Quantum Many-body States Research Articles

Entanglement of Formation of Mixed Many-Body Quantum States via Tree Tensor Operators.

We present a numerical strategy to efficiently estimate bipartite entanglement measures, and in particular the entanglement of formation, for many-body quantum systems on a lattice. Our approach exploits the tree tensor operator tensor network Ansatz, a positive loopless representation for density matrices which, as we demonstrate, efficiently encodes information on bipartite entanglement, enabling the upscaling of entanglement estimation. Employing this technique, we observe a finite-size scaling law for the entanglement of formation in 1D critical lattice models at finite temperature for up to 128 spins, extending to mixed states the scaling law for the entanglement entropy.

Quantum entanglement control of electron–phonon systems by light irradiation

We numerically study the dynamics of quantum entanglement between interacting electron-phonon and qubit-spin systems under photoirradiation, employing a model of multiple spins and boson modes. The interplay of the antiferromagnetic exchange and electron-phonon interactions provides us with a phase diagram, wherein each phase is characterized by the ground state property of the electron-phonon system. Light irradiation of the electron-phonon system facilitates the generation of quantum entanglement, according to the spin configuration and the phonon state in the ground state. Analyses using the quantum mutual information and the singular values of the reduced density matrix indicate that the quantum mechanical effect of the irradiated light appears in the state of the material.

Anti-Unruh effect in the thermal background

We study the influence of the thermal background on the existence of the anti-Unruh effect. For the massless scalar field, we present that the anti-Unruh effect can appear when the detector is accelerated in the thermal field, which is shown to be absent when accelerating in the Minkowski vacuum. For the massive scalar field, it is found that the anti-Unruh effect which exists in the case for accelerated detectors in the vacuum can disappear when the background temperature increases. We use the many-body entangled quantum state to study the situation of the massive scalar field. When the many-body state is accelerated, its entanglement is decreasing with the increase of the background temperature and the phenomenon of the sudden death for entanglement occurs. This provides a valuable indication that the anti-Unruh effect can exist below some specific background temperature, dependent on the actual experimental environment.

Stability of scar states in the two-dimensional PXP model against random disorder

Recently a class of quantum systems exhibiting weak ergodicity breaking has attracted much attention. These systems feature a special band of eigenstates called quantum many-body scar states in the energy spectrum. In this work we study the fate of quantum many-body scar states in a two-dimensional lattice against random disorders. We show that in both the square lattice and the honeycomb lattice the scar states can persist up to a finite disorder strength, before eventually being erased by the disorder. We further study the localization properties of the system in the presence of even stronger disorders and show that whether a full localization transition occurs depends on the type of disorder we introduce. Our study thus reveals the fascinating interplay between disorder and quantum many-body scarring in a two-dimensional system.

Digraph states and their neural network representations

With the rapid development of machine learning, artificial neural networks provide a powerful tool to represent or approximate many-body quantum states. It was proved that every graph state can be generated by a neural network. Here, we introduce digraph states and explore their neural network representations (NNRs). Based on some discussions about digraph states and neural network quantum states (NNQSs), we construct explicitly an NNR for any digraph state, implying every digraph state is an NNQS. The obtained results will provide a theoretical foundation for solving the quantum many-body problem with machine learning method whenever the wave-function is known as an unknown digraph state or it can be approximated by digraph states.

Quantum Circuit Approximations and Entanglement Renormalization for the Dirac Field in 1+1 Dimensions

The multiscale entanglement renormalization ansatz describes quantum many-body states by a hierarchical entanglement structure organized by length scale. Numerically, it has been demonstrated to capture critical lattice models and the data of the corresponding conformal field theories with high accuracy. However, a rigorous understanding of its success and precise relation to the continuum is still lacking. To address this challenge, we provide an explicit construction of entanglement-renormalization quantum circuits that rigorously approximate correlation functions of the massless Dirac conformal field theory. We directly target the continuum theory: discreteness is introduced by our choice of how to probe the system, not by any underlying short-distance lattice regulator. To achieve this, we use multiresolution analysis from wavelet theory to obtain an approximation scheme and to implement entanglement renormalization in a natural way. This could be a starting point for constructing quantum circuit approximations for more general conformal field theories.

Quantum Circuits Assisted by Local Operations and Classical Communication: Transformations and Phases of Matter.

We introduce deterministic state-transformation protocols between many-body quantum states that can be implemented by low-depth quantum circuits followed by local operations and classical communication. We show that this gives rise to a classification of phases in which topologically ordered states or other paradigmatic entangled states become trivial. We also investigate how the set of unitary operations is enhanced by local operations and classical communication in this scenario, allowing one to perform certain large-depth quantum circuits in terms of low-depth ones.

Attention-based quantum tomography

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.

Quantum many-body states and Green's functions of nonequilibrium electron-magnon systems: Localized spin operators versus their mapping to Holstein-Primakoff bosons

The operators of localized spins within a magnetic material commute at different sites of its lattice and anticommute on the same site, so they are neither fermionic nor bosonic operators. Thus, to construct diagrammatic many-body perturbation theory, the spin operators are usually mapped to the bosonic ones with Holstein-Primakoff (HP) transformation being the most widely used in magnonics and spintronics literature. However, to make calculations tractable, the square root of operators in the HP transformation is expanded into a Taylor series truncated to some low order. This poses a question on the range of validity of truncated HP transformation when describing nonequilibrium dynamics of localized spins interacting with each other or with conduction electron spins. Here we apply exact diagonalization techniques to Hamiltonian of fermions (i.e., electrons) interacting with HP bosons vs. Hamiltonian of fermions interacting with the original localized spin operators in order to compare their many-body states and one-particle equilibrium or nonequilibrium Green functions. The Hamiltonian of fermions interacting with HP bosons gives incorrect ground state and electronic spectral function, unless large number of terms are retained in truncated HP transformation. Furthermore, tracking nonequilibrium dynamics of localized spins over longer time intervals requires progressively larger number of terms in truncated HP transformation. Finally, we show that recently proposed [M. Vogl et al., Phys. Rev. Research 2, 043243 (2020); J. K\"{o}nig et al., SciPost Phys. 10, 007 (2021)] resummed HP transformation resolves the trouble with truncated HP transformation, while allowing us to derive an exact (manifestly Hermitian) Hamiltonian consisting of finite and fixed number of boson-boson and electron-boson interacting terms.

Fast-forwarding with NISQ processors without feedback loop

Simulating quantum dynamics is expected to be performed more easily on a quantum computer than on a classical computer. However, the currently available quantum devices lack the capability to implement fault-tolerant quantum algorithms for quantum simulation. Hybrid classical quantum algorithms such as the variational quantum algorithms have been proposed to effectively use current term quantum devices. One promising approach to quantum simulation in the noisy intermediate-scale quantum (NISQ) era is the diagonalisation based approach, with some of the promising examples being the subspace variational quantum simulator (SVQS), variational fast forwarding (VFF), fixed-state variational fast forwarding (fs-VFF), and the variational Hamiltonian diagonalisation (VHD) algorithms. However, these algorithms require a feedback loop between the classical and quantum computers, which can be a crucial bottleneck in practical application. Here, we present the classical quantum fast forwarding (CQFF) as an alternative diagonalisation based algorithm for quantum simulation. CQFF shares some similarities with SVQS, VFF, fs-VFF and VHD but removes the need for a classical-quantum feedback loop and controlled multi-qubit unitaries. The CQFF algorithm does not suffer from the barren plateau problem and the accuracy can be systematically increased. Furthermore, if the Hamiltonian to be simulated is expressed as a linear combination of tensored-Pauli matrices, the CQFF algorithm reduces to the task of sampling some many-body quantum state in a set of Pauli-rotated bases, which is easy to do in the NISQ era. We run the CQFF algorithm on existing quantum processors and demonstrate the promise of the CQFF algorithm for current-term quantum hardware. We compare CQFF with Trotterization for a XY spin chain model Hamiltonian and find that the CQFF algorithm can simulate the dynamics more than 105 times longer than Trotterization on current-term quantum hardware. This provides a 104 times improvement over the previous record.

Quantum Variational Learning of the Entanglement Hamiltonian.

Learning the structure of the entanglement Hamiltonian (EH) is central to characterizing quantum many-body states in analog quantum simulation. We describe a protocol where spatial deformations of the many-body Hamiltonian, physically realized on the quantum device, serve as an efficient variational ansatz for a local EH. Optimal variational parameters are determined in a feedback loop, involving quench dynamics with the deformed Hamiltonian as a quantum processing step, and classical optimization. We simulate the protocol for the ground state of Fermi-Hubbard models in quasi-1D geometries, finding excellent agreement of the EH with Bisognano-Wichmann predictions. Subsequent on-device spectroscopy enables a direct measurement of the entanglement spectrum, which we illustrate for a Fermi Hubbard model in a topological phase.

Dimerization of Many-Body Subradiant States in Waveguide Quantum Electrodynamics.

We theoretically study subradiant states in an array of atoms coupled to photons propagating in a one-dimensional waveguide focusing on the strongly interacting many-body regime with large excitation fill factor f. We introduce a generalized many-body entropy of entanglement based on exact numerical diagonalization followed by a high-order singular value decomposition. This approach has allowed us to visualize and understand the structure of a many-body quantum state. We reveal the breakdown of fermionized subradiant states with increase of f with the emergence of short-ranged dimerized antiferromagnetic correlations at the critical point f=1/2 and the complete disappearance of subradiant states at f>1/2.

Deep Learning Quantum States for Hamiltonian EstimationSupported by the National Natural Science Foundation of China (Grant Nos. 12004266, 11834014 and 11975050), the Beijing Natural Science Foundation (Grant Nos. 1192005 and Z180013), the Foundation of Beijing Education Committees (Grant No. KM202010028013), and the Academy for Multidisciplinary Studies, Capital Normal University.

Human experts cannot efficiently access physical information of a quantum many-body states by simply “reading” its coefficients, but have to reply on the previous knowledge such as order parameters and quantum measurements. We demonstrate that convolutional neural network (CNN) can learn from coefficients of many-body states or reduced density matrices to estimate the physical parameters of the interacting Hamiltonians, such as coupling strengths and magnetic fields, provided the states as the ground states. We propose QubismNet that consists of two main parts: the Qubism map that visualizes the ground states (or the purified reduced density matrices) as images, and a CNN that maps the images to the target physical parameters. By assuming certain constraints on the training set for the sake of balance, QubismNet exhibits impressive powers of learning and generalization on several quantum spin models. While the training samples are restricted to the states from certain ranges of the parameters, QubismNet can accurately estimate the parameters of the states beyond such training regions. For instance, our results show that QubismNet can estimate the magnetic fields near the critical point by learning from the states away from the critical vicinity. Our work provides a data-driven way to infer the Hamiltonians that give the designed ground states, and therefore would benefit the existing and future generations of quantum technologies such as Hamiltonian-based quantum simulations and state tomography.

Symmetry enriched phases of quantum circuits

Quantum circuits consisting of random unitary gates and subject to local measurements have been shown to undergo a phase transition, tuned by the rate of measurement, from a state with volume-law entanglement to an area-law state. From a broader perspective, these circuits generate a novel ensemble of quantum many-body states at their output. In this paper, we characterize this ensemble and classify the phases that can be established as steady states. Symmetry plays a nonstandard role in that the physical symmetry imposed on the circuit elements does not on its own dictate the possible phases. Instead, it is extended by dynamical symmetries associated with this ensemble to form an enlarged symmetry. Thus, we predict phases that have no equilibrium counterpart and could not have been supported by the physical circuit symmetry alone. We give the following examples. First, we classify the phases of a circuit operating on qubit chains with Z2 symmetry. One striking prediction, corroborated with numerical simulation, is the existence of distinct volume-law phases in one dimension, which nonetheless support true long-range order. We furthermore argue that owing to the enlarged symmetry, this system can in principle support a topological area-law phase, protected by the combination of the circuit symmetry and a dynamical permutation symmetry. Second, we consider a Gaussian fermionic circuit that only conserves fermion parity. Here the enlarged symmetry gives rise to a U(1) critical phase at moderate measurement rates and a Kosterlitz–Thouless transition to area-law phases. We comment on the interpretation of the different phases in terms of the capacity to encode quantum information. We discuss close analogies to the theory of spin glasses pioneered by Edwards and Anderson as well as crucial differences that stem from the quantum nature of the circuit ensemble.

Dynamical preparation of stripe states in spin-orbit-coupled gases

In spinor Bose-Einstein condensates, spin-changing collisions are a remarkable proxy to coherently realize macroscopic many-body quantum states. These processes have been, e.g., exploited to generate entanglement, to study dynamical quantum phase transitions, and proposed for realizing nematic phases in atomic condensates. In the same systems dressed by Raman beams, the coupling between spin and momentum induces a spin dependence in the scattering processes taking place in the gas. Here we show that, at weak couplings, such modulation of the collisions leads to an effective Hamiltonian which is equivalent to the one of an artificial spinor gas with spin-changing collisions that are tunable with the Raman intensity. By exploiting this dressed-basis description, we propose a robust protocol to coherently drive the spin-orbit coupled condensate into the ferromagnetic stripe phase via crossing a quantum phase transition of the effective low-energy model in an excited-state.

Symmetry-protected dissipative preparation of matrix product states

We propose and analyze a method for efficient dissipative preparation of matrix product states that exploits their symmetry properties. Specifically, we construct an explicit protocol that makes use of driven-dissipative dynamics to prepare a many-body quantum state that features symmetry-protected topological order and nontrivial edge excitations. The preparation protocol is protected from errors that respect the symmetry, allowing for robust experimental implementation without fine-tuned control. Numerical simulations show that the preparation time scales polynomially in system size $n$. Furthermore, we demonstrate that this scaling can be improved to $O({log}^{2}n)$ by using parallel preparation of individual segments and fusing them via quantum feedback. A concrete scheme using excitation of trapped neutral atoms into the Rydberg state via electromagnetically induced transparency is proposed and generalizations to a broader class of matrix product states are discussed.

Topological quantum many-body scars in quantum dimer models on the kagome lattice

We present a class of quantum dimer models on the kagome lattice with full translational invariance that feature a quantum many-body scar state of analytically known entanglement properties within their spectra. Using exact diagonalization on lattices of up to 60 sites, we show that non-scar states conform to the eigenstate thermalization hypothesis. Specifically, we show that energies are distributed according to the Gaussian ensemble expected of their respective symmetry sector, illustrate the existence of the scar from bipartite entanglement properties, and demonstrate revival phenomena in studies of fidelity dynamics.

Representations of hypergraph states with neural networks* *Supported by the National Natural Science Foundation of China (Nos. 12001480, 11871318), the Applied Basic Research Program of Shanxi Province (No. 201901D211461), the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (No. 2020L0554), the Excellent Doctoral Research Project of Shanxi Province (No. QZX-2020001), and the PhD Start-up Project of Yuncheng

The quantum many-body problem (QMBP) has become a hot topic in high-energy physics and condensed-matter physics. With an exponential increase in the dimensions of Hilbert space, it becomes very challenging to solve the QMBP, even with the most powerful computers. With the rapid development of machine learning, artificial neural networks provide a powerful tool that can represent or approximate quantum many-body states. In this paper, we aim to explicitly construct the neural network representations of hypergraph states. We construct the neural network representations for any k-uniform hypergraph state and any hypergraph state, respectively, without stochastic optimization of the network parameters. Our method constructively shows that all hypergraph states can be represented precisely by the appropriate neural networks introduced in [Science 355 (2017) 602] and formulated in [Sci. China-Phys. Mech. Astron. 63 (2020) 210312].

Optimal parent Hamiltonians for time-dependent states

Given a generic time-dependent many-body quantum state, we determine the associated parent Hamiltonian. This procedure may require, in general, interactions of any sort. Enforcing the requirement of a fixed set of engineerable Hamiltonians, we find the optimal Hamiltonian once a set of realistic elementary interactions is defined. We provide three examples of this approach. We first apply the optimization protocol to the ground states of the one-dimensional Ising model and a ferromagnetic $p$-spin model but with time-dependent coefficients. We also consider a time-dependent state that interpolates between a product state and the ground state of a $p$-spin model. We determine the time-dependent optimal parent Hamiltonian for these states and analyze the capability of this Hamiltonian of generating the state evolution. Finally, we discuss the connections of our approach to shortcuts to adiabaticity.

Quantum entanglement recognition

Entanglement constitutes a key characteristic feature of quantum matter. Its detection, however, still faces major challenges. In this letter, we formulate a framework for probing entanglement based on machine learning techniques. The central element is a protocol for the generation of statistical images from quantum many-body states, with which we perform image classification by means of convolutional neural networks. We show that the resulting quantum entanglement recognition task is accurate and can be assigned a well-controlled error across a wide range of quantum states. We discuss the potential use of our scheme to quantify quantum entanglement in experiments. Our developed scheme provides a generally applicable strategy for quantum entanglement recognition in both equilibrium and nonequilibrium quantum matter.