we go beyond this and characterize the patterns of many-body entanglement that can be deterministically created from measurement. Focusing on one spatial dimension, a framework is developed for the case where a single round of measurements is the only entangling operation. We show this creates matrix-product states and identify necessary and sufficient tensor conditions for preparability, which uniquely determine the preparation protocol. We use these conditions to both classify preparable quantum states and characterize their physical constraints. In particular, we find a trade-off between the richness of the preparable entanglement spectrum and correlation functions, which leads to a no-go theorem for preparing certain quantum states. More broadly, we connect properties of the preparation protocol to the resulting phase of matter, including trivial, symmetry-breaking, and symmetry-protected topological phases—for both uniform and modulated symmetries. This work offers a resource-theoretic perspective on preparable quantum entanglement and shows how to systematically create states of matter, away from their fixed points, in quantum devices.

We investigate the quantum electrodynamics of many-body systems, and discuss the circumstances under which the results are identical to the results of classical electrodynamics. This classical behavior is shown to occur when the motion of a large number of the particles is correlated. The correlation manifests itself in a form of coherence which is characterized by the many-body quantum states being very nearly an eigenstate of the current operator. We present several explicit examples of quantum states which exhibit such coherence and show that they give the expected classical results. Nowhere do we assume h $Yields$ 0. (AIP)

Dissipation generally leads to the decoherence of a quantum state. In contrast, numerous recent proposals have illustrated that dissipation can also be tailored to stabilize many-body entangled quantum states. While the focus of these works has been primarily on engineering the nonequilibrium steady state, we investigate the buildup of entanglement in the quantum trajectories. Specifically, we analyze the competition between two different dissipation channels arising from two incompatible continuous monitoring protocols. The first protocol locks the phase of neighboring sites upon registering a quantum jump, thereby generating a long-range entanglement through the system, while the second destroys the coherence via a dephasing mechanism. By studying the unraveling of stochastic quantum trajectories associated with the continuous monitoring protocols, we present a transition for the scaling of the averaged trajectory entanglement entropies, from critical scaling to area-law behavior. Our work provides an alternative perspective on the measurement-induced phase transition: the measurement can be viewed as monitoring and registering quantum jumps, offering an intriguing extension of these phase transitions through the long-established realm of quantum optics.

Optimization of quantum states using the variational principle has recently seen an upsurge due to developments of increasingly expressive wave functions. In order to improve on the accuracy of the Ansatze, it is a time-honored strategy to impose the systems' symmetries. We present an Ansatz where global non-Abelian symmetries are inherently embedded in its structure. We extend the model to incorporate lattice symmetries as well. We consider the prototypical example of the frustrated two-dimensional J(1)-J(2) model on a square lattice, for which eigenstates have been hard to model variationally. Our novel approach guarantees that the obtained ground state will have total spin zero. Benchmarks on the 2D J(1)-J(2) model demonstrate its state-of-the-art performance in representing the ground state. Furthermore, our methodology permits to find the wave functions of excited states with definite quantum numbers associated to the considered symmetries (including the non-Abelian ones), without modifying the architecture of the network.

Controlling the evolution of photonic quantum states is crucial for most quantum information processing and metrology tasks. Due to its importance, many mechanisms of quantum state evolution have been tested in detail and are well understood; however, the fundamental phase anomaly of evolving waves, called the Gouy phase, has had a limited number of studies in the context of elementary quantum states of light, especially in the case of photon number states. Here we outline a simple method for calculating the quantum state evolution upon propagation and demonstrate experimentally how this quantum Gouy phase affects two-photon quantum states. Our results show that the increased phase sensitivity of multi-photon states also extends to this fundamental phase anomaly and has to be taken into account to fully understand the state evolution. We further demonstrate how the Gouy phase can be used as a tool for manipulating quantum states of any bosonic system in future quantum technologies, outline a possible application in quantum-enhanced sensing, and dispel a common misconception attributing the increased phase sensitivity of multi-photon quantum states solely to an effective de Broglie wavelength.