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Quantum computing holds the unparalleled potentials to enhance, speed up or innovate machine learning. However, an unambiguous demonstration of quantum learning advantage has not been achieved so far. Here, we rigorously establish a noise-robust, unconditional quantum learning advantage in terms of expressivity, inference speed, and training efficiency, compared to commonly-used classical machine learning models. Our proof is information-theoretic and pinpoints the origin of this advantage: quantum entanglement can be used to reduce the communication required by non-local machine learning tasks. In particular, we design a fully classical task that can be solved with unit accuracy by a quantum model with a constant number of variational parameters using entanglement resources, whereas commonly-used classical models must scale at least linearly with the size of the task to achieve a larger-than-exponentially-small accuracy. We further show that the quantum model can be trained with constant time and a number of samples inversely proportional to the problem size. We prove that this advantage is robust against constant depolarization noise. We show through numerical simulations that even though the classical models can have improved performance as their sizes are increased, they would suffer from overfitting. The constant-versus-linear separation, bolstered by the overfitting problem, makes it possible to demonstrate the quantum advantage with relatively small system sizes. We demonstrate, through both numerical simulations and trapped-ion experiments on IonQ Aria, the desired quantum-classical learning separation. Our results provide a valuable guide for demonstrating quantum learning advantages in practical applications with current noisy intermediate-scale quantum devices., Comment: 7 pages, 2 figures + 13-page supplementary materials

Quantum learning models hold the potential to bring computational advantages over the classical realm. As powerful quantum servers become available on the cloud, ensuring the protection of clients' private data becomes crucial. By incorporating quantum homomorphic encryption schemes, we present a general framework that enables quantum delegated and federated learning with a computation-theoretical data privacy guarantee. We show that learning and inference under this framework feature substantially lower communication complexity compared with schemes based on blind quantum computing. In addition, in the proposed quantum federated learning scenario, there is less computational burden on local quantum devices from the client side, since the server can operate on encrypted quantum data without extracting any information. We further prove that certain quantum speedups in supervised learning carry over to private delegated learning scenarios employing quantum kernel methods. Our results provide a valuable guide toward privacy-guaranteed quantum learning on the cloud, which may benefit future studies and security-related applications., Comment: 5 pages, 1 figure, 1 table

Large language models (LLMs) have demonstrated significant potential in the development of intelligent applications and systems such as LLM-based agents and agent operating systems (AIOS). However, when these applications and systems interact with the underlying file system, the file system still remains the traditional paradigm: reliant on manual navigation through precise commands. This paradigm poses a bottleneck to the usability of these systems as users are required to navigate complex folder hierarchies and remember cryptic file names. To address this limitation, we propose an LLM-based semantic file system ( LSFS ) for prompt-driven file management. Unlike conventional approaches, LSFS incorporates LLMs to enable users or agents to interact with files through natural language prompts, facilitating semantic file management. At the macro-level, we develop a comprehensive API set to achieve semantic file management functionalities, such as semantic file retrieval, file update monitoring and summarization, and semantic file rollback). At the micro-level, we store files by constructing semantic indexes for them, design and implement syscalls of different semantic operations (e.g., CRUD, group by, join) powered by vector database. Our experiments show that LSFS offers significant improvements over traditional file systems in terms of user convenience, the diversity of supported functions, and the accuracy and efficiency of file operations. Additionally, with the integration of LLM, our system enables more intelligent file management tasks, such as content summarization and version comparison, further enhancing its capabilities.

Quantum computers may outperform classical computers on machine learning tasks. In recent years, a variety of quantum algorithms promising unparalleled potential to enhance, speed up, or innovate machine learning have been proposed. Yet, quantum learning systems, similar to their classical counterparts, may likewise suffer from the catastrophic forgetting problem, where training a model with new tasks would result in a dramatic performance drop for the previously learned ones. This problem is widely believed to be a crucial obstacle to achieving continual learning of multiple sequential tasks. Here, we report an experimental demonstration of quantum continual learning on a fully programmable superconducting processor. In particular, we sequentially train a quantum classifier with three tasks, two about identifying real-life images and the other on classifying quantum states, and demonstrate its catastrophic forgetting through experimentally observed rapid performance drops for prior tasks. To overcome this dilemma, we exploit the elastic weight consolidation strategy and show that the quantum classifier can incrementally learn and retain knowledge across the three distinct tasks, with an average prediction accuracy exceeding 92.3%. In addition, for sequential tasks involving quantum-engineered data, we demonstrate that the quantum classifier can achieve a better continual learning performance than a commonly used classical feedforward network with a comparable number of variational parameters. Our results establish a viable strategy for empowering quantum learning systems with desirable adaptability to multiple sequential tasks, marking an important primary experimental step towards the long-term goal of achieving quantum artificial general intelligence., Comment: 21 pages, 14 figures

Recent breakthroughs have ushered the quantum network into a new era, where quantum information can be stored, transferred, and processed across multiple nodes on a metropolitan scale. A key challenge in this new era is enhancing the capabilities of individual nodes, providing precise and robust control over multiple qubits and advanced functionality for scalable quantum networks. Here, we report on precise and complex control in a hybrid quantum node based on a diamond color center. We demonstrate hybrid coherent control by entangling three types of qubits: an electron spin as an interface qubit, a nuclear spin with long memory time, and a flying photonic qubit, with their qubit frequencies spanning three distinct regimes from the optical domain to the rf domain. By incorporating two additional memory qubits, we encode three memory qubits into a logical state using the three-qubit repetition code and entangle this logical qubit with a photonic qubit. Leveraging hybrid qubits and precise control, we repeatedly read out the error syndromes of memory qubits through the electron spin, serving as an auxiliary qubit, then apply a real-time feedback operation to correct bit-flip errors. We execute and verify active error correction for up to twelve rounds and demonstrate the improvement over the uncorrected counterpart. Our results demonstrate the feasibility of several key functionalities for next-generation quantum repeaters, paving the way towards full-fledged metropolitan-scale quantum networks for a wide range of practical applications., Comment: 8 pages, 3 figures

Quantum state tomography, a process that reconstructs a quantum state from measurements on an ensemble of identically prepared copies, plays a crucial role in benchmarking quantum devices. However, brute-force approaches to quantum state tomography would become impractical for large systems, as the required resources scale exponentially with the system size. Here, we explore a machine learning approach and report an experimental demonstration of reconstructing quantum states based on neural network generative models with an array of programmable superconducting transmon qubits. In particular, we experimentally prepare the Greenberger-Horne-Zeilinger states and random states up to five qubits and demonstrate that the machine learning approach can efficiently reconstruct these states with the number of required experimental samples scaling linearly with system size. Our results experimentally showcase the intriguing potential for exploiting machine learning techniques in validating and characterizing complex quantum devices, offering a valuable guide for the future development of quantum technologies.

Bell nonlocality is an intrinsic feature of quantum mechanics, which can be certified via the violation of Bell inequalities. It is therefore a fundamental question to certify Bell nonlocality from experimental data. Here, we present an optimization scheme to improve nonlocality certification by exploring flexible mappings between Bell inequalities and Hamiltonians corresponding to the Bell operators. We show that several Hamiltonian models can be mapped to new inequalities with improved classical bounds than the original one, enabling a more robust detection of nonlocality. From the other direction, we investigate the mapping from fixed Bell inequalities to Hamiltonians, aiming to maximize quantum violations while considering experimental imperfections. As a practical demonstration, we apply this method to an XXZ-like honeycomb-lattice model utilizing over 70 superconducting qubits. The successful application of this technique, as well as combining the two directions to form an optimization loop, may open new avenues for developing more practical and noise-resilient nonlocality certification techniques and enable broader experimental explorations., Comment: 11 pages, 5 figures, 1 table

Quantum nonlocality describes a stronger form of quantum correlation than that of entanglement. It refutes Einstein's belief of local realism and is among the most distinctive and enigmatic features of quantum mechanics. It is a crucial resource for achieving quantum advantages in a variety of practical applications, ranging from cryptography and certified random number generation via self-testing to machine learning. Nevertheless, the detection of nonlocality, especially in quantum many-body systems, is notoriously challenging. Here, we report an experimental certification of genuine multipartite Bell correlations, which signal nonlocality in quantum many-body systems, up to 24 qubits with a fully programmable superconducting quantum processor. In particular, we employ energy as a Bell correlation witness and variationally decrease the energy of a many-body system across a hierarchy of thresholds, below which an increasing Bell correlation depth can be certified from experimental data. As an illustrating example, we variationally prepare the low-energy state of a two-dimensional honeycomb model with 73 qubits and certify its Bell correlations by measuring an energy that surpasses the corresponding classical bound with up to 48 standard deviations. In addition, we variationally prepare a sequence of low-energy states and certify their genuine multipartite Bell correlations up to 24 qubits via energies measured efficiently by parity oscillation and multiple quantum coherence techniques. Our results establish a viable approach for preparing and certifying multipartite Bell correlations, which provide not only a finer benchmark beyond entanglement for quantum devices, but also a valuable guide towards exploiting multipartite Bell correlation in a wide spectrum of practical applications., Comment: 11 pages,6 figures + 14 pages, 6 figures

We demonstrate the existence of prethermal discrete time crystals whose sub-harmonic response is entirely localized to zero-dimensional corner modes. Within the exponentially long prethermal regime, we show that the robustness of these corner modes arises from two related, yet distinct mechanisms: the presence of a higher-order symmetry-protected topological phase in the effective Hamiltonian, or the emergence of a dynamical constraint that prevents the decay of the corner mode. While the first mechanism ensures the stability of the sub-harmonic response throughout the entirety of the prethermal regime, it is restricted to initial states in the ground state manifold of the effective Hamiltonian. By contrast, the second mechanism enables the observation of the prethermal time-crystalline order for arbitrary initial states, albeit with a time scale that is not only determined by the frequency of the drive, but also the relative energy scale across the system's sublattices. We characterize these two mechanisms by simulating the dynamics of a periodically driven two-dimensional spin model, and discuss natural extensions of our model to all other dimensions., Comment: 7+8 pages, 3+5 figures

We study quantum-classical separations between classical and quantum supervised learning models based on constant depth (i.e., shallow) circuits, in scenarios with and without noises. We construct a classification problem defined by a noiseless shallow quantum circuit and rigorously prove that any classical neural network with bounded connectivity requires logarithmic depth to output correctly with a larger-than-exponentially-small probability. This unconditional near-optimal quantum-classical separation originates from the quantum nonlocality property that distinguishes quantum circuits from their classical counterparts. We further derive the noise thresholds for demonstrating such a separation on near-term quantum devices under the depolarization noise model. We prove that this separation will persist if the noise strength is upper bounded by an inverse polynomial with respect to the system size, and vanish if the noise strength is greater than an inverse polylogarithmic function. In addition, for quantum devices with constant noise strength, we prove that no super-polynomial classical-quantum separation exists for any classification task defined by shallow Clifford circuits, independent of the structures of the circuits that specify the learning models., Comment: 14 pages, 3 figures

Non-Abelian topological orders offer an intriguing path towards fault-tolerant quantum computation, where information can be encoded and manipulated in a topologically protected manner immune to arbitrary local noises and perturbations. However, realizing non-Abelian topologically ordered states is notoriously challenging in both condensed matter and programmable quantum systems, and it was not until recently that signatures of non-Abelian statistics were observed through digital quantum simulation approaches. Despite these exciting progresses, none of them has demonstrated the appropriate type of topological orders and associated non-Abelian anyons whose braidings alone support universal quantum computation. Here, we report the realization of non-Abelian topologically ordered states of the Fibonacci string-net model and demonstrate braidings of Fibonacci anyons featuring universal computational power, with a superconducting quantum processor. We exploit efficient quantum circuits to prepare the desired states and verify their nontrivial topological nature 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 versatile digital approach to exploring exotic non-Abelian topological states and their associated braiding statistics with current noisy intermediate-scale quantum processors.

Locality-sensitive hashing (LSH) is a fundamental algorithmic technique widely employed in large-scale data processing applications, such as nearest-neighbor search, entity resolution, and clustering. However, its applicability in some real-world scenarios is limited due to the need for careful design of hashing functions that align with specific metrics. Existing LSH-based Entity Blocking solutions primarily rely on generic similarity metrics such as Jaccard similarity, whereas practical use cases often demand complex and customized similarity rules surpassing the capabilities of generic similarity metrics. Consequently, designing LSH functions for these customized similarity rules presents considerable challenges. In this research, we propose a neuralization approach to enhance locality-sensitive hashing by training deep neural networks to serve as hashing functions for complex metrics. We assess the effectiveness of this approach within the context of the entity resolution problem, which frequently involves the use of task-specific metrics in real-world applications. Specifically, we introduce NLSHBlock (Neural-LSH Block), a novel blocking methodology that leverages pre-trained language models, fine-tuned with a novel LSH-based loss function. Through extensive evaluations conducted on a diverse range of real-world datasets, we demonstrate the superiority of NLSHBlock over existing methods, exhibiting significant performance improvements. Furthermore, we showcase the efficacy of NLSHBlock in enhancing the performance of the entity matching phase, particularly within the semi-supervised setting.

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., Comment: 8 pages (main text), 16 pages (supplementary information)

Quantum tangent kernel methods provide an efficient approach to analyzing the performance of quantum machine learning models in the infinite-width limit, which is of crucial importance in designing appropriate circuit architectures for certain learning tasks. Recently, they have been adapted to describe the convergence rate of training errors in quantum neural networks in an analytical manner. Here, we study the connections between the trainability and expressibility of quantum tangent kernel models. In particular, for global loss functions, we rigorously prove that high expressibility of both the global and local quantum encodings can lead to exponential concentration of quantum tangent kernel values to zero. Whereas for local loss functions, such issue of exponential concentration persists owing to the high expressibility, but can be partially mitigated. We further carry out extensive numerical simulations to support our analytical theories. Our discoveries unveil a pivotal characteristic of quantum neural tangent kernels, offering valuable insights for the design of wide quantum variational circuit models in practical applications., Comment: 23 pages,6 figures

Quantum mechanics features a variety of distinct properties such as coherence and entanglement, which could be explored to showcase potential advantages over classical counterparts in information processing. In general, legitimate quantum operations must adhere to principles of quantum mechanics, particularly the requirements of complete positivity and trace preservation. Nonetheless, non-physical maps, especially Hermitian-preserving maps, play a crucial role in quantum information science. To date, there exists no effective method for implementing these non-physical maps with quantum devices. In this work, we introduce the Hermitian-preserving map exponentiation algorithm, which can effectively realize the action of an arbitrary Hermitian-preserving map by encoding its output into a quantum process. We analyze the performances of this algorithm, including its sample complexity and robustness, and prove its optimality in certain cases. When combined with algorithms such as the Hadamard test and quantum phase estimation, it allows for the extraction of information and generation of states from outputs of Hermitian-preserving maps, enabling various applications. Utilizing positive but not completely positive maps, this algorithm provides exponential advantages in entanglement detection and quantification compared to protocols based on single-copy operations. In addition, it facilitates the recovery of noiseless quantum states from multiple copies of noisy states by implementing the inverse map of the corresponding noise channel, offering an intriguing approach to handling quantum errors. Our findings present a pathway for systematically and efficiently implementing non-physical actions with quantum devices, thereby boosting the exploration of potential quantum advantages across a wide range of information processing tasks., Comment: 34 pages, 10 figures, comments are welcome

We discover a variety of exact quantum many-body scars in higher-spin kinetically constrained models, through the recently developed DMRG-S algorithm [Zhang et al., Phys. Rev. Lett. 131, 020402]. Specifically, for the higher-spin PXP model on arbitrary bipartite lattices of any spatial dimension, we find exact many-body scars that are equidistantly spaced in the energy spectrum and exhibit similar structures to the ground state of the Affleck-Kennedy-Lieb-Tasaki model. For the one-dimensional Fermi-Hubbard model with a tilted potential in a certain parameter regime, whose effective model is equivalent to a kinetically constrained spin model with four degrees of freedom on each site, we find several many-body scars at energy $E=0$ and $E=\pm \sqrt{2}$ that can be exactly represented as matrix product states with finite bond dimensions. Our results demonstrate that larger local degrees of freedom in the kinetically constrained models provide a much broader space for the emergence of quantum many-body scars and weak ergodicity breaking., Comment: 10 pages, 3 figures

Quantum many-body scars are non-thermal excited eigenstates of non-integrable Hamiltonians, which could support coherent revival dynamics from special initial states when scars form an equally spaced tower in the energy spectrum. For open quantum systems, engineering many-body scarred dynamics by a controlled coupling to the environment remains largely unexplored. In this paper, we provide a general framework to exactly embed quantum many-body scars into the decoherence-free subspaces of Lindblad master equations. The dissipative scarred dynamics manifest persistent periodic oscillations for generic initial states, and can be practically utilized to prepare scar states with potential quantum metrology applications.We construct the Liouvillian dissipators with the local projectors that annihilate the whole scar towers, and utilize the Hamiltonian part to rotate the undesired states out of the null space of dissipators. We demonstrate our protocol through several typical models hosting many-body scar towers, and propose an experimental scheme to observe the dissipative scarred dynamics based on digital quantum simulations and resetting ancilla qubits., Comment: 7+11 pages, 3+4 figures

Amidst the rapid advancements in experimental technology, noise-intermediate-scale quantum (NISQ) devices have become increasingly programmable, offering versatile opportunities to leverage quantum computational advantage. Here we explore the intricate dynamics of programmable NISQ devices for quantum reservoir computing. Using a genetic algorithm to configure the quantum reservoir dynamics, we systematically enhance the learning performance. Remarkably, a single configured quantum reservoir can simultaneously learn multiple tasks, including a synthetic oscillatory network of transcriptional regulators, chaotic motifs in gene regulatory networks, and the fractional-order Chua's circuit. Our configured quantum reservoir computing yields highly precise predictions for these learning tasks, outperforming classical reservoir computing. We also test the configured quantum reservoir computing in foreign exchange (FX) market applications and demonstrate its capability to capture the stochastic evolution of the exchange rates with significantly greater accuracy than classical reservoir computing approaches. Through comparison with classical reservoir computing, we highlight the unique role of quantum coherence in the quantum reservoir, which underpins its exceptional learning performance. Our findings suggest the exciting potential of configured quantum reservoir computing for exploiting the quantum computation power of NISQ devices in developing artificial general intelligence.

Tensor network methods are powerful tools for studying quantum many-body systems. In this paper, we investigate the emergent statistical properties of random high-dimensional tensor-network states and the trainability of variational tensor networks. We utilize diagrammatic methods and map our problems to the calculations of different partition functions for high-dimensional Ising models with special structures. To address the notorious difficulty in cracking these models, we develop a combinatorial method based on solving the ``puzzle of polyominoes". With this method, we are able to rigorously study statistical properties of the high dimensional random tensor networks. We prove: (a) the entanglement entropy approaches the maximal volume law, except for a small probability that is bounded by an inverse polynomial of the bond dimension; (b) the typicality occurs for the expectation value of a local observable when the bond dimension increases. In addition, we investigate the barren plateaus (i.e., exponentially vanishing gradients) for the high-dimensional tensor network models. We prove that such variational models suffer from barren plateaus for global loss functions, rendering their training processes inefficient in general. Whereas, for local loss functions, we prove that the gradient is independent of the system size (thus no barren plateau occurs), but decays exponentially with the distance between the region where the local observable acts and the site that hosts the derivative parameter. Our results uncover in a rigorous fashion some fundamental properties for variational high-dimensional tensor networks, which paves a way for their future theoretical studies and practical applications., Comment: 19 pages, 8 figures

The interplay between quantum physics and machine learning gives rise to the emergent frontier of quantum machine learning, where advanced quantum learning models may outperform their classical counterparts in solving certain challenging problems. However, quantum learning systems are vulnerable to adversarial attacks: adding tiny carefully-crafted perturbations on legitimate input samples can cause misclassifications. To address this issue, we propose a general scheme to protect quantum learning systems from adversarial attacks by randomly encoding the legitimate data samples through unitary or quantum error correction encoders. In particular, we rigorously prove that both global and local random unitary encoders lead to exponentially vanishing gradients (i.e. barren plateaus) for any variational quantum circuits that aim to add adversarial perturbations, independent of the input data and the inner structures of adversarial circuits and quantum classifiers. In addition, we prove a rigorous bound on the vulnerability of quantum classifiers under local unitary adversarial attacks. We show that random black-box quantum error correction encoders can protect quantum classifiers against local adversarial noises and their robustness increases as we concatenate error correction codes. To quantify the robustness enhancement, we adapt quantum differential privacy as a measure of the prediction stability for quantum classifiers. Our results establish versatile defense strategies for quantum classifiers against adversarial perturbations, which provide valuable guidance to enhance the reliability and security for both near-term and future quantum learning technologies.

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 the first experimental demonstration of training deep quantum neural networks via the backpropagation algorithm with a six-qubit programmable superconducting processor. 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 explicitly showcase the advantages of deep quantum neural networks, including quantum analogue of the backpropagation algorithm and less stringent coherence-time requirement for their constituting physical qubits, thus providing a valuable guide for quantum machine learning applications with both near-term and future quantum devices., Comment: 7 pages (main text) + 11 pages (Supplementary Information), 10 figures

Non-Abelian anyons are exotic quasiparticle excitations hosted by certain topological phases of matter. They break the fermion-boson dichotomy and obey non-Abelian braiding statistics: their interchanges yield unitary operations, rather than merely a phase factor, in a space spanned by topologically degenerate wavefunctions. They are the building blocks of topological quantum computing. However, experimental observation of non-Abelian anyons and their characterizing braiding statistics is notoriously challenging and has remained elusive hitherto, in spite of various theoretical proposals. Here, we report an experimental quantum digital simulation of projective non-Abelian anyons and their braiding statistics with up to 68 programmable superconducting qubits arranged on a two-dimensional lattice. By implementing the ground states of the toric-code model with twists through quantum circuits, we demonstrate that twists exchange electric and magnetic charges and behave as a particular type of non-Abelian anyons, i.e., the Ising anyons. In particular, we show experimentally that these twists follow the fusion rules and non-Abelian braiding statistics of the Ising type, and can be explored to encode topological logical qubits. Furthermore, we demonstrate how to implement both single- and two-qubit logic gates through applying a sequence of elementary Pauli gates on the underlying physical qubits. Our results demonstrate a versatile quantum digital approach for simulating non-Abelian anyons, offering a new lens into the study of such peculiar quasiparticles.

Quantum many-body scarred systems host nonthermal excited eigenstates immersed in a sea of thermal ones. In cases where exact expressions for these special eigenstates are not known, it is computationally demanding to distinguish them from their exponentially many thermal neighbors. We propose a matrix-product-state (MPS) algorithm, dubbed DMRG-S, to extract such states at system sizes far beyond the scope of exact diagonalization. Using this technique, we obtain scarred eigenstates in Rydberg-blockaded chains of up to 80 sites and perform a finite-size scaling study to address the lingering question of the stability for the N\'eel state revivals in the thermodynamic limit. Our method also provides a systematic way to obtain exact MPS representations for scarred eigenstates near the target energy without a priori knowledge. In particular, we find several new scarred eigenstates with exact MPS representations in kinetically constrained spin and clock models. The combination of numerical and analytical investigations in our work provides a new methodology for future studies of quantum many-body scars., Comment: 7+14 pages, 3+9 figures

Machine learning has achieved dramatic success over the past decade, with applications ranging from face recognition to natural language processing. Meanwhile, rapid progress has been made in the field of quantum computation including developing both powerful quantum algorithms and advanced quantum devices. The interplay between machine learning and quantum physics holds the intriguing potential for bringing practical applications to the modern society. Here, we focus on quantum neural networks in the form of parameterized quantum circuits. We will mainly discuss different structures and encoding strategies of quantum neural networks for supervised learning tasks, and benchmark their performance utilizing Yao.jl, a quantum simulation package written in Julia Language. The codes are efficient, aiming to provide convenience for beginners in scientific works such as developing powerful variational quantum learning models and assisting the corresponding experimental demonstrations., Comment: 30 pages, 5 figures, 6 tables

Quantum computing promises to enhance machine learning and artificial intelligence. Different quantum algorithms have been proposed to improve a wide spectrum of machine learning tasks. Yet, recent theoretical works show that, similar to traditional classifiers based on deep classical neural networks, quantum classifiers would suffer from the vulnerability problem: adding tiny carefully-crafted perturbations to the legitimate original data samples would facilitate incorrect predictions at a notably high confidence level. This will pose serious problems for future quantum machine learning applications in safety and security-critical scenarios. Here, we report the first experimental demonstration of quantum adversarial learning with programmable superconducting qubits. We train quantum classifiers, which are built upon variational quantum circuits consisting of ten transmon qubits featuring average lifetimes of 150 $\mu$s, and average fidelities of simultaneous single- and two-qubit gates above 99.94% and 99.4% respectively, with both real-life images (e.g., medical magnetic resonance imaging scans) and quantum data. We demonstrate that these well-trained classifiers (with testing accuracy up to 99%) can be practically deceived by small adversarial perturbations, whereas an adversarial training process would significantly enhance their robustness to such perturbations. Our results reveal experimentally a crucial vulnerability aspect of quantum learning systems under adversarial scenarios and demonstrate an effective defense strategy against adversarial attacks, which provide a valuable guide for quantum artificial intelligence applications with both near-term and future quantum devices., Comment: 26 pages, 17 figures, 8 algorithms

Machine learning can be substantially powered by a quantum computer owing to its huge Hilbert space and inherent quantum parallelism. In the pursuit of quantum advantages for machine learning with noisy intermediate-scale quantum devices, it was proposed that the learning model can be designed in an end-to-end fashion, i.e., the quantum ansatz is parameterized by directly manipulable control pulses without circuit design and compilation. Such gate-free models are hardware friendly and can fully exploit limited quantum resources. Here, we report the first experimental realization of quantum end-to-end machine learning on a superconducting processor. The trained model can achieve 98% recognition accuracy for two handwritten digits (via two qubits) and 89% for four digits (via three qubits) in the MNIST (Mixed National Institute of Standards and Technology) database. The experimental results exhibit the great potential of quantum end-to-end learning for resolving complex real-world tasks when more qubits are available., Comment: 6 pages, 4 figures and supplement

Quantum many-body scarred systems host special non-thermal eigenstates that support periodic revival dynamics and weakly break the ergodicity. Here, we study the quantum information scrambling dynamics in quantum many-body scarred systems, with a focus on the "PXP" model. We use the out-of-time-ordered correlator (OTOC) and Holevo information as measures of the information scrambling, and apply an efficient numerical method based on matrix product operators to compute them up to 41 spins. We find that both the OTOC and Holevo information exhibit a linear light cone and periodic oscillations inside the light cone for initial states within the scarred subspace, which is in sharp contrast to thermal or many-body localized systems. The periodic revivals of OTOCs and Holevo information signify unusual breakdown of quantum chaos and are not equivalent to the revival dynamics of state fidelity or local observables studied in the previous literature. To explain the formation of the linear light cone structure, we provide a perturbation-type calculation based on a phenomenological model. In addition, we demonstrate that the OTOC and Holevo information dynamics of the "PXP" model can be measured using the Rydberg-atom quantum simulators with current experimental technologies, and numerically identify the measurable signatures using experimental parameters., Comment: 14 pages, 11 figures

Capsule networks, which incorporate the paradigms of connectionism and symbolism, have brought fresh insights into artificial intelligence. The capsule, as the building block of capsule networks, is a group of neurons represented by a vector to encode different features of an entity. The information is extracted hierarchically through capsule layers via routing algorithms. Here, we introduce a quantum capsule network (dubbed QCapsNet) together with an efficient quantum dynamic routing algorithm. To benchmark the performance of the QCapsNet, we carry out extensive numerical simulations on the classification of handwritten digits and symmetry-protected topological phases, and show that the QCapsNet can achieve an enhanced accuracy and outperform conventional quantum classifiers evidently. We further unpack the output capsule state and find that a particular subspace may correspond to a human-understandable feature of the input data, which indicates the potential explainability of such networks. Our work reveals an intriguing prospect of quantum capsule networks in quantum machine learning, which may provide a valuable guide towards explainable quantum artificial intelligence., Comment: 7 pages (main text) + 8 pages (supplementary information), 8 figures

Non-Hermiticity has widespread applications in quantum physics. It brings about distinct topological phases without Hermitian counterparts, and gives rise to the fundamental challenge of phase classification from both theoretical and experimental aspects. Here we report the first experimental demonstration of unsupervised learning of non-Hermitian topological phases with the nitrogen-vacancy center platform. In particular, we implement the non-Hermitian twister model, which hosts peculiar knotted topological phases, with a solid-state quantum simulator consisting of an electron spin and a nearby $^{13}$C nuclear spin in a nitrogen-vacancy center in diamond. By tuning the microwave pulses, we efficiently generate a set of experimental data without phase labels. Furthermore, based on the diffusion map method, we cluster this set of experimental raw data into three different knotted phases in an unsupervised fashion without a priori knowledge of the system, which is in sharp contrast to the previously implemented supervised learning phases of matter. Our results showcase the intriguing potential for autonomous classification of exotic unknown topological phases with experimental raw data., Comment: 7+20 pages

Classification and identification of different phases and the transitions between them is a central task in condensed matter physics. Machine learning, which has achieved dramatic success in a wide range of applications, holds the promise to bring unprecedented perspectives for this challenging task. However, despite the exciting progress made along this direction, the reliability of machine-learning approaches likewise demands further investigation. Here, with the nitrogen-vacancy center platform, we report the first proof-of-principle experimental demonstration of adversarial examples in learning topological phases. We show that, after adding a tiny amount of carefully-designed perturbations, the experimentally observed adversarial examples can successfully deceive a splendid phase classifier, whose prediction accuracy is larger than $99.2\%$ on legitimate samples, with a notably high confidence. Our results explicitly showcase the crucial vulnerability aspect of applying machine learning techniques in classifying phases of matter, which provides an indispensable guide for future studies in this interdisciplinary field.

We propose a general and systematic strategy to compile arbitrary quantum channels without using ancillary qubits, based on proximal policy optimization -- a powerful deep reinforcement learning algorithm. We rigorously prove that, in sharp contrast to the case of compiling unitary gates, it is impossible to compile an arbitrary channel to arbitrary precision with any given finite elementary channel set, regardless of the length of the decomposition sequence. However, for a fixed accuracy $\epsilon$ one can construct a universal set with constant number of $\epsilon$-dependent elementary channels, such that an arbitrary quantum channel can be decomposed into a sequence of these elementary channels followed by a unitary gate, with the sequence length bounded by $O(\frac{1}{\epsilon}\log\frac{1}{\epsilon})$. Through a concrete example concerning topological compiling of Majorana fermions, we show that our proposed algorithm can conveniently and effectively reduce the use of expensive elementary gates through adding the weighted cost into the reward function of the proximal policy optimization., Comment: 14 pages, 4 figures

We report the observation of a symmetry-protected topological time crystal, which is implemented with an array of programmable superconducting qubits. Unlike the time crystals reported in previous experiments, where spontaneous breaking of the discrete time translational symmetry occurs for local observables throughout the whole system, the topological time crystal observed in our experiment breaks the time translational symmetry only at the boundaries and has trivial dynamics in the bulk. More concretely, we observe robust long-lived temporal correlations and sub-harmonic temporal response for the edge spins up to 40 driving cycles. We demonstrate that the sub-harmonic response is independent of whether the initial states are random product states or symmetry-protected topological states, and experimentally map out the phase boundary between the time crystalline and thermal phases. Our work paves the way to exploring peculiar non-equilibrium phases of matter emerged from the interplay between topology and localization as well as periodic driving, with current noisy intermediate-scale quantum processors., Comment: 8 pages main text, and 11 pages for supplementary materials