Jun Qi, Chao-Han Huck Yang, Pin-Yu Chen et al. · nvidia
The noisy intermediate-scale quantum (NISQ) devices enable the implementation of the variational quantum circuit (VQC) for quantum neural networks (QNN). Although the VQC-based QNN has succeeded in many machine learning tasks, the representation and generalization powers of VQC still require further investigation, particularly when the dimensionality of classical inputs is concerned. In this work, we first put forth an end-to-end quantum neural network, TTN-VQC, which consists of a quantum tensor network based on a tensor-train network (TTN) for dimensionality reduction and a VQC for functional regression. Then, we aim at the error performance analysis for the TTN-VQC in terms of representation and generalization powers. We also characterize the optimization properties of TTN-VQC by leveraging the Polyak-Lojasiewicz (PL) condition. Moreover, we conduct the experiments of functional regression on a handwritten digit classification dataset to justify our theoretical analysis.
Quantum computers are next-generation devices that hold promise to perform calculations beyond the reach of classical computers. A leading method towards achieving this goal is through quantum machine learning, especially quantum generative learning. Due to the intrinsic probabilistic nature of quantum mechanics, it is reasonable to postulate that quantum generative learning models (QGLMs) may surpass their classical counterparts. As such, QGLMs are receiving growing attention from the quantum physics and computer science communities, where various QGLMs that can be efficiently implemented on near-term quantum machines with potential computational advantages are proposed. In this paper, we review the current progress of QGLMs from the perspective of machine learning. Particularly, we interpret these QGLMs, covering quantum circuit born machines, quantum generative adversarial networks, quantum Boltzmann machines, and quantum autoencoders, as the quantum extension of classical generative learning models. In this context, we explore their intrinsic relation and their fundamental differences. We further summarize the potential applications of QGLMs in both conventional machine learning tasks and quantum physics. Last, we discuss the challenges and further research directions for QGLMs.
Variational quantum circuits have been widely employed in quantum simulation and quantum machine learning in recent years. However, quantum circuits with random structures have poor trainability due to the exponentially vanishing gradient with respect to the circuit depth and the qubit number. This result leads to a general standpoint that deep quantum circuits would not be feasible for practical tasks. In this work, we propose an initialization strategy with theoretical guarantees for the vanishing gradient problem in general deep quantum circuits. Specifically, we prove that under proper Gaussian initialized parameters, the norm of the gradient decays at most polynomially when the qubit number and the circuit depth increase. Our theoretical results hold for both the local and the global observable cases, where the latter was believed to have vanishing gradients even for very shallow circuits. Experimental results verify our theoretical findings in the quantum simulation and quantum chemistry.
Quantum neural networks (QNNs) have become an important tool for understanding the physical world, but their advantages and limitations are not fully understood. Some QNNs with specific encoding methods can be efficiently simulated by classical surrogates, while others with quantum memory may perform better than classical classifiers. Here we systematically investigate the problem-dependent power of quantum neural classifiers (QCs) on multi-class classification tasks. Through the analysis of expected risk, a measure that weighs the training loss and the generalization error of a classifier jointly, we identify two key findings: first, the training loss dominates the power rather than the generalization ability; second, QCs undergo a U-shaped risk curve, in contrast to the double-descent risk curve of deep neural classifiers. We also reveal the intrinsic connection between optimal QCs and the Helstrom bound and the equiangular tight frame. Using these findings, we propose a method that uses loss dynamics to probe whether a QC may be more effective than a classical classifier on a particular learning task. Numerical results demonstrate the effectiveness of our approach to explain the superiority of QCs over multilayer Perceptron on parity datasets and their limitations over convolutional neural networks on image datasets. Our work sheds light on the problem-dependent power of QNNs and offers a practical tool for evaluating their potential merit.
Factorization Machine (FM) is the most commonly used model to build a recommendation system since it can incorporate side information to improve performance. However, producing item suggestions for a given user with a trained FM is time-consuming. It requires a run-time of $O((N_m \log N_m)^2)$, where $N_m$ is the number of items in the dataset. To address this problem, we propose a quadratic unconstrained binary optimization (QUBO) scheme to combine with FM and apply quantum annealing (QA) computation. Compared to classical methods, this hybrid algorithm provides a faster than quadratic speedup in finding good user suggestions. We then demonstrate the aforementioned computational advantage on current NISQ hardware by experimenting with a real example on a D-Wave annealer.
Cross-platform verification, a critical undertaking in the realm of early-stage quantum computing, endeavors to characterize the similarity of two imperfect quantum devices executing identical algorithms, utilizing minimal measurements. While the random measurement approach has been instrumental in this context, the quasi-exponential computational demand with increasing qubit count hurdles its feasibility in large-qubit scenarios. To bridge this knowledge gap, here we introduce an innovative multimodal learning approach, recognizing that the formalism of data in this task embodies two distinct modalities: measurement outcomes and classical description of compiled circuits on explored quantum devices, both enriched with unique information. Building upon this insight, we devise a multimodal neural network to independently extract knowledge from these modalities, followed by a fusion operation to create a comprehensive data representation. The learned representation can effectively characterize the similarity between the explored quantum devices when executing new quantum algorithms not present in the training data. We evaluate our proposal on platforms featuring diverse noise models, encompassing system sizes up to 50 qubits. The achieved results demonstrate a three-orders-of-magnitude improvement in prediction accuracy compared to the random measurements and offer compelling evidence of the complementary roles played by each modality in cross-platform verification. These findings pave the way for harnessing the power of multimodal learning to overcome challenges in wider quantum system learning tasks.
Recent advancements in understanding the impulse response of the first arrival position (FAP) channel in molecular communication (MC) have illuminated its Shannon capacity. While Lee et al. shed light on FAP channel capacities with vertical drifts, the zero-drift scenario remains a conundrum, primarily due to the challenges associated with the heavy-tailed Cauchy distributions whose first and second moments do not exist, rendering traditional mutual information constraints ineffective. This paper unveils a novel characterization of the zero drift FAP channel capacity for both 2D and 3D. Interestingly, our results reveal a 3D FAP channel capacity that is double its 2D counterpart, hinting at a capacity increase with spatial dimension growth. Furthermore, our approach, which incorporates a modified logarithmic constraint and an output signal constraint, offers a simplified and more intuitive formula (similar to the well-known Gaussian case) for estimating FAP channel capacity.
The vast and complicated large-qubit state space forbids us to comprehensively capture the dynamics of modern quantum computers via classical simulations or quantum tomography. Recent progress in quantum learning theory prompts a crucial question: can linear properties of a large-qubit circuit with d tunable RZ gates and G-d Clifford gates be efficiently learned from measurement data generated by varying classical inputs? In this work, we prove that the sample complexity scaling linearly in $d$ is required to achieve a small prediction error, while the corresponding computational complexity may scale exponentially in d. To address this challenge, we propose a kernel-based method leveraging classical shadows and truncated trigonometric expansions, enabling a controllable trade-off between prediction accuracy and computational overhead. Our results advance two crucial realms in quantum computation: the exploration of quantum algorithms with practical utilities and learning-based quantum system certification. We conduct numerical simulations to validate our proposals across diverse scenarios, encompassing quantum information processing protocols, Hamiltonian simulation, and variational quantum algorithms up to 60 qubits.
Yu-Fang Chen, Kai-Min Chung, Min-Hsiu Hsieh et al.
We present a verifier of quantum programs called AutoQ 2.0. Quantum programs extend quantum circuits (the domain of AutoQ 1.0) by classical control flow constructs, which enable users to describe advanced quantum algorithms in a formal and precise manner. The extension is highly non-trivial, as we needed to tackle both theoretical challenges (such as the treatment of measurement, the normalization problem, and lifting techniques for verification of classical programs with loops to the quantum world), and engineering issues (such as extending the input format with a~support for specifying loop invariants). We have successfully used AutoQ 2.0 to verify two types of advanced quantum programs that cannot be expressed using only quantum circuits: the \emph{repeat-until-success} (RUS) algorithm and the weak-measurement-based version of Grover's search algorithm. AutoQ 2.0 can efficiently verify all our benchmarks: all RUS algorithms were verified instantly and, for the weak-measurement-based version of Grover's search, we were able to handle the case of 100 qubits in $\sim$20 minutes.
Quantum machine learning, which involves running machine learning algorithms on quantum devices, may be one of the most significant flagship applications for these devices. Unlike its classical counterparts, the role of data in quantum machine learning has not been fully understood. In this work, we quantify the performances of quantum machine learning in the landscape of quantum data. Provided that the encoding of quantum data is sufficiently random, the performance, we find that the training efficiency and generalization capabilities in quantum machine learning will be exponentially suppressed with the increase in the number of qubits, which we call "the curse of random quantum data". Our findings apply to both the quantum kernel method and the large-width limit of quantum neural networks. Conversely, we highlight that through meticulous design of quantum datasets, it is possible to avoid these curses, thereby achieving efficient convergence and robust generalization. Our conclusions are corroborated by extensive numerical simulations.
This tutorial intends to introduce readers with a background in AI to quantum machine learning (QML) -- a rapidly evolving field that seeks to leverage the power of quantum computers to reshape the landscape of machine learning. For self-consistency, this tutorial covers foundational principles, representative QML algorithms, their potential applications, and critical aspects such as trainability, generalization, and computational complexity. In addition, practical code demonstrations are provided in https://qml-tutorial.github.io/ to illustrate real-world implementations and facilitate hands-on learning. Together, these elements offer readers a comprehensive overview of the latest advancements in QML. By bridging the gap between classical machine learning and quantum computing, this tutorial serves as a valuable resource for those looking to engage with QML and explore the forefront of AI in the quantum era.
Variational Quantum Computing (VQC) faces fundamental scalability barriers, primarily due to the presence of barren plateaus and its sensitivity to quantum noise. To address these challenges, we introduce TensorHyper-VQC, a novel tensor-train (TT)-guided hypernetwork framework that significantly improves the robustness and scalability of VQC. Our framework fully delegates the generation of quantum circuit parameters to a classical TT network, effectively decoupling optimization from quantum hardware. This innovative parameterization mitigates gradient vanishing, enhances noise resilience through structured low-rank representations, and facilitates efficient gradient propagation. Grounded in Neural Tangent Kernel and statistical learning theory, our rigorous theoretical analyses establish strong guarantees on approximation capability, optimization stability, and generalization performance. Extensive empirical results across quantum dot classification, Max-Cut optimization, and molecular quantum simulation tasks demonstrate that TensorHyper-VQC consistently achieves superior performance and robust noise tolerance, including hardware-level validation on a 156-qubit IBM Heron processor. These results position TensorHyper-VQC as a scalable and noise-resilient framework for advancing practical quantum machine learning on near-term devices.
Variational quantum circuits (VQCs) hold promise for quantum machine learning but face challenges in expressivity, trainability, and noise resilience. We propose VQC-MLPNet, a hybrid architecture where a VQC generates the first-layer weights of a classical multilayer perceptron during training, while inference is performed entirely classically. This design preserves scalability, reduces quantum resource demands, and enables practical deployment. We provide a theoretical analysis based on statistical learning and neural tangent kernel theory, establishing explicit risk bounds and demonstrating improved expressivity and trainability compared to purely quantum or existing hybrid approaches. These theoretical insights demonstrate exponential improvements in representation capacity relative to quantum circuit depth and the number of qubits, providing clear computational advantages over standalone quantum circuits and existing hybrid quantum architectures. Empirical results on diverse datasets, including quantum-dot classification and genomic sequence analysis, show that VQC-MLPNet achieves high accuracy and robustness under realistic noise models, outperforming classical and quantum baselines while using significantly fewer trainable parameters.
Data encoding remains a fundamental bottleneck in quantum machine learning, where amplitude encoding of high-dimensional classical vectors into quantum states incurs exponential cost. In this work, we propose a pre-trained tensor-train (TT) encoding network (Pre-TT-Encoder) that significantly reduces the computational complexity of amplitude encoding while preserving essential data structure. The Pre-TT-Encoder exploits low-rank TT decompositions learned from classical data, enabling polynomial-time state preparation in the number of qubits and TT-ranks. We provide a theoretical analysis of the encoding complexity and establish fidelity bounds that quantify the trade-off between TT-rank and approximation error. Empirical evaluations on classical (MNIST) and quantum-native (semiconductor quantum dot) datasets demonstrate that our approach achieves substantial gains in encoding efficiency over direct amplitude encoding and PCA-based dimensionality reduction, while maintaining competitive performance in downstream variational quantum circuit classification tasks. The proposed method highlights the role of tensor networks as scalable intermediaries between classical data and quantum processors.
Training quantum neural networks (QNNs) using gradient-based or gradient-free classical optimisation approaches is severely impacted by the presence of barren plateaus in the cost landscapes. In this paper, we devise a framework for leveraging quantum optimisation algorithms to find optimal parameters of QNNs for certain tasks. To achieve this, we coherently encode the cost function of QNNs onto relative phases of a superposition state in the Hilbert space of the network parameters. The parameters are tuned with an iterative quantum optimisation structure using adaptively selected Hamiltonians. The quantum mechanism of this framework exploits hidden structure in the QNN optimisation problem and hence is expected to provide beyond-Grover speed up, mitigating the barren plateau issue.
Variational quantum algorithms (VQAs) are expected to be a path to quantum advantages on noisy intermediate-scale quantum devices. However, both empirical and theoretical results exhibit that the deployed ansatz heavily affects the performance of VQAs such that an ansatz with a larger number of quantum gates enables a stronger expressivity, while the accumulated noise may render a poor trainability. To maximally improve the robustness and trainability of VQAs, here we devise a resource and runtime efficient scheme termed quantum architecture search (QAS). In particular, given a learning task, QAS automatically seeks a near-optimal ansatz (i.e., circuit architecture) to balance benefits and side-effects brought by adding more noisy quantum gates to achieve a good performance. We implement QAS on both the numerical simulator and real quantum hardware, via the IBM cloud, to accomplish data classification and quantum chemistry tasks. In the problems studied, numerical and experimental results show that QAS can not only alleviate the influence of quantum noise and barren plateaus, but also outperforms VQAs with pre-selected ansatze.
The eligibility of various advanced quantum algorithms will be questioned if they can not guarantee privacy. To fill this knowledge gap, here we devise an efficient quantum differentially private (QDP) Lasso estimator to solve sparse regression tasks. Concretely, given $N$ $d$-dimensional data points with $N\ll d$, we first prove that the optimal classical and quantum non-private Lasso requires $Ω(N+d)$ and $Ω(\sqrt{N}+\sqrt{d})$ runtime, respectively. We next prove that the runtime cost of QDP Lasso is \textit{dimension independent}, i.e., $O(N^{5/2})$, which implies that the QDP Lasso can be faster than both the optimal classical and quantum non-private Lasso. Last, we exhibit that the QDP Lasso attains a near-optimal utility bound $\tilde{O}(N^{-2/3})$ with privacy guarantees and discuss the chance to realize it on near-term quantum chips with advantages.
Noise in quantum information processing is often viewed as a disruptive and difficult-to-avoid feature, especially in near-term quantum technologies. However, noise has often played beneficial roles, from enhancing weak signals in stochastic resonance to protecting the privacy of data in differential privacy. It is then natural to ask, can we harness the power of quantum noise that is beneficial to quantum computing? An important current direction for quantum computing is its application to machine learning, such as classification problems. One outstanding problem in machine learning for classification is its sensitivity to adversarial examples. These are small, undetectable perturbations from the original data where the perturbed data is completely misclassified in otherwise extremely accurate classifiers. They can also be considered as `worst-case' perturbations by unknown noise sources. We show that by taking advantage of depolarisation noise in quantum circuits for classification, a robustness bound against adversaries can be derived where the robustness improves with increasing noise. This robustness property is intimately connected with an important security concept called differential privacy which can be extended to quantum differential privacy. For the protection of quantum data, this is the first quantum protocol that can be used against the most general adversaries. Furthermore, we show how the robustness in the classical case can be sensitive to the details of the classification model, but in the quantum case the details of classification model are absent, thus also providing a potential quantum advantage for classical data that is independent of quantum speedups. This opens the opportunity to explore other ways in which quantum noise can be used in our favour, as well as identifying other ways quantum algorithms can be helpful that is independent of quantum speedups.
In this paper, we present a new framework to obtain tail inequalities for sums of random matrices. Compared with existing works, our tail inequalities have the following characteristics: 1) high feasibility--they can be used to study the tail behavior of various matrix functions, e.g., arbitrary matrix norms, the absolute value of the sum of the sum of the $j$ largest singular values (resp. eigenvalues) of complex matrices (resp. Hermitian matrices); and 2) independence of matrix dimension --- they do not have the matrix-dimension term as a product factor, and thus are suitable to the scenario of high-dimensional or infinite-dimensional random matrices. The price we pay to obtain these advantages is that the convergence rate of the resulting inequalities will become slow when the number of summand random matrices is large. We also develop the tail inequalities for matrix random series and matrix martingale difference sequence. We also demonstrate usefulness of our tail bounds in several fields. In compressed sensing, we employ the resulted tail inequalities to achieve a proof of the restricted isometry property when the measurement matrix is the sum of random matrices without any assumption on the distributions of matrix entries. In probability theory, we derive a new upper bound to the supreme of stochastic processes. In machine learning, we prove new expectation bounds of sums of random matrices matrix and obtain matrix approximation schemes via random sampling. In quantum information, we show a new analysis relating to the fractional cover number of quantum hypergraphs. In theoretical computer science, we obtain randomness-efficient samplers using matrix expander graphs that can be efficiently implemented in time without dependence on matrix dimensions.
We present an efficient quantum algorithm aiming to find the negative curvature direction for escaping the saddle point, which is the critical subroutine for many second-order non-convex optimization algorithms. We prove that our algorithm could produce the target state corresponding to the negative curvature direction with query complexity O(polylog(d) /ε), where d is the dimension of the optimization function. The quantum negative curvature finding algorithm is exponentially faster than any known classical method which takes time at least O(d /\sqrtε). Moreover, we propose an efficient quantum algorithm to achieve the classical read-out of the target state. Our classical read-out algorithm runs exponentially faster on the degree of d than existing counterparts.
A wide range of fundamental machine learning tasks that are addressed by the maximum a posteriori estimation can be reduced to a general minimum conical hull problem. The best-known solution to tackle general minimum conical hull problems is the divide-and-conquer anchoring learning scheme (DCA), whose runtime complexity is polynomial in size. However, big data is pushing these polynomial algorithms to their performance limits. In this paper, we propose a sublinear classical algorithm to tackle general minimum conical hull problems when the input has stored in a sample-based low-overhead data structure. The algorithm's runtime complexity is polynomial in the rank and polylogarithmic in size. The proposed algorithm achieves the exponential speedup over DCA and, therefore, provides advantages for high dimensional problems.
The exploration of quantum algorithms that possess quantum advantages is a central topic in quantum computation and quantum information processing. One potential candidate in this area is quantum generative adversarial learning (QuGAL), which conceptually has exponential advantages over classical adversarial networks. However, the corresponding learning algorithm remains obscured. In this paper, we propose the first quantum generative adversarial learning algorithm-- the quantum multiplicative matrix weight algorithm (QMMW)-- which enables the efficient processing of fundamental tasks. The computational complexity of QMMW is polynomially proportional to the number of training rounds and logarithmically proportional to the input size. The core concept of the proposed algorithm combines QuGAL with online learning. We exploit the implementation of QuGAL with parameterized quantum circuits, and numerical experiments for the task of entanglement test for pure state are provided to support our claims.
In classical machine learning, a set of weak classifiers can be adaptively combined to form a strong classifier for improving the overall performance, a technique called adaptive boosting (or AdaBoost). However, constructing the strong classifier for a large data set is typically resource consuming. Here we propose a quantum extension of AdaBoost, demonstrating a quantum algorithm that can output the optimal strong classifier with a quadratic speedup in the number of queries of the weak classifiers. Our results also include a generalization of the standard AdaBoost to the cases where the output of each classifier may be probabilistic even for the same input. We prove that the update rules and the query complexity of the non-deterministic classifiers are the same as those of deterministic classifiers, which may be of independent interest to the classical machine-learning community. Furthermore, the AdaBoost algorithm can also be applied to data encoded in the form of quantum states; we show how the training set can be simplified by using the tools of t-design. Our approach describes a model of quantum machine learning where quantum speedup is achieved in finding the optimal classifier, which can then be applied for classical machine-learning applications.
The vicinal risk minimization (VRM) principle, first proposed by \citet{vapnik1999nature}, is an empirical risk minimization (ERM) variant that replaces Dirac masses with vicinal functions. Although there is strong numerical evidence showing that VRM outperforms ERM if appropriate vicinal functions are chosen, a comprehensive theoretical understanding of VRM is still lacking. In this paper, we study the generalization bounds for VRM. Our results support Vapnik's original arguments and additionally provide deeper insights into VRM. First, we prove that the complexity of function classes convolving with vicinal functions can be controlled by that of the original function classes under the assumption that the function class is composed of Lipschitz-continuous functions. Then, the resulting generalization bounds for VRM suggest that the generalization performance of VRM is also effected by the choice of vicinity function and the quality of function classes. These findings can be used to examine whether the choice of vicinal function is appropriate for the VRM-based learning setting. Finally, we provide a theoretical explanation for existing VRM models, e.g., uniform distribution-based models, Gaussian distribution-based models, and mixup models.
The hybrid quantum-classical learning scheme provides a prominent way to achieve quantum advantages on near-term quantum devices. A concrete example towards this goal is the quantum neural network (QNN), which has been developed to accomplish various supervised learning tasks such as classification and regression. However, there are two central issues that remain obscure when QNN is exploited to accomplish classification tasks. First, a quantum classifier that can well balance the computational cost such as the number of measurements and the learning performance is unexplored. Second, it is unclear whether quantum classifiers can be applied to solve certain problems that outperform their classical counterparts. Here we devise a Grover-search based quantum learning scheme (GBLS) to address the above two issues. Notably, most existing QNN-based quantum classifiers can be seamlessly embedded into the proposed scheme. The key insight behind our proposal is reformulating the classification tasks as the search problem. Numerical simulations exhibit that GBLS can achieve comparable performance with other quantum classifiers under various noise settings, while the required number of measurements is dramatically reduced. We further demonstrate a potential quantum advantage of GBLS over classical classifiers in the measure of query complexity. Our work provides guidance to develop advanced quantum classifiers on near-term quantum devices and opens up an avenue to explore potential quantum advantages in various classification tasks.
Quantum machine learning has received significant attention in recent years, and promising progress has been made in the development of quantum algorithms to speed up traditional machine learning tasks. In this work, however, we focus on investigating the information-theoretic upper bounds of sample complexity - how many training samples are sufficient to predict the future behaviour of an unknown target function. This kind of problem is, arguably, one of the most fundamental problems in statistical learning theory and the bounds for practical settings can be completely characterised by a simple measure of complexity. Our main result in the paper is that, for learning an unknown quantum measurement, the upper bound, given by the fat-shattering dimension, is linearly proportional to the dimension of the underlying Hilbert space. Learning an unknown quantum state becomes a dual problem to ours, and as a byproduct, we can recover Aaronson's famous result [Proc. R. Soc. A 463:3089-3144 (2007)] solely using a classical machine learning technique. In addition, other famous complexity measures like covering numbers and Rademacher complexities are derived explicitly. We are able to connect measures of sample complexity with various areas in quantum information science, e.g. quantum state/measurement tomography, quantum state discrimination and quantum random access codes, which may be of independent interest. Lastly, with the assistance of general Bloch-sphere representation, we show that learning quantum measurements/states can be mathematically formulated as a neural network. Consequently, classical ML algorithms can be applied to efficiently accomplish the two quantum learning tasks.