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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 has recently become highly desirable in the applications of machine-learning techniques to quantum many-body physics. In this paper, we explore the data structures that encode the physical features in the network states by studying the quantum entanglement properties, with a focus on the restricted-Boltzmann-machine (RBM) architecture. We prove that the entanglement entropy 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 quantum states with massive entanglement. Strikingly, the neural-network representation for these states is remarkably efficient, in the sense that the number of nonzero parameters scales only linearly with the system size. We further examine the entanglement properties of generic RBM states by randomly sampling the weight parameters of the RBM. We find that their averaged entanglement entropy obeys volume-law scaling, and the 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 a 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 regardless of how much entanglement they possess, which paves a novel way to bridge computer-science-based machine-learning techniques to outstanding quantum condensed-matter physics problems.

Machine learning has achieved dramatic success in a broad spectrum of applications. Its interplay with quantum physics may lead to unprecedented perspectives for both fundamental research and commercial applications, giving rise to an emergent research frontier of quantum machine learning. Along this line, quantum classifiers, which are quantum devices that aim to solve classification problems in machine learning, have attracted tremendous attention recently. In this review, we give a relatively comprehensive overview for the studies of quantum classifiers, with a focus on recent advances. First, we will review a number of quantum classification algorithms, including quantum support vector machines, quantum kernel methods, quantum decision tree classifiers, quantum nearest neighbor algorithms, and quantum annealing based classifiers. Then, we move on to introduce the variational quantum classifiers, which are essentially variational quantum circuits for classifications. We will review different architectures for constructing variational quantum classifiers and introduce the barren plateau problem, where the training of quantum classifiers might be hindered by the exponentially vanishing gradient. In addition, the vulnerability aspect of quantum classifiers in the setting of adversarial learning and the recent experimental progress on different quantum classifiers will also be discussed.

Private distributed learning studies the problem of how multiple distributed entities collaboratively train a shared deep network with their private data unrevealed. With the security provided by the protocols of blind quantum computation, the cooperation between quantum physics and machine learning may lead to unparalleled prospect for solving private distributed learning tasks. In this paper, we introduce a quantum protocol for distributed learning that is able to utilize the computational power of the remote quantum servers while keeping the private data safe. For concreteness, we first introduce a protocol for private single-party delegated training of variational quantum classifiers based on blind quantum computing and then extend this protocol to multiparty private distributed learning incorporated with differential privacy. We carry out extensive numerical simulations with different real-life datasets and encoding strategies to benchmark the effectiveness of our protocol. We find that our protocol is robust to experimental imperfections and is secure under the gradient attack after the incorporation of differential privacy. Our results show the potential for handling computationally expensive distributed learning tasks with privacy guarantees, thus providing a valuable guide for exploring quantum advantages from the security perspective in the field of machine learning with real-life applications.

Quantum many-body systems away from equilibrium host a rich variety of exotic phenomena that are forbidden by equilibrium thermodynamics. A prominent example is that of discrete time crystals 1 – 8 , in which time-translational symmetry is spontaneously broken in periodically driven systems. Pioneering experiments have observed signatures of time crystalline phases with trapped ions 9 , 10 , solid-state spin systems 11 – 15 , ultracold atoms 16 , 17 and superconducting qubits 18 – 20 . Here we report the observation of a distinct type of non-equilibrium state of matter, Floquet symmetry-protected topological phases, which are implemented through digital quantum simulation with an array of programmable superconducting qubits. We observe robust long-lived temporal correlations and subharmonic temporal response for the edge spins over up to 40 driving cycles using a circuit of depth exceeding 240 and acting on 26 qubits. We demonstrate that the subharmonic response is independent of the initial state, and experimentally map out a phase boundary between the Floquet symmetry-protected topological and thermal phases. Our results establish a versatile digital simulation approach to exploring exotic non-equilibrium phases of matter with current noisy intermediate-scale quantum processors 21 . Signatures of non-equilibrium Floquet SPT phases with a programmable superconducting quantum processor are observed in which the discrete time translational symmetry only breaks at the boundaries and not in the bulk.

Deep learning and quantum computing have achieved dramatic progresses in recent years. The interplay between these two fast-growing fields gives rise to a new research frontier of quantum machine learning. In this work, we report an experimental demonstration of training deep quantum neural networks via the backpropagation algorithm with a six-qubit programmable superconducting processor. We experimentally perform the forward process of the backpropagation algorithm and classically simulate the backward process. In particular, we show that three-layer deep quantum neural networks can be trained efficiently to learn two-qubit quantum channels with a mean fidelity up to 96.0% and the ground state energy of molecular hydrogen with an accuracy up to 93.3% compared to the theoretical value. In addition, six-layer deep quantum neural networks can be trained in a similar fashion to achieve a mean fidelity up to 94.8% for learning single-qubit quantum channels. Our experimental results indicate that the number of coherent qubits required to maintain does not scale with the depth of the deep quantum neural network, thus providing a valuable guide for quantum machine learning applications with both near-term and future quantum devices. Experimental studies about the trainability and generalization capacities of quantum neural networks are highly in need. Here, the authors implement a previously proposed parametrization and training scheme using a 6-qubit superconducting quantum processor.

The glucose transporter GLUT1 catalyses facilitative diffusion of glucose into erythrocytes and is responsible for glucose supply to the brain and other organs. Dysfunctional mutations may lead to GLUT1 deficiency syndrome, whereas overexpression of GLUT1 is a prognostic indicator for cancer. Despite decades of investigation, the structure of GLUT1 remains unknown. Here we report the crystal structure of human GLUT1 at 3.2 Å resolution. The full-length protein, which has a canonical major facilitator superfamily fold, is captured in an inward-open conformation. This structure allows accurate mapping and potential mechanistic interpretation of disease-associated mutations in GLUT1. Structure-based analysis of these mutations provides an insight into the alternating access mechanism of GLUT1 and other members of the sugar porter subfamily. Structural comparison of the uniporter GLUT1 with its bacterial homologue XylE, a proton-coupled xylose symporter, allows examination of the transport mechanisms of both passive facilitators and active transporters. The structure of human GLUT1 in an inward-open conformation is reported; access to the structure of the human protein, instead of just a bacterial homologue, made it possible to map (inactivating) mutations associated with GLUT1 deficiency syndrome onto the structure. Human GLUT1 glucose transporter structure GLUT1 is a membrane protein that is responsible for the uptake of glucose into erythrocytes and other cells. The structure of a proton-coupled xylose symporter that is a bacterial homologue of GLUT1 has been reported previously and here Nieng Yan and colleagues report the structure of human GLUT1 in an inward-open conformation. Having access to the structure of the human protein, the authors were able to map inactivating mutations associated with GLUT1 deficiency syndrome — also known as De Vivo syndrome — onto their structure. Because elevated expression levels of GLUT1 have been observed in several cancer types, access to this structure may facilitate the development of new anticancer agents.

Andradite-rich garnet is a common U-bearing mineral in a variety of alkalic igneous rocks and skarn deposits, but has been largely neglected as a U–Pb chronometer. In situ laser ablation-inductively coupled plasma mass spectrometry U–Pb dates of andradite-rich garnet from a syenite pluton and two iron skarn deposits in the North China craton demonstrate the suitability and reliability of the mineral in accurately dating magmatic and hydrothermal processes. Two hydrothermal garnets from the iron skarn deposits have homogenous cores and zoned rims (Ad 86 Gr 11 to Ad 98 Gr 1 ) with 22–118 ppm U, whereas one magmatic garnet from the syenite is texturally and compositionally homogenous (Ad 70 Gr 22 to Ad 77 Gr 14 ) and has 0.1–20 ppm U. All three garnets have flat time-resolved signals obtained from depth profile analyses for U, indicating structurally bound U. Uranium is correlated with REE in both magmatic and hydrothermal garnets, indicating that the incorporation of U into the garnet is largely controlled by substitution mechanisms. Two hydrothermal garnets yielded U–Pb dates of 129 ± 2 (2 σ ; MSWD = 0.7) and 130 ± 1 Ma (2 σ ; MSWD = 0.5), indistinguishable from zircon U–Pb dates of 131 ± 1 and 129 ± 1 Ma for their respective ore-related intrusions. The magmatic garnet has a U–Pb age of 389 ± 3 Ma (2 σ ; MSWD = 0.6), consistent with a U–Pb zircon date of 388 ± 2 Ma for the syenite. The consistency between the garnet and zircon U–Pb dates confirms the reliability and accuracy of garnet U–Pb dating. Given the occurrence of andradite-rich garnet in alkaline and ultramafic magmatic rocks and hydrothermal ore deposits, our results highlight the potential utilization of garnet as a powerful U–Pb geochronometer for dating magmatism and skarn-related mineralization.

Quantum many-body systems with a non-Abelian topological order can host anyonic quasiparticles. It has been proposed that anyons could be used to encode and manipulate information in a topologically protected manner that is immune to local noise, with quantum gates performed by braiding and fusing anyons. Unfortunately, realizing non-Abelian topologically ordered states is challenging, and it was not until recently that the signatures of non-Abelian statistics were observed through digital quantum simulation approaches. However, not all forms of topological order can be used to realize universal quantum computation. Here we use a superconducting quantum processor to simulate non-Abelian topologically ordered states of the Fibonacci string-net model and demonstrate braidings of Fibonacci anyons featuring universal computational power. We demonstrate the non-trivial topological nature of the quantum states by measuring the topological entanglement entropy. In addition, we create two pairs of Fibonacci anyons and demonstrate their fusion rule and non-Abelian braiding statistics by applying unitary gates on the underlying physical qubits. Our results establish a digital approach to explore non-Abelian topological states and their associated braiding statistics with current noisy intermediate-scale quantum processors. Superconducting qubits have been used to realize a quantum many-body state that is capable of universal topological quantum computation.

Topologically ordered phases of matter elude Landau’s symmetry-breaking theory, featuring a variety of intriguing properties such as long-range entanglement and intrinsic robustness against local perturbations. Their extension to periodically driven systems gives rise to exotic new phenomena that are forbidden in thermal equilibrium. Here, we report the observation of signatures of such a phenomenon—a prethermal topologically ordered time crystal—with programmable superconducting qubits arranged on a square lattice. By periodically driving the superconducting qubits with a surface code Hamiltonian, we observe discrete time-translation symmetry breaking dynamics that is only manifested in the subharmonic temporal response of nonlocal logical operators. We further connect the observed dynamics to the underlying topological order by measuring a nonzero topological entanglement entropy and studying its subsequent dynamics. Our results demonstrate the potential to explore exotic topologically ordered nonequilibrium phases of matter with noisy intermediate-scale quantum processors. Recently, there have been proposals to extend the concept of time crystals to topological order. Here the authors observe a prethermal topologically ordered time crystal on a superconducting quantum processor, where discrete time-translation symmetry breaking manifests for nonlocal rather than local observables.

Characterization of qubit couplings in many-body quantum systems is essential for benchmarking quantum computation and simulation. We propose a tomographic measurement scheme to determine all the coupling terms in a general many-body Hamiltonian with arbitrary long-range interactions, provided the energy density of the Hamiltonian remains finite. Different from quantum process tomography, our scheme is fully scalable with the number of qubits as the required rounds of measurements increase only linearly with the number of coupling terms in the Hamiltonian. The scheme makes use of synchronized dynamical decoupling pulses to simplify the many-body dynamics so that the unknown parameters in the Hamiltonian can be retrieved one by one. We simulate the performance of the scheme under the influence of various pulse errors and show that it is robust to typical noise and experimental imperfections.