No-Free-Lunch Theories for Tensor-Network Machine Learning Models

Tensor network machine learning models have shown remarkable versatility in tackling complex data-driven tasks, ranging from quantum many-body problems to classical pattern recognitions. Despite their promising performance, a comprehensive understanding of the underlying assumptions and limitations of these models is still lacking. In this work, we focus on the rigorous formulation of their no-free-lunch theorem -- essential yet notoriously challenging to formalize for specific tensor network machine learning models. In particular, we rigorously analyze the generalization risks of learning target output functions from input data encoded in tensor network states. We first prove a no-free-lunch theorem for machine learning models based on matrix product states, i.e., the one-dimensional tensor network states. Furthermore, we circumvent the challenging issue of calculating the partition function for two-dimensional Ising model, and prove the no-free-lunch theorem for the case of two-dimensional projected entangled-pair state, by introducing the combinatorial method associated to the "puzzle of polyominoes". Our findings reveal the intrinsic limitations of tensor network-based learning models in a rigorous fashion, and open up an avenue for future analytical exploration of both the strengths and limitations of quantum-inspired machine learning frameworks.

Entanglement-induced provable and robust quantum learning advantages

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.

Quantum delegated and federated learning via quantum homomorphic encryption

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.

Probing many-body Bell correlation depth with superconducting qubits

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.

Quantum-Classical Separations in Shallow-Circuit-Based Learning with and without Noises

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.

Expressibility-induced Concentration of Quantum Neural Tangent Kernels

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.

Enhancing Quantum Adversarial Robustness by Randomized Encodings

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.

Quantum Neural Network Classifiers: A Tutorial

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.

Experimental quantum adversarial learning with programmable superconducting qubits

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.

Quantum Capsule Networks

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 a quantum dynamic routing algorithm. Our model enjoys an exponential speedup in the dynamic routing process and exhibits an enhanced representation power. 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 the state-of-the-art accuracy and outperforms 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.