Machine learning, one of today's most rapidly growing interdisciplinary fields, promises an unprecedented perspective for solving intricate quantum many-body problems. Understanding the physical aspects of the representative artificial neural-network states is recently becoming highly desirable in the applications of machine learning techniques to quantum many-body physics. Here, we study the quantum entanglement properties of neural-network states, with a focus on the restricted-Boltzmann-machine (RBM) architecture. We prove that the entanglement of all short-range RBM states satisfies an area law for arbitrary dimensions and bipartition geometry. For long-range RBM states we show by using an exact construction that such states could exhibit volume-law entanglement, implying a notable capability of RBM in representing efficiently quantum states with massive entanglement. We further examine generic RBM states with random weight parameters. We find that their averaged entanglement entropy obeys volume-law scaling and meantime strongly deviates from the Page-entropy of the completely random pure states. We show that their entanglement spectrum has no universal part associated with random matrix theory and bears a Poisson-type level statistics. Using reinforcement learning, we demonstrate that RBM is capable of finding the ground state (with power-law entanglement) of a model Hamiltonian with long-range interaction. In addition, we show, through a concrete example of the one-dimensional symmetry-protected topological cluster states, that the RBM representation may also be used as a tool to analytically compute the entanglement spectrum. Our results uncover the unparalleled power of artificial neural networks in representing quantum many-body states, which paves a novel way to bridge computer science based machine learning techniques to outstanding quantum condensed matter physics problems.Comment: 17 pages, 8 figures. Version published in Phys. Rev.

Artificial neural networks and machine learning have now reached a new era after several decades of improvement where applications are to explode in many fields of science, industry, and technology. Here, we use artificial neural networks to study an intriguing phenomenon in quantum physics-the topological phases of matter. We find that certain topological states, either symmetry-protected or with intrinsic topological order, can be represented with classical artificial neural networks. This is demonstrated by using three concrete spin systems, the one-dimensional (1D) symmetry-protected topological cluster state and the 2D and 3D toric code states with intrinsic topological orders. For all three cases we show rigorously that the topological ground states can be represented by shortrange neural networks in an exact and efficient fashion-the required number of hidden neurons is as small as the number of physical spins and the number of parameters scales only linearly with the system size. For the 2D toric-code model, we find that the proposed short-range neural networks can describe the excited states with abelain anyons and their nontrivial mutual statistics as well. In addition, by using reinforcement learning we show that neural networks are capable of finding the topological ground states of non-integrable Hamiltonians with strong interactions and studying their topological phase transitions. Our results demonstrate explicitly the exceptional power of neural networks in describing topological quantum states, and at the same time provide valuable guidance to machine learning of topological phases in generic lattice models.

Three-dimensional (3D) topological insulators in general need to be protected by certain kinds of symmetries other than the presumed U (1) charge conservation. A peculiar exception is the Hopf insulators which are 3D topological insulators characterized by an integer Hopf index. To demonstrate the existence and physical relevance of the Hopf insulators, we construct a class of tight-binding model Hamiltonians which realize all kinds of Hopf insulators with arbitrary integer Hopf index. These Hopf insulator phases have topologically protected surface states and we numerically demonstrate the robustness of these topologically protected states under general random perturbations without any symmetry other than the U (1) charge conservation that is implicit in all kinds of topological insulators.Topological phases of matter may be divided into two classes: the intrinsic ones and the symmetry protected ones 1 . Symmetry protected topological (SPT) phases are gapped quantum phases that are protected by symmetries of the Hamiltonian and cannot be smoothly connected to the trivial phases under perturbations that respect the same kind of symmetries. Intrinsic topological (IT) phases, on the other hand, do not require symmetry protection and are topologically stable under arbitrary perturbations. Unlike SPT phases, IT phases may have exotic excitations bearing fractional or even non-Abelian statistics in the bulk 2 . Fractional 3 quantum Hall states and spin liquids 4 belong to these IT phases. Remarkable examples of the SPT phases include the well known 2D and 3D topological insulators and superconductors protected by time reversal symmetry 5-7 , and the Haldane phase of the spin-1 chain protected by the SO(3) spin rotational symmetry 8 . For interacting bosonic systems with on-site symmetry G, distinct SPT phases can be systematically classified by group cohomology of G 1 , while for free fermions, the SPT phases can be systematically described by K-theory or homotopy group theory 9 , which leads to the well known periodic table for topological insulators and superconductors 10,11 .Most 3D topological insulators have to be protected by some other symmetries 10,11 , such as time reversal, particle hole or chrial symmetry, and the U (1) charge conservation symmetry 12 . A peculiar exception occurs when the Hamiltonian has just two effective bands. In this case, interesting topological phases, the so-called Hopf insulators 13 , may exist. These Hopf insulator phases have no symmetry other than the prerequisite U (1) charge conservation. To elucidate why this happens, let us consider a generic band Hamiltonian in 3D with m filled bands and n empty bands. Without symmetry constraint, the space of such Hamiltonians is topologically equivalent to the Grassmannian manifold G m,m+n and can be classified by the homotopy group of this Grassmannian 11 . Since the homotopy group π 3 (G m,m+n ) = {0} for all (m, n) = (1, 1), there exists no nontrivial topological phase in general. However, when m = n = 1, G 1,2 is topologically equival...

The marriage of the two fields may give birth to a new research frontier that could transform them both.

We study the many-body localization aspects of single-particle mobility edges in fermionic systems. We investigate incommensurate lattices and random disorder Anderson models. Many-body localization and quantum nonergodic properties are studied by comparing entanglement and thermal entropy, and by calculating the scaling of subsystem particle number fluctuations, respectively. We establish a nonergodic extended phase as a generic intermediate phase (between purely ergodic extended and nonergodic localized phases) for the many-body localization transition of non-interacting fermions where the entanglement entropy manifests a volume law (hence, 'extended'), but there are large fluctuations in the subsystem particle numbers (hence, 'nonergodic'). Based on the numerical results, we expect such an intermediate phase scenario may continue to hold even for the manybody localization in the presence of interactions as well. We find for many-body fermionic states in non-interacting one dimensional Aubry-André and three dimensional Anderson models that the entanglement entropy density and the normalized particle-number fluctuation have discontinuous jumps at the localization transition where the entanglement entropy is sub-thermal but obeys the "volume law". In the vicinity of the localization transition we find that both the entanglement entropy and the particle number fluctuations obey a single parameter scaling based on the diverging localization length. We argue using numerical and theoretical results that such a critical scaling behavior should persist for the interacting many-body localization problem with important observable consequences. Our work provides persuasive evidence in favor of there being two transitions in many-body systems with single-particle mobility edges, the first one indicating a transition from the purely localized nonergodic many-body localized phase to a nonergodic extended many-body metallic phase, and the second one being a transition eventually to the usual ergodic many-body extended phase.

We propose a systematic magnetic-flux-free approach to detect, manipulate and braid Majorana fermions in a semiconductor nanowire-based topological Josephson junction by utilizing the Majorana spin degree of freedom. We find an intrinsic π-phase difference between spin-triplet pairings enforced by the Majorana zeros modes (MZMs) at the two ends of a one-dimensional spinful topological superconductor. This π-phase is identified to be a spin-dependent superconducting phase, referred to as the spin-phase, which we show to be tunable by controlling spin-orbit coupling strength via electric gates. This electric controllable spin-phase not only affects the coupling energy between MZMs but also leads to a fractional Josephson effect in the absence of any applied magnetic flux, which enables the efficient topological qubit readout. We thus propose an all-electrically controlled superconductor-semiconductor hybrid circuit to manipulate MZMs and to detect their non-Abelian braiding statistics properties. Our work on spin properties of topological Josephson effects potentially opens up a new thrust for spintronic applications with Majorana-based semiconductor quantum circuits.

A quantum version of generative adversarial learning is experimentally demonstrated with a superconducting circuit.

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