Showing 1–50 of 52 results for author: Tao, D

Quantum circuit optimization using deep reinforcement learning: Applications across multiple gate sets

Authors: Khoa Dang Tao, Sumin Jin, Muhammad Raza, Changhyoup Lee

Abstract: The practical implementation of quantum algorithms on noisy intermediate-scale quantum devices encounters operational limitations due to decoherence and other sources of noise inherent in real hardware. To mitigate these errors while preserving the original functionality of the algorithm, shorter quantum circuits are therefore preferred. This motivates the development of effective quantum circuit… ▽ More The practical implementation of quantum algorithms on noisy intermediate-scale quantum devices encounters operational limitations due to decoherence and other sources of noise inherent in real hardware. To mitigate these errors while preserving the original functionality of the algorithm, shorter quantum circuits are therefore preferred. This motivates the development of effective quantum circuit optimization algorithms. Learning-based approaches have emerged as a leading candidate, yet existing autonomous agents remain inefficient, spending most of their training capacity rediscovering elementary reductions that deterministic rule-based methods already handle reliably. To address this challenge, we propose a reinforcement learning framework that embeds a deterministic Commutation-and-Reduction (CR) algorithm directly into the training environment. After every agent action, the CR algorithm automatically resolves elementary commutations and cancellations, enabling the agent to focus its learning capacity on the non-trivial optimizations where reinforcement learning adds real value. Empirical evaluation across two gate sets, the universal Clifford+T basis and the CNOT+Pauli basis, shows that RL+CR produces shorter circuits than a standard RL agent at all tested scales. We demonstrate that RL trained on smaller quantum circuits can be applied to larger quantum circuits. On 20-qubit Clifford+T circuits, five times larger than the training circuits, RL+CR removes twice as many gates as standard RL. This work provides a robust approach that could accelerate the compilation and optimization processes for future fault-tolerant and utility-scale quantum systems. △ Less

Submitted 19 August, 2026; originally announced August 2026.

Comments: 11 pages, comments welcome

Learnable yet not simulable: a quantum resource theory of learning models

Authors: Xinbiao Wang, Yuxuan Du, Dacheng Tao

Abstract: Quantum resource theory has sharpened our understanding of the intrinsic complexity of quantum systems, particularly their classical simulability. However, it remains unclear which quantum resource governs the classical learnability of quantum circuits, especially beyond the regime of efficient classical simulation. Here we close this knowledge gap by studying the expectation-value functions of fa… ▽ More Quantum resource theory has sharpened our understanding of the intrinsic complexity of quantum systems, particularly their classical simulability. However, it remains unclear which quantum resource governs the classical learnability of quantum circuits, especially beyond the regime of efficient classical simulation. Here we close this knowledge gap by studying the expectation-value functions of families of tunable quantum circuits, with many applications in digital quantum simulation, quantum metrology, and quantum-system characterization. Specifically, we introduce a new resource measure, the dynamical stabilizer entropy (\DSE), which quantifies how broadly an expectation-value function is distributed across its frequency modes. By relating \DSE to operator stabilizer entropy, we establish a computational phase diagram that compares classical simulators with quantum-data-assisted classical surrogates. We first determine the \DSE-dependent learnability boundary of this diagram by deriving bounds on the sample complexity and runtime of classical surrogates, and by developing a \DSE-guided surrogate. We then complete the diagram by proving, under standard complexity-theoretic assumptions, the existence of circuit families that can be efficiently learned by this surrogate but cannot be efficiently emulated from their circuit descriptions alone. Numerical experiments on random and structured circuits with up to 80 qubits support the predicted \DSE-dependent computational landscape. These results establish a quantitative resource-theoretic framework for delineating the boundary between classical simulation and learning, motivate resource measures linking quantum resources to learnability, and guide the design of learning-based algorithms for scalable quantum systems beyond the reach of direct classical simulation. △ Less

Submitted 3 August, 2026; originally announced August 2026.

Comments: 61 Pages, 7 figures

Stochastic Pauli-path simulator for large-scale quantum optimization

Authors: Kaining Zhang, Xinbiao Wang, Kunsheng Li, Qixin Zhang, Yuxuan Du, Min-Hsiu Hsieh, Dacheng Tao

Abstract: Pauli-based simulators offer a promising route to large-scale classical simulation of quantum circuits in the low-magic regime. Yet their applicability remains largely limited to forward simulation, making them inadequate for optimization-driven quantum tasks such as variational state preparation and parameter initialization. Existing approaches either lack native support for gradient-based optimi… ▽ More Pauli-based simulators offer a promising route to large-scale classical simulation of quantum circuits in the low-magic regime. Yet their applicability remains largely limited to forward simulation, making them inadequate for optimization-driven quantum tasks such as variational state preparation and parameter initialization. Existing approaches either lack native support for gradient-based optimization or suffer from severe gradient bias. Here we propose the stochastic Pauli-path simulator (SPPS), a computational framework for large-scale quantum optimization that enables unbiased stochastic gradient estimation via Pauli-path sampling across optimization iterations. Our theoretical analysis shows that the proposed simulator yields unbiased gradient estimates and admits provable convergence guarantees. We systematically evaluate our proposal, including quantum eigensolver benchmarks with up to 100 qubits and quantum neural network benchmarks with up to 40 qubits. Across these tasks, SPPS faithfully tracks optimization dynamics, converges within minutes, and broadens the role of Pauli-based simulation from forward estimation to large-scale quantum optimization. △ Less

Submitted 20 July, 2026; originally announced July 2026.

Comments: 43 pages, 8 figures

AQER: a scalable and efficient data loader for digital quantum computers

Authors: Kaining Zhang, Xinbiao Wang, Yuxuan Du, Min-Hsiu Hsieh, Dacheng Tao

Abstract: Digital quantum computing promises to offer computational capabilities beyond the reach of classical systems, yet its capabilities are often challenged by scarce quantum resources. A critical bottleneck in this context is how to load classical or quantum data into quantum circuits efficiently. Approximate quantum loaders (AQLs) provide a viable solution to this problem by balancing fidelity and ci… ▽ More Digital quantum computing promises to offer computational capabilities beyond the reach of classical systems, yet its capabilities are often challenged by scarce quantum resources. A critical bottleneck in this context is how to load classical or quantum data into quantum circuits efficiently. Approximate quantum loaders (AQLs) provide a viable solution to this problem by balancing fidelity and circuit complexity. However, most existing AQL methods are either heuristic or provide guarantees only for specific input types, and a general theoretical framework is still lacking. To address this gap, here we reformulate most AQL methods into a unified framework and establish information-theoretic bounds on their approximation error. Our analysis reveals that the achievable infidelity between the prepared state and target state scales linearly with the total entanglement entropy across subsystems when the loading circuit is applied to the target state. In light of this, we develop AQER, a scalable AQL method that constructs the loading circuit by systematically reducing entanglement in target states. We conduct systematic experiments to evaluate the effectiveness of AQER, using synthetic datasets, classical image and language datasets, and a quantum many-body state datasets with up to 50 qubits. The results show that AQER consistently outperforms existing methods in both accuracy and gate efficiency. Our work paves the way for scalable quantum data processing and real-world quantum computing applications. △ Less

Submitted 2 February, 2026; originally announced February 2026.

Comments: 45 pages, 19 figures

Journal ref: Fourteenth International Conference on Learning Representations (ICLR 2026)

Mitigating Barren Plateaus in Quantum Denoising Diffusion Probabilistic Model

Authors: Haipeng Cao, Kaining Zhang, Dacheng Tao, Zhaofeng Su

Abstract: Quantum generative models exploit quantum superposition and entanglement to enhance learning efficiency for both classical and quantum data. Recently, inspired by classical diffusion frameworks, the quantum denoising diffusion probabilistic model has emerged as a powerful tool for learning correlated noise models, many-body phases, and topological data structures. However, we demonstrate that this… ▽ More Quantum generative models exploit quantum superposition and entanglement to enhance learning efficiency for both classical and quantum data. Recently, inspired by classical diffusion frameworks, the quantum denoising diffusion probabilistic model has emerged as a powerful tool for learning correlated noise models, many-body phases, and topological data structures. However, we demonstrate that this framework is currently restricted to small-scale systems. As the system size increases, a severe barren plateau problem emerges, fundamentally limiting the model's scalability. We provide rigorous theoretical proofs and experimental validation to identify the origin of this barren plateau, distinct from previously known causes. To restore trainability, we introduce an enhanced architecture that effectively mitigates the barren plateau phenomenon and guarantees the model's trainability in the tested settings. Building on this architecture, we further propose a conditional quantum denoising diffusion probabilistic model, capable of generating ground states based on Hamiltonian parameters, expanding the utility of quantum generative models for complex quantum state preparation to a certain extent. Our approach not only holds the potential to address the scalability and trainability bottlenecks of quantum diffusion models, but also provides a robust tool for exploring complex quantum matter and state preparation in the NISQ era. △ Less

Submitted 10 August, 2026; v1 submitted 7 December, 2025; originally announced December 2025.

Comments: 17 pages, 8 figures

Journal ref: Physical Review A 114, 022418, August, 2026

AiDE-Q: Synthetic Labeled Datasets Can Enhance Learning Models for Quantum Property Estimation

Authors: Xinbiao Wang, Yuxuan Du, Zihan Lou, Yang Qian, Kaining Zhang, Yong Luo, Bo Du, Dacheng Tao

Abstract: Quantum many-body problems are central to various scientific disciplines, yet their ground-state properties are intrinsically challenging to estimate. Recent advances in deep learning (DL) offer potential solutions in this field, complementing prior purely classical and quantum approaches. However, existing DL-based models typically assume access to a large-scale and noiseless labeled dataset coll… ▽ More Quantum many-body problems are central to various scientific disciplines, yet their ground-state properties are intrinsically challenging to estimate. Recent advances in deep learning (DL) offer potential solutions in this field, complementing prior purely classical and quantum approaches. However, existing DL-based models typically assume access to a large-scale and noiseless labeled dataset collected by infinite sampling. This idealization raises fundamental concerns about their practical utility, especially given the limited availability of quantum hardware in the near term. To unleash the power of these DL-based models, we propose AiDE-Q (\underline{a}utomat\underline{i}c \underline{d}ata \underline{e}ngine for \underline{q}uantum property estimation), an effective framework that addresses this challenge by iteratively generating high-quality synthetic labeled datasets. Specifically, AiDE-Q utilizes a consistency-check method to assess the quality of synthetic labels and continuously improves the employed DL models with the identified high-quality synthetic dataset. To verify the effectiveness of AiDE-Q, we conduct extensive numerical simulations on a diverse set of quantum many-body and molecular systems, with up to 50 qubits. The results show that AiDE-Q enhances prediction performance for various reference learning models, with improvements of up to $14.2\%$. Moreover, we exhibit that a basic supervised learning model integrated with AiDE-Q outperforms advanced reference models, highlighting the importance of a synthetic dataset. Our work paves the way for more efficient and practical applications of DL for quantum property estimation. △ Less

Submitted 30 September, 2025; originally announced September 2025.

Comments: 24 pages, 6 figures

Artificial intelligence for representing and characterizing quantum systems

Authors: Yuxuan Du, Yan Zhu, Yuan-Hang Zhang, Min-Hsiu Hsieh, Patrick Rebentrost, Weibo Gao, Ya-Dong Wu, Jens Eisert, Giulio Chiribella, Dacheng Tao, Barry C. Sanders

Abstract: Efficient characterization of large-scale quantum systems, especially those produced by quantum analog simulators and megaquop quantum computers, poses a central challenge in quantum science due to the exponential scaling of the Hilbert space with respect to system size. Recent advances in artificial intelligence (AI), with its aptitude for high-dimensional pattern recognition and function approxi… ▽ More Efficient characterization of large-scale quantum systems, especially those produced by quantum analog simulators and megaquop quantum computers, poses a central challenge in quantum science due to the exponential scaling of the Hilbert space with respect to system size. Recent advances in artificial intelligence (AI), with its aptitude for high-dimensional pattern recognition and function approximation, have emerged as a powerful tool to address this challenge. A growing body of research has leveraged AI to represent and characterize scalable quantum systems, spanning from theoretical foundations to experimental realizations. Depending on how prior knowledge and learning architectures are incorporated, the integration of AI into quantum system characterization can be categorized into three synergistic paradigms: machine learning, and, in particular, deep learning and language models. This review discusses how each of these AI paradigms contributes to two core tasks in quantum systems characterization: quantum property prediction and the construction of surrogates for quantum states. These tasks underlie diverse applications, from quantum certification and benchmarking to the enhancement of quantum algorithms and the understanding of strongly correlated phases of matter. Key challenges and open questions are also discussed, together with future prospects at the interface of AI and quantum science. △ Less

Submitted 5 September, 2025; originally announced September 2025.

Comments: 32 pages. Comments are welcome

Journal ref: Nature Reviews Physics (2026)

Demonstration of Efficient Predictive Surrogates for Large-scale Quantum Processors

Authors: Wei-You Liao, Yuxuan Du, Xinbiao Wang, Tian-Ci Tian, Yong Luo, Bo Du, Dacheng Tao, He-Liang Huang

Abstract: The ongoing development of quantum processors is driving breakthroughs in scientific discovery. Despite this progress, the formidable cost of fabricating large-scale quantum processors means they will remain rare for the foreseeable future, limiting their widespread application. To address this bottleneck, we introduce the concept of predictive surrogates, which are classical learning models desig… ▽ More The ongoing development of quantum processors is driving breakthroughs in scientific discovery. Despite this progress, the formidable cost of fabricating large-scale quantum processors means they will remain rare for the foreseeable future, limiting their widespread application. To address this bottleneck, we introduce the concept of predictive surrogates, which are classical learning models designed to emulate the mean-value behavior of a given quantum processor with provably computational efficiency. In particular, we propose two predictive surrogates that can substantially reduce the need for quantum processor access in diverse practical scenarios. To demonstrate their potential in advancing digital quantum simulation, we use these surrogates to emulate a quantum processor with up to 20 programmable superconducting qubits, enabling efficient pre-training of variational quantum eigensolvers for families of transverse-field Ising models and identification of non-equilibrium Floquet symmetry-protected topological phases. Experimental results reveal that the predictive surrogates not only reduce measurement overhead by orders of magnitude, but can also surpass the performance of conventional, quantum-resource-intensive approaches. Collectively, these findings establish predictive surrogates as a practical pathway to broadening the impact of advanced quantum processors. △ Less

Submitted 23 July, 2025; originally announced July 2025.

Comments: 53 pages, 15 figures, comments are welcome

Quantum Machine Learning: A Hands-on Tutorial for Machine Learning Practitioners and Researchers

Authors: Yuxuan Du, Xinbiao Wang, Naixu Guo, Zhan Yu, Yang Qian, Kaining Zhang, Min-Hsiu Hsieh, Patrick Rebentrost, Dacheng Tao

Abstract: 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,… ▽ More 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. △ Less

Submitted 3 February, 2025; originally announced February 2025.

Comments: 260 pages; Comments are welcome

MG-Net: Learn to Customize QAOA with Circuit Depth Awareness

Authors: Yang Qian, Xinbiao Wang, Yuxuan Du, Yong Luo, Dacheng Tao

Abstract: Quantum Approximate Optimization Algorithm (QAOA) and its variants exhibit immense potential in tackling combinatorial optimization challenges. However, their practical realization confronts a dilemma: the requisite circuit depth for satisfactory performance is problem-specific and often exceeds the maximum capability of current quantum devices. To address this dilemma, here we first analyze the c… ▽ More Quantum Approximate Optimization Algorithm (QAOA) and its variants exhibit immense potential in tackling combinatorial optimization challenges. However, their practical realization confronts a dilemma: the requisite circuit depth for satisfactory performance is problem-specific and often exceeds the maximum capability of current quantum devices. To address this dilemma, here we first analyze the convergence behavior of QAOA, uncovering the origins of this dilemma and elucidating the intricate relationship between the employed mixer Hamiltonian, the specific problem at hand, and the permissible maximum circuit depth. Harnessing this understanding, we introduce the Mixer Generator Network (MG-Net), a unified deep learning framework adept at dynamically formulating optimal mixer Hamiltonians tailored to distinct tasks and circuit depths. Systematic simulations, encompassing Ising models and weighted Max-Cut instances with up to 64 qubits, substantiate our theoretical findings, highlighting MG-Net's superior performance in terms of both approximation ratio and efficiency. △ Less

Submitted 27 September, 2024; originally announced September 2024.

Comments: 29 pages, 16 figures

Efficient Learning for Linear Properties of Bounded-Gate Quantum Circuits

Authors: Yuxuan Du, Min-Hsiu Hsieh, Dacheng Tao

Abstract: 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… ▽ More 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. △ Less

Submitted 19 September, 2025; v1 submitted 22 August, 2024; originally announced August 2024.

Comments: Published version in Nature Communications. 61 Pages, 20 figures

Journal ref: Nat Commun 16, 3790 (2025)

The curse of random quantum data

Authors: Kaining Zhang, Junyu Liu, Liu Liu, Liang Jiang, Min-Hsiu Hsieh, Dacheng Tao

Abstract: 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 th… ▽ More 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. △ Less

Submitted 19 August, 2024; originally announced August 2024.

Comments: 40 pages, 8 figures

Separable Power of Classical and Quantum Learning Protocols Through the Lens of No-Free-Lunch Theorem

Authors: Xinbiao Wang, Yuxuan Du, Kecheng Liu, Yong Luo, Bo Du, Dacheng Tao

Abstract: The No-Free-Lunch (NFL) theorem, which quantifies problem- and data-independent generalization errors regardless of the optimization process, provides a foundational framework for comprehending diverse learning protocols' potential. Despite its significance, the establishment of the NFL theorem for quantum machine learning models remains largely unexplored, thereby overlooking broader insights int… ▽ More The No-Free-Lunch (NFL) theorem, which quantifies problem- and data-independent generalization errors regardless of the optimization process, provides a foundational framework for comprehending diverse learning protocols' potential. Despite its significance, the establishment of the NFL theorem for quantum machine learning models remains largely unexplored, thereby overlooking broader insights into the fundamental relationship between quantum and classical learning protocols. To address this gap, we categorize a diverse array of quantum learning algorithms into three learning protocols designed for learning quantum dynamics under a specified observable and establish their NFL theorem. The exploited protocols, namely Classical Learning Protocols (CLC-LPs), Restricted Quantum Learning Protocols (ReQu-LPs), and Quantum Learning Protocols (Qu-LPs), offer varying levels of access to quantum resources. Our derived NFL theorems demonstrate quadratic reductions in sample complexity across CLC-LPs, ReQu-LPs, and Qu-LPs, contingent upon the orthogonality of quantum states and the diagonality of observables. We attribute this performance discrepancy to the unique capacity of quantum-related learning protocols to indirectly utilize information concerning the global phases of non-orthogonal quantum states, a distinctive physical feature inherent in quantum mechanics. Our findings not only deepen our understanding of quantum learning protocols' capabilities but also provide practical insights for the development of advanced quantum learning algorithms. △ Less

Submitted 12 May, 2024; originally announced May 2024.

Optical Quantum Sensing for Agnostic Environments via Deep Learning

Authors: Zeqiao Zhou, Yuxuan Du, Xu-Fei Yin, Shanshan Zhao, Xinmei Tian, Dacheng Tao

Abstract: Optical quantum sensing promises measurement precision beyond classical sensors termed the Heisenberg limit (HL). However, conventional methodologies often rely on prior knowledge of the target system to achieve HL, presenting challenges in practical applications. Addressing this limitation, we introduce an innovative Deep Learning-based Quantum Sensing scheme (DQS), enabling optical quantum senso… ▽ More Optical quantum sensing promises measurement precision beyond classical sensors termed the Heisenberg limit (HL). However, conventional methodologies often rely on prior knowledge of the target system to achieve HL, presenting challenges in practical applications. Addressing this limitation, we introduce an innovative Deep Learning-based Quantum Sensing scheme (DQS), enabling optical quantum sensors to attain HL in agnostic environments. DQS incorporates two essential components: a Graph Neural Network (GNN) predictor and a trigonometric interpolation algorithm. Operating within a data-driven paradigm, DQS utilizes the GNN predictor, trained on offline data, to unveil the intrinsic relationships between the optical setups employed in preparing the probe state and the resulting quantum Fisher information (QFI) after interaction with the agnostic environment. This distilled knowledge facilitates the identification of optimal optical setups associated with maximal QFI. Subsequently, DQS employs a trigonometric interpolation algorithm to recover the unknown parameter estimates for the identified optical setups. Extensive experiments are conducted to investigate the performance of DQS under different settings up to eight photons. Our findings not only offer a new lens through which to accelerate optical quantum sensing tasks but also catalyze future research integrating deep learning and quantum mechanics. △ Less

Submitted 13 November, 2023; originally announced November 2023.

Multimodal deep representation learning for quantum cross-platform verification

Authors: Yang Qian, Yuxuan Du, Zhenliang He, Min-hsiu Hsieh, Dacheng Tao

Abstract: 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 fea… ▽ More 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. △ Less

Submitted 6 November, 2023; originally announced November 2023.

MEMQSim: Highly Memory-Efficient and Modularized Quantum State-Vector Simulation

Authors: Boyuan Zhang, Bo Fang, Qiang Guan, Ang Li, Dingwen Tao

Abstract: In this extended abstract, we have introduced a highly memory-efficient state vector simulation of quantum circuits premised on data compression, harnessing the capabilities of both CPUs and GPUs. We have elucidated the inherent challenges in architecting this system, while concurrently proposing our tailored solutions. Moreover, we have delineated our preliminary implementation and deliberated up… ▽ More In this extended abstract, we have introduced a highly memory-efficient state vector simulation of quantum circuits premised on data compression, harnessing the capabilities of both CPUs and GPUs. We have elucidated the inherent challenges in architecting this system, while concurrently proposing our tailored solutions. Moreover, we have delineated our preliminary implementation and deliberated upon the potential for integration with other GPU-oriented simulators. In forthcoming research, we aim to present a more comprehensive set of results, bolstering the assertion of the efficacy and performance of our approach. △ Less

Submitted 29 September, 2023; originally announced September 2023.

ShadowNet for Data-Centric Quantum System Learning

Authors: Yuxuan Du, Yibo Yang, Tongliang Liu, Zhouchen Lin, Bernard Ghanem, Dacheng Tao

Abstract: Understanding the dynamics of large quantum systems is hindered by the curse of dimensionality. Statistical learning offers new possibilities in this regime through neural network protocols and classical shadows, while both methods have limitations: the former suffers from incompatible dataset construction rules, resulting in substantial computational demands for data collection when addressing di… ▽ More Understanding the dynamics of large quantum systems is hindered by the curse of dimensionality. Statistical learning offers new possibilities in this regime through neural network protocols and classical shadows, while both methods have limitations: the former suffers from incompatible dataset construction rules, resulting in substantial computational demands for data collection when addressing different tasks; the latter lacks the ability to distill knowledge from prior data to enhance subsequent learning endeavors. In this study, we propose a data-centric learning paradigm combining the strengths of these two approaches to advance quantum system learning (QSL). Central to our paradigm lies a unified dataset construction rule, achieved by classical shadows along with other easily obtainable information of quantum systems. To illustrate our approach, we present ShadowNet, implemented under both convolutional and attention mechanisms, to efficiently and faithfully tackle two pivotal QSL tasks: quantum state tomography (QST) and direct fidelity estimation (DFE). Numerical simulations on QST and DFE up to 60 qubits validate the efficacy of our proposal, showcasing how ShadowNet advances classical shadows with limited state copies, and highlighting how the varied neural networks impact the performance. Our work underscores the immense potential of a data-centric approach in comprehending novel and large quantum systems. △ Less

Submitted 15 August, 2026; v1 submitted 22 August, 2023; originally announced August 2023.

Comments: Accepted to IEEE Transactions on Pattern Analysis and Machine Intelligence. 20 pages. 12 Figures

Carbon Emissions of Quantum Circuit Simulation: More than You Would Think

Authors: Jinyang Li, Qiang Guan, Dingwen Tao, Weiwen Jiang

Abstract: The rapid advancement of quantum hardware brings a host of research opportunities and the potential for quantum advantages across numerous fields. In this landscape, quantum circuit simulations serve as an indispensable tool by emulating quantum behavior on classical computers. They offer easy access, noise-free environments, and real-time observation of quantum states. However, the sustainability… ▽ More The rapid advancement of quantum hardware brings a host of research opportunities and the potential for quantum advantages across numerous fields. In this landscape, quantum circuit simulations serve as an indispensable tool by emulating quantum behavior on classical computers. They offer easy access, noise-free environments, and real-time observation of quantum states. However, the sustainability aspect of quantum circuit simulation is yet to be explored. In this paper, we introduce for the first time the concept of environmental impact from quantum circuit simulation. We present a preliminary model to compute the CO2e emissions derived from quantum circuit simulations. Our results indicate that large quantum circuit simulations (43 qubits) could lead to CO2e emissions 48 times greater than training a transformer machine learning model. △ Less

Submitted 4 July, 2023; originally announced July 2023.

Transition Role of Entangled Data in Quantum Machine Learning

Authors: Xinbiao Wang, Yuxuan Du, Zhuozhuo Tu, Yong Luo, Xiao Yuan, Dacheng Tao

Abstract: Entanglement serves as the resource to empower quantum computing. Recent progress has highlighted its positive impact on learning quantum dynamics, wherein the integration of entanglement into quantum operations or measurements of quantum machine learning (QML) models leads to substantial reductions in training data size, surpassing a specified prediction error threshold. However, an analytical un… ▽ More Entanglement serves as the resource to empower quantum computing. Recent progress has highlighted its positive impact on learning quantum dynamics, wherein the integration of entanglement into quantum operations or measurements of quantum machine learning (QML) models leads to substantial reductions in training data size, surpassing a specified prediction error threshold. However, an analytical understanding of how the entanglement degree in data affects model performance remains elusive. In this study, we address this knowledge gap by establishing a quantum no-free-lunch (NFL) theorem for learning quantum dynamics using entangled data. Contrary to previous findings, we prove that the impact of entangled data on prediction error exhibits a dual effect, depending on the number of permitted measurements. With a sufficient number of measurements, increasing the entanglement of training data consistently reduces the prediction error or decreases the required size of the training data to achieve the same prediction error. Conversely, when few measurements are allowed, employing highly entangled data could lead to an increased prediction error. The achieved results provide critical guidance for designing advanced QML protocols, especially for those tailored for execution on early-stage quantum computers with limited access to quantum resources. △ Less

Submitted 12 May, 2024; v1 submitted 6 June, 2023; originally announced June 2023.

Comments: Accept to Nature Communications 15, 3716 (2024)

Quantum Imitation Learning

Authors: Zhihao Cheng, Kaining Zhang, Li Shen, Dacheng Tao

Abstract: Despite remarkable successes in solving various complex decision-making tasks, training an imitation learning (IL) algorithm with deep neural networks (DNNs) suffers from the high computation burden. In this work, we propose quantum imitation learning (QIL) with a hope to utilize quantum advantage to speed up IL. Concretely, we develop two QIL algorithms, quantum behavioural cloning (Q-BC) and qua… ▽ More Despite remarkable successes in solving various complex decision-making tasks, training an imitation learning (IL) algorithm with deep neural networks (DNNs) suffers from the high computation burden. In this work, we propose quantum imitation learning (QIL) with a hope to utilize quantum advantage to speed up IL. Concretely, we develop two QIL algorithms, quantum behavioural cloning (Q-BC) and quantum generative adversarial imitation learning (Q-GAIL). Q-BC is trained with a negative log-likelihood loss in an off-line manner that suits extensive expert data cases, whereas Q-GAIL works in an inverse reinforcement learning scheme, which is on-line and on-policy that is suitable for limited expert data cases. For both QIL algorithms, we adopt variational quantum circuits (VQCs) in place of DNNs for representing policies, which are modified with data re-uploading and scaling parameters to enhance the expressivity. We first encode classical data into quantum states as inputs, then perform VQCs, and finally measure quantum outputs to obtain control signals of agents. Experiment results demonstrate that both Q-BC and Q-GAIL can achieve comparable performance compared to classical counterparts, with the potential of quantum speed-up. To our knowledge, we are the first to propose the concept of QIL and conduct pilot studies, which paves the way for the quantum era. △ Less

Submitted 4 April, 2023; originally announced April 2023.

Comments: Manuscript submitted to a journal for review on January 5, 2022

TeD-Q: a tensor network enhanced distributed hybrid quantum machine learning framework

Authors: Yaocheng Chen, Chung-Yun Kuo, Yuxuan Du, Dacheng Tao, Xingyao Wu

Abstract: TeD-Q is an open-source software framework for quantum machine learning, variational quantum algorithm (VQA), and simulation of quantum computing. It seamlessly integrates classical machine learning libraries with quantum simulators, giving users the ability to leverage the power of classical machine learning while training quantum machine learning models. TeD-Q supports auto-differentiation that… ▽ More TeD-Q is an open-source software framework for quantum machine learning, variational quantum algorithm (VQA), and simulation of quantum computing. It seamlessly integrates classical machine learning libraries with quantum simulators, giving users the ability to leverage the power of classical machine learning while training quantum machine learning models. TeD-Q supports auto-differentiation that provides backpropagation, parameters shift, and finite difference methods to obtain gradients. With tensor contraction, simulation of quantum circuits with large number of qubits is possible. TeD-Q also provides a graphical mode in which the quantum circuit and the training progress can be visualized in real-time. △ Less

Submitted 7 December, 2024; v1 submitted 13 January, 2023; originally announced January 2023.

Comments: 20 pages, 15 figures

Problem-Dependent Power of Quantum Neural Networks on Multi-Class Classification

Authors: Yuxuan Du, Yibo Yang, Dacheng Tao, Min-Hsiu Hsieh

Abstract: 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… ▽ More 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. △ Less

Submitted 30 October, 2023; v1 submitted 29 December, 2022; originally announced January 2023.

Comments: Updated version. Published on PRL

Journal ref: Phys. Rev. Lett. 131, 140601 (2023)

Shuffle-QUDIO: accelerate distributed VQE with trainability enhancement and measurement reduction

Authors: Yang Qian, Yuxuan Du, Dacheng Tao

Abstract: The variational quantum eigensolver (VQE) is a leading strategy that exploits noisy intermediate-scale quantum (NISQ) machines to tackle chemical problems outperforming classical approaches. To gain such computational advantages on large-scale problems, a feasible solution is the QUantum DIstributed Optimization (QUDIO) scheme, which partitions the original problem into $K$ subproblems and allocat… ▽ More The variational quantum eigensolver (VQE) is a leading strategy that exploits noisy intermediate-scale quantum (NISQ) machines to tackle chemical problems outperforming classical approaches. To gain such computational advantages on large-scale problems, a feasible solution is the QUantum DIstributed Optimization (QUDIO) scheme, which partitions the original problem into $K$ subproblems and allocates them to $K$ quantum machines followed by the parallel optimization. Despite the provable acceleration ratio, the efficiency of QUDIO may heavily degrade by the synchronization operation. To conquer this issue, here we propose Shuffle-QUDIO to involve shuffle operations into local Hamiltonians during the quantum distributed optimization. Compared with QUDIO, Shuffle-QUDIO significantly reduces the communication frequency among quantum processors and simultaneously achieves better trainability. Particularly, we prove that Shuffle-QUDIO enables a faster convergence rate over QUDIO. Extensive numerical experiments are conducted to verify that Shuffle-QUDIO allows both a wall-clock time speedup and low approximation error in the tasks of estimating the ground state energy of molecule. We empirically demonstrate that our proposal can be seamlessly integrated with other acceleration techniques, such as operator grouping, to further improve the efficacy of VQE. △ Less

Submitted 26 September, 2022; originally announced September 2022.

Symmetric Pruning in Quantum Neural Networks

Authors: Xinbiao Wang, Junyu Liu, Tongliang Liu, Yong Luo, Yuxuan Du, Dacheng Tao

Abstract: Many fundamental properties of a quantum system are captured by its Hamiltonian and ground state. Despite the significance of ground states preparation (GSP), this task is classically intractable for large-scale Hamiltonians. Quantum neural networks (QNNs), which exert the power of modern quantum machines, have emerged as a leading protocol to conquer this issue. As such, how to enhance the perfor… ▽ More Many fundamental properties of a quantum system are captured by its Hamiltonian and ground state. Despite the significance of ground states preparation (GSP), this task is classically intractable for large-scale Hamiltonians. Quantum neural networks (QNNs), which exert the power of modern quantum machines, have emerged as a leading protocol to conquer this issue. As such, how to enhance the performance of QNNs becomes a crucial topic in GSP. Empirical evidence showed that QNNs with handcraft symmetric ansatzes generally experience better trainability than those with asymmetric ansatzes, while theoretical explanations have not been explored. To fill this knowledge gap, here we propose the effective quantum neural tangent kernel (EQNTK) and connect this concept with over-parameterization theory to quantify the convergence of QNNs towards the global optima. We uncover that the advance of symmetric ansatzes attributes to their large EQNTK value with low effective dimension, which requests few parameters and quantum circuit depth to reach the over-parameterization regime permitting a benign loss landscape and fast convergence. Guided by EQNTK, we further devise a symmetric pruning (SP) scheme to automatically tailor a symmetric ansatz from an over-parameterized and asymmetric one to greatly improve the performance of QNNs when the explicit symmetry information of Hamiltonian is unavailable. Extensive numerical simulations are conducted to validate the analytical results of EQNTK and the effectiveness of SP. △ Less

Submitted 7 February, 2023; v1 submitted 30 August, 2022; originally announced August 2022.

Comments: Accepted to International Conference on Learning Representations (ICLR) 2023

Journal ref: International Conference on Learning Representations (ICLR) 2023

Recent Advances for Quantum Neural Networks in Generative Learning

Authors: Jinkai Tian, Xiaoyu Sun, Yuxuan Du, Shanshan Zhao, Qing Liu, Kaining Zhang, Wei Yi, Wanrong Huang, Chaoyue Wang, Xingyao Wu, Min-Hsiu Hsieh, Tongliang Liu, Wenjing Yang, Dacheng Tao

Abstract: 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 sur… ▽ More 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. △ Less

Submitted 7 June, 2022; originally announced June 2022.

Comments: The first two authors contributed equally to this work

QAOA-in-QAOA: solving large-scale MaxCut problems on small quantum machines

Authors: Zeqiao Zhou, Yuxuan Du, Xinmei Tian, Dacheng Tao

Abstract: The design of fast algorithms for combinatorial optimization greatly contributes to a plethora of domains such as logistics, finance, and chemistry. Quantum approximate optimization algorithms (QAOAs), which utilize the power of quantum machines and inherit the spirit of adiabatic evolution, are novel approaches to tackle combinatorial problems with potential runtime speedups. However, hurdled by… ▽ More The design of fast algorithms for combinatorial optimization greatly contributes to a plethora of domains such as logistics, finance, and chemistry. Quantum approximate optimization algorithms (QAOAs), which utilize the power of quantum machines and inherit the spirit of adiabatic evolution, are novel approaches to tackle combinatorial problems with potential runtime speedups. However, hurdled by the limited quantum resources nowadays, QAOAs are infeasible to manipulate large-scale problems. To address this issue, here we revisit the MaxCut problem via the divide-and-conquer heuristic: seek the solutions of subgraphs in parallel and then merge these solutions to obtain the global solution. Due to the $\mathbb{Z}_2$ symmetry in MaxCut, we prove that the merging process can be further cast into a new MaxCut problem and thus be addressed by QAOAs or other MaxCut solvers. With this regard, we propose QAOA-in-QAOA ($\text{QAOA}^2$) to solve arbitrary large-scale MaxCut problems using small quantum machines. We also prove that the approximation ratio of $\text{QAOA}^2$ is lower bounded by 1/2. Experiment results illustrate that under different graph settings, $\text{QAOA}^2$ attains a competitive or even better performance over the best known classical algorithms when the node count is around 2000. Our method can be seamlessly embedded into other advanced strategies to enhance the capability of QAOAs in large-scale combinatorial optimization problems. △ Less

Submitted 23 May, 2022; originally announced May 2022.

Power of Quantum Generative Learning

Authors: Yuxuan Du, Zhuozhuo Tu, Bujiao Wu, Xiao Yuan, Dacheng Tao

Abstract: The intrinsic probabilistic nature of quantum mechanics invokes endeavors of designing quantum generative learning models (QGLMs). Despite the empirical achievements, the foundations and the potential advantages of QGLMs remain largely obscure. To narrow this knowledge gap, here we explore the generalization property of QGLMs, the capability to extend the model from learned to unknown data. We con… ▽ More The intrinsic probabilistic nature of quantum mechanics invokes endeavors of designing quantum generative learning models (QGLMs). Despite the empirical achievements, the foundations and the potential advantages of QGLMs remain largely obscure. To narrow this knowledge gap, here we explore the generalization property of QGLMs, the capability to extend the model from learned to unknown data. We consider two prototypical QGLMs, quantum circuit Born machines and quantum generative adversarial networks, and explicitly give their generalization bounds. The result identifies superiorities of QGLMs over classical methods when quantum devices can directly access the target distribution and quantum kernels are employed. We further employ these generalization bounds to exhibit potential advantages in quantum state preparation and Hamiltonian learning. Numerical results of QGLMs in loading Gaussian distribution and estimating ground states of parameterized Hamiltonians accord with the theoretical analysis. Our work opens the avenue for quantitatively understanding the power of quantum generative learning models. △ Less

Submitted 4 August, 2022; v1 submitted 10 May, 2022; originally announced May 2022.

Comments: 28 pages, 9 figures

Escaping from the Barren Plateau via Gaussian Initializations in Deep Variational Quantum Circuits

Authors: Kaining Zhang, Liu Liu, Min-Hsiu Hsieh, Dacheng Tao

Abstract: 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… ▽ More 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. △ Less

Submitted 19 February, 2025; v1 submitted 17 March, 2022; originally announced March 2022.

Comments: Accepted by the Thirty-Sixth Conference on Neural Information Processing Systems (NeurIPS 2022)

Journal ref: Advances in Neural Information Processing Systems, 2022, 35: 18612-18627

Efficient Bipartite Entanglement Detection Scheme with a Quantum Adversarial Solver

Authors: Xu-Fei Yin, Yuxuan Du, Yue-Yang Fei, Rui Zhang, Li-Zheng Liu, Yingqiu Mao, Tongliang Liu, Min-Hsiu Hsieh, Li Li, Nai-Le Liu, Dacheng Tao, Yu-Ao Chen, Jian-Wei Pan

Abstract: The recognition of entanglement states is a notoriously difficult problem when no prior information is available. Here, we propose an efficient quantum adversarial bipartite entanglement detection scheme to address this issue. Our proposal reformulates the bipartite entanglement detection as a two-player zero-sum game completed by parameterized quantum circuits, where a two-outcome measurement can… ▽ More The recognition of entanglement states is a notoriously difficult problem when no prior information is available. Here, we propose an efficient quantum adversarial bipartite entanglement detection scheme to address this issue. Our proposal reformulates the bipartite entanglement detection as a two-player zero-sum game completed by parameterized quantum circuits, where a two-outcome measurement can be used to query a classical binary result about whether the input state is bipartite entangled or not. In principle, for an $N$-qubit quantum state, the runtime complexity of our proposal is $O(\text{poly}(N)T)$ with $T$ being the number of iterations. We experimentally implement our protocol on a linear optical network and exhibit its effectiveness to accomplish the bipartite entanglement detection for 5-qubit quantum pure states and 2-qubit quantum mixed states. Our work paves the way for using near-term quantum machines to tackle entanglement detection on multipartite entangled quantum systems. △ Less

Submitted 15 March, 2022; originally announced March 2022.

Comments: 7 pages, 3 figures

Journal ref: Phys. Rev. Lett. 128, 110501 (2022)

Quantum circuit architecture search on a superconducting processor

Authors: Kehuan Linghu, Yang Qian, Ruixia Wang, Meng-Jun Hu, Zhiyuan Li, Xuegang Li, Huikai Xu, Jingning Zhang, Teng Ma, Peng Zhao, Dong E. Liu, Min-Hsiu Hsieh, Xingyao Wu, Yuxuan Du, Dacheng Tao, Yirong Jin, Haifeng Yu

Abstract: Variational quantum algorithms (VQAs) have shown strong evidences to gain provable computational advantages for diverse fields such as finance, machine learning, and chemistry. However, the heuristic ansatz exploited in modern VQAs is incapable of balancing the tradeoff between expressivity and trainability, which may lead to the degraded performance when executed on the noisy intermediate-scale q… ▽ More Variational quantum algorithms (VQAs) have shown strong evidences to gain provable computational advantages for diverse fields such as finance, machine learning, and chemistry. However, the heuristic ansatz exploited in modern VQAs is incapable of balancing the tradeoff between expressivity and trainability, which may lead to the degraded performance when executed on the noisy intermediate-scale quantum (NISQ) machines. To address this issue, here we demonstrate the first proof-of-principle experiment of applying an efficient automatic ansatz design technique, i.e., quantum architecture search (QAS), to enhance VQAs on an 8-qubit superconducting quantum processor. In particular, we apply QAS to tailor the hardware-efficient ansatz towards classification tasks. Compared with the heuristic ansatze, the ansatz designed by QAS improves test accuracy from 31% to 98%. We further explain this superior performance by visualizing the loss landscape and analyzing effective parameters of all ansatze. Our work provides concrete guidance for developing variable ansatze to tackle various large-scale quantum learning problems with advantages. △ Less

Submitted 3 January, 2022; originally announced January 2022.

Toward Trainability of Deep Quantum Neural Networks

Authors: Kaining Zhang, Min-Hsiu Hsieh, Liu Liu, Dacheng Tao

Abstract: Quantum Neural Networks (QNNs) with random structures have poor trainability due to the exponentially vanishing gradient as the circuit depth and the qubit number increase. This result leads to a general belief that a deep QNN will not be feasible. In this work, we provide the first viable solution to the vanishing gradient problem for deep QNNs with theoretical guarantees. Specifically, we prove… ▽ More Quantum Neural Networks (QNNs) with random structures have poor trainability due to the exponentially vanishing gradient as the circuit depth and the qubit number increase. This result leads to a general belief that a deep QNN will not be feasible. In this work, we provide the first viable solution to the vanishing gradient problem for deep QNNs with theoretical guarantees. Specifically, we prove that for circuits with controlled-layer architectures, the expectation of the gradient norm can be lower bounded by a value that is independent of the qubit number and the circuit depth. Our results follow from a careful analysis of the gradient behaviour on parameter space consisting of rotation angles, as employed in almost any QNNs, instead of relying on impractical 2-design assumptions. We explicitly construct examples where only our QNNs are trainable and converge, while others in comparison cannot. △ Less

Submitted 26 September, 2022; v1 submitted 30 December, 2021; originally announced December 2021.

Comments: 16 pages, 9 figures

On exploring the potential of quantum auto-encoder for learning quantum systems

Authors: Yuxuan Du, Dacheng Tao

Abstract: The frequent interactions between quantum computing and machine learning revolutionize both fields. One prototypical achievement is the quantum auto-encoder (QAE), as the leading strategy to relieve the curse of dimensionality ubiquitous in the quantum world. Despite its attractive capabilities, practical applications of QAE have yet largely unexplored. To narrow this knowledge gap, here we devise… ▽ More The frequent interactions between quantum computing and machine learning revolutionize both fields. One prototypical achievement is the quantum auto-encoder (QAE), as the leading strategy to relieve the curse of dimensionality ubiquitous in the quantum world. Despite its attractive capabilities, practical applications of QAE have yet largely unexplored. To narrow this knowledge gap, here we devise three effective QAE-based learning protocols to address three classically computational hard learning problems when learning quantum systems, which are low-rank state fidelity estimation, quantum Fisher information estimation, and Gibbs state preparation. Attributed to the versatility of QAE, our proposals can be readily executed on near-term quantum machines. Besides, we analyze the error bounds of the trained protocols and showcase the necessary conditions to provide practical utility from the perspective of complexity theory. We conduct numerical simulations to confirm the effectiveness of the proposed three protocols. Our work sheds new light on developing advanced quantum learning algorithms to accomplish hard quantum physics and quantum information processing tasks. △ Less

Submitted 2 October, 2024; v1 submitted 29 June, 2021; originally announced June 2021.

Comments: Accepted to IEEE Transactions on Neural Networks and Learning Systems

Accelerating variational quantum algorithms with multiple quantum processors

Authors: Yuxuan Du, Yang Qian, Dacheng Tao

Abstract: Variational quantum algorithms (VQAs) have the potential of utilizing near-term quantum machines to gain certain computational advantages over classical methods. Nevertheless, modern VQAs suffer from cumbersome computational overhead, hampered by the tradition of employing a solitary quantum processor to handle large-volume data. As such, to better exert the superiority of VQAs, it is of great sig… ▽ More Variational quantum algorithms (VQAs) have the potential of utilizing near-term quantum machines to gain certain computational advantages over classical methods. Nevertheless, modern VQAs suffer from cumbersome computational overhead, hampered by the tradition of employing a solitary quantum processor to handle large-volume data. As such, to better exert the superiority of VQAs, it is of great significance to improve their runtime efficiency. Here we devise an efficient distributed optimization scheme, called QUDIO, to address this issue. Specifically, in QUDIO, a classical central server partitions the learning problem into multiple subproblems and allocate them to multiple local nodes where each of them consists of a quantum processor and a classical optimizer. During the training procedure, all local nodes proceed parallel optimization and the classical server synchronizes optimization information among local nodes timely. In doing so, we prove a sublinear convergence rate of QUDIO in terms of the number of global iteration under the ideal scenario, while the system imperfection may incur divergent optimization. Numerical results on standard benchmarks demonstrate that QUDIO can surprisingly achieve a superlinear runtime speedup with respect to the number of local nodes. Our proposal can be readily mixed with other advanced VQAs-based techniques to narrow the gap between the state of the art and applications with quantum advantage. △ Less

Submitted 24 June, 2021; originally announced June 2021.

The dilemma of quantum neural networks

Authors: Yang Qian, Xinbiao Wang, Yuxuan Du, Xingyao Wu, Dacheng Tao

Abstract: The core of quantum machine learning is to devise quantum models with good trainability and low generalization error bound than their classical counterparts to ensure better reliability and interpretability. Recent studies confirmed that quantum neural networks (QNNs) have the ability to achieve this goal on specific datasets. With this regard, it is of great importance to understand whether these… ▽ More The core of quantum machine learning is to devise quantum models with good trainability and low generalization error bound than their classical counterparts to ensure better reliability and interpretability. Recent studies confirmed that quantum neural networks (QNNs) have the ability to achieve this goal on specific datasets. With this regard, it is of great importance to understand whether these advantages are still preserved on real-world tasks. Through systematic numerical experiments, we empirically observe that current QNNs fail to provide any benefit over classical learning models. Concretely, our results deliver two key messages. First, QNNs suffer from the severely limited effective model capacity, which incurs poor generalization on real-world datasets. Second, the trainability of QNNs is insensitive to regularization techniques, which sharply contrasts with the classical scenario. These empirical results force us to rethink the role of current QNNs and to design novel protocols for solving real-world problems with quantum advantages. △ Less

Submitted 9 June, 2021; originally announced June 2021.

Efficient measure for the expressivity of variational quantum algorithms

Authors: Yuxuan Du, Zhuozhuo Tu, Xiao Yuan, Dacheng Tao

Abstract: The superiority of variational quantum algorithms (VQAs) such as quantum neural networks (QNNs) and variational quantum eigen-solvers (VQEs) heavily depends on the expressivity of the employed ansatze. Namely, a simple ansatze is insufficient to capture the optimal solution, while an intricate ansatze leads to the hardness of the trainability. Despite its fundamental importance, an effective strat… ▽ More The superiority of variational quantum algorithms (VQAs) such as quantum neural networks (QNNs) and variational quantum eigen-solvers (VQEs) heavily depends on the expressivity of the employed ansatze. Namely, a simple ansatze is insufficient to capture the optimal solution, while an intricate ansatze leads to the hardness of the trainability. Despite its fundamental importance, an effective strategy of measuring the expressivity of VQAs remains largely unknown. Here, we exploit an advanced tool in statistical learning theory, i.e., covering number, to study the expressivity of VQAs. In particular, we first exhibit how the expressivity of VQAs with an arbitrary ansatze is upper bounded by the number of quantum gates and the measurement observable. We next explore the expressivity of VQAs on near-term quantum chips, where the system noise is considered. We observe an exponential decay of the expressivity with increasing circuit depth. We also utilize the achieved expressivity to analyze the generalization of QNNs and the accuracy of VQE. We numerically verify our theory employing VQAs with different levels of expressivity. Our work opens the avenue for quantitative understanding of the expressivity of VQAs. △ Less

Submitted 27 February, 2022; v1 submitted 20 April, 2021; originally announced April 2021.

Comments: Updated version. Published on PRL

Journal ref: Phys. Rev. Lett. 128, 080506 (2022)

Towards understanding the power of quantum kernels in the NISQ era

Authors: Xinbiao Wang, Yuxuan Du, Yong Luo, Dacheng Tao

Abstract: A key problem in the field of quantum computing is understanding whether quantum machine learning (QML) models implemented on noisy intermediate-scale quantum (NISQ) machines can achieve quantum advantages. Recently, Huang et al. [Nat Commun 12, 2631] partially answered this question by the lens of quantum kernel learning. Namely, they exhibited that quantum kernels can learn specific datasets wit… ▽ More A key problem in the field of quantum computing is understanding whether quantum machine learning (QML) models implemented on noisy intermediate-scale quantum (NISQ) machines can achieve quantum advantages. Recently, Huang et al. [Nat Commun 12, 2631] partially answered this question by the lens of quantum kernel learning. Namely, they exhibited that quantum kernels can learn specific datasets with lower generalization error over the optimal classical kernel methods. However, most of their results are established on the ideal setting and ignore the caveats of near-term quantum machines. To this end, a crucial open question is: does the power of quantum kernels still hold under the NISQ setting? In this study, we fill this knowledge gap by exploiting the power of quantum kernels when the quantum system noise and sample error are considered. Concretely, we first prove that the advantage of quantum kernels is vanished for large size of datasets, few number of measurements, and large system noise. With the aim of preserving the superiority of quantum kernels in the NISQ era, we further devise an effective method via indefinite kernel learning. Numerical simulations accord with our theoretical results. Our work provides theoretical guidance of exploring advanced quantum kernels to attain quantum advantages on NISQ devices. △ Less

Submitted 26 August, 2021; v1 submitted 30 March, 2021; originally announced March 2021.

Journal ref: Quantum 5, 531 (2021)

Toward Trainability of Quantum Neural Networks

Authors: Kaining Zhang, Min-Hsiu Hsieh, Liu Liu, Dacheng Tao

Abstract: Quantum Neural Networks (QNNs) have been recently proposed as generalizations of classical neural networks to achieve the quantum speed-up. Despite the potential to outperform classical models, serious bottlenecks exist for training QNNs; namely, QNNs with random structures have poor trainability due to the vanishing gradient with rate exponential to the input qubit number. The vanishing gradient… ▽ More Quantum Neural Networks (QNNs) have been recently proposed as generalizations of classical neural networks to achieve the quantum speed-up. Despite the potential to outperform classical models, serious bottlenecks exist for training QNNs; namely, QNNs with random structures have poor trainability due to the vanishing gradient with rate exponential to the input qubit number. The vanishing gradient could seriously influence the applications of large-size QNNs. In this work, we provide a viable solution with theoretical guarantees. Specifically, we prove that QNNs with tree tensor and step controlled architectures have gradients that vanish at most polynomially with the qubit number. We numerically demonstrate QNNs with tree tensor and step controlled structures for the application of binary classification. Simulations show faster convergent rates and better accuracy compared to QNNs with random structures. △ Less

Submitted 4 December, 2020; v1 submitted 12 November, 2020; originally announced November 2020.

Comments: 11 pages, 5 figures, 1 table

Quantum circuit architecture search for variational quantum algorithms

Authors: Yuxuan Du, Tao Huang, Shan You, Min-Hsiu Hsieh, Dacheng Tao

Abstract: 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 trainabilit… ▽ More 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. △ Less

Submitted 30 May, 2022; v1 submitted 20 October, 2020; originally announced October 2020.

Comments: Final version. See also a concurrent paper [arXiv:2010.08561]

Journal ref: npj Quantum Information volume 8, Article number: 62 (2022)

Experimental Quantum Generative Adversarial Networks for Image Generation

Authors: He-Liang Huang, Yuxuan Du, Ming Gong, Youwei Zhao, Yulin Wu, Chaoyue Wang, Shaowei Li, Futian Liang, Jin Lin, Yu Xu, Rui Yang, Tongliang Liu, Min-Hsiu Hsieh, Hui Deng, Hao Rong, Cheng-Zhi Peng, Chao-Yang Lu, Yu-Ao Chen, Dacheng Tao, Xiaobo Zhu, Jian-Wei Pan

Abstract: Quantum machine learning is expected to be one of the first practical applications of near-term quantum devices. Pioneer theoretical works suggest that quantum generative adversarial networks (GANs) may exhibit a potential exponential advantage over classical GANs, thus attracting widespread attention. However, it remains elusive whether quantum GANs implemented on near-term quantum devices can ac… ▽ More Quantum machine learning is expected to be one of the first practical applications of near-term quantum devices. Pioneer theoretical works suggest that quantum generative adversarial networks (GANs) may exhibit a potential exponential advantage over classical GANs, thus attracting widespread attention. However, it remains elusive whether quantum GANs implemented on near-term quantum devices can actually solve real-world learning tasks. Here, we devise a flexible quantum GAN scheme to narrow this knowledge gap, which could accomplish image generation with arbitrarily high-dimensional features, and could also take advantage of quantum superposition to train multiple examples in parallel. For the first time, we experimentally achieve the learning and generation of real-world hand-written digit images on a superconducting quantum processor. Moreover, we utilize a gray-scale bar dataset to exhibit the competitive performance between quantum GANs and the classical GANs based on multilayer perceptron and convolutional neural network architectures, respectively, benchmarked by the Fréchet Distance score. Our work provides guidance for developing advanced quantum generative models on near-term quantum devices and opens up an avenue for exploring quantum advantages in various GAN-related learning tasks. △ Less

Submitted 7 September, 2021; v1 submitted 13 October, 2020; originally announced October 2020.

Comments: This work was completed in 2019, and the first version of manuscript was submitted to the journal in January 2020

Journal ref: Phys. Rev. Applied 16, 024051 (2021)

On the learnability of quantum neural networks

Authors: Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, Shan You, Dacheng Tao

Abstract: We consider the learnability of the quantum neural network (QNN) built on the variational hybrid quantum-classical scheme, which remains largely unknown due to the non-convex optimization landscape, the measurement error, and the unavoidable gate errors introduced by noisy intermediate-scale quantum (NISQ) machines. Our contributions in this paper are multi-fold. First, we derive the utility bound… ▽ More We consider the learnability of the quantum neural network (QNN) built on the variational hybrid quantum-classical scheme, which remains largely unknown due to the non-convex optimization landscape, the measurement error, and the unavoidable gate errors introduced by noisy intermediate-scale quantum (NISQ) machines. Our contributions in this paper are multi-fold. First, we derive the utility bounds of QNN towards empirical risk minimization, and show that large gate noise, few quantum measurements, and deep circuit depth will lead to the poor utility bounds. This result also applies to the variational quantum circuits with gradient-based classical optimization, and can be of independent interest. We then prove that QNN can be treated as a differentially private (DP) model. Thirdly, we show that if a concept class can be efficiently learned by QNN, then it can also be effectively learned by QNN even with gate noise. This result implies the same learnability of QNN whether it is implemented on noiseless or noisy quantum machines. We last exhibit that the quantum statistical query (QSQ) model can be effectively simulated by noisy QNN. Since the QSQ model can tackle certain tasks with runtime speedup, our result suggests that the modified QNN implemented on NISQ devices will retain the quantum advantage. Numerical simulations support the theoretical results. △ Less

Submitted 24 July, 2020; originally announced July 2020.

Quantum Differentially Private Sparse Regression Learning

Authors: Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, Shan You, Dacheng Tao

Abstract: 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… ▽ More 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. △ Less

Submitted 30 May, 2022; v1 submitted 23 July, 2020; originally announced July 2020.

Comments: Final Version. Accepted in IEEE Transactions on Information Theory on 19 March, 2022

Quantum Geometric Machine Learning for Quantum Circuits and Control

Authors: Elija Perrier, Christopher Ferrie, Dacheng Tao

Abstract: The application of machine learning techniques to solve problems in quantum control together with established geometric methods for solving optimisation problems leads naturally to an exploration of how machine learning approaches can be used to enhance geometric approaches to solving problems in quantum information processing. In this work, we review and extend the application of deep learning to… ▽ More The application of machine learning techniques to solve problems in quantum control together with established geometric methods for solving optimisation problems leads naturally to an exploration of how machine learning approaches can be used to enhance geometric approaches to solving problems in quantum information processing. In this work, we review and extend the application of deep learning to quantum geometric control problems. Specifically, we demonstrate enhancements in time-optimal control in the context of quantum circuit synthesis problems by applying novel deep learning algorithms in order to approximate geodesics (and thus minimal circuits) along Lie group manifolds relevant to low-dimensional multi-qubit systems, such as SU(2), SU(4) and SU(8). We demonstrate the superior performance of greybox models, which combine traditional blackbox algorithms with prior domain knowledge of quantum mechanics, as means of learning underlying quantum circuit distributions of interest. Our results demonstrate how geometric control techniques can be used to both (a) verify the extent to which geometrically synthesised quantum circuits lie along geodesic, and thus time-optimal, routes and (b) synthesise those circuits. Our results are of interest to researchers in quantum control and quantum information theory seeking to combine machine learning and geometric techniques for time-optimal control problems. △ Less

Submitted 7 July, 2020; v1 submitted 19 June, 2020; originally announced June 2020.

Comments: 28 pages, 14 figures. Code and select datasets available at https://github.com/eperrier/quant-geom-machine-learning

Quantum Gram-Schmidt Processes and Their Application to Efficient State Read-out for Quantum Algorithms

Authors: Kaining Zhang, Min-Hsiu Hsieh, Liu Liu, Dacheng Tao

Abstract: Many quantum algorithms that claim speed-up over their classical counterparts only generate quantum states as solutions instead of their final classical description. The additional step to decode quantum states into classical vectors normally will destroy the quantum advantage in most scenarios because all existing tomographic methods require runtime that is polynomial with respect to the state di… ▽ More Many quantum algorithms that claim speed-up over their classical counterparts only generate quantum states as solutions instead of their final classical description. The additional step to decode quantum states into classical vectors normally will destroy the quantum advantage in most scenarios because all existing tomographic methods require runtime that is polynomial with respect to the state dimension. In this work, we present an efficient read-out protocol that yields the classical vector form of the generated state, so it will achieve the end-to-end advantage for those quantum algorithms. Our protocol suits the case that the output state lies in the row space of the input matrix, of rank $r$, that is stored in the quantum random access memory. The quantum resources for decoding the state in $\ell^2$ norm with $ε$ error require $\poly(r,1/ε)$ copies of the output state and $\poly(r, κ^r,1/ε)$ queries to the input oracles, where $κ$ is the condition number of the input matrix. With our read-out protocol, we completely characterise the end-to-end resources for quantum linear equation solvers and quantum singular value decomposition. One of our technical tools is an efficient quantum algorithm for performing the Gram-Schmidt orthonormal procedure, which we believe, will be of independent interest. △ Less

Submitted 30 May, 2022; v1 submitted 14 April, 2020; originally announced April 2020.

Comments: Final version

Journal ref: Physical Review Research 3, 04395 (2021)

Quantum noise protects quantum classifiers against adversaries

Authors: Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, Dacheng Tao, Nana Liu

Abstract: 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 t… ▽ More 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. △ Less

Submitted 20 March, 2020; originally announced March 2020.

Comments: 16 pages, 8 figures

Journal ref: Phys. Rev. Research 3, 023153 (2021)

On Dimension-free Tail Inequalities for Sums of Random Matrices and Applications

Authors: Chao Zhang, Min-Hsiu Hsieh, Dacheng Tao

Abstract: 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. eigenv… ▽ More 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. △ Less

Submitted 8 October, 2019; originally announced October 2019.

Quantum algorithm for finding the negative curvature direction in non-convex optimization

Authors: Kaining Zhang, Min-Hsiu Hsieh, Liu Liu, Dacheng Tao

Abstract: 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 optimiz… ▽ More 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. △ Less

Submitted 17 September, 2019; originally announced September 2019.

Comments: 29 pages, 2 figures

A Quantum-inspired Algorithm for General Minimum Conical Hull Problems

Authors: Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, Dacheng Tao

Abstract: 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 algorith… ▽ More 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. △ Less

Submitted 15 July, 2019; originally announced July 2019.

Journal ref: Phys. Rev. Research 2, 033199 (2020)

Efficient Online Quantum Generative Adversarial Learning Algorithms with Applications

Authors: Yuxuan Du, Min-Hsiu Hsieh, Dacheng Tao

Abstract: 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, w… ▽ More 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. △ Less

Submitted 21 April, 2019; originally announced April 2019.

Report number: Paper superseded by arXiv:2203.07749

Journal ref: Phys. Rev. Lett. 128, 110501 (2022)

The Expressive Power of Parameterized Quantum Circuits

Authors: Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, Dacheng Tao

Abstract: Parameterized quantum circuits (PQCs) have been broadly used as a hybrid quantum-classical machine learning scheme to accomplish generative tasks. However, whether PQCs have better expressive power than classical generative neural networks, such as restricted or deep Boltzmann machines, remains an open issue. In this paper, we prove that PQCs with a simple structure already outperform any classica… ▽ More Parameterized quantum circuits (PQCs) have been broadly used as a hybrid quantum-classical machine learning scheme to accomplish generative tasks. However, whether PQCs have better expressive power than classical generative neural networks, such as restricted or deep Boltzmann machines, remains an open issue. In this paper, we prove that PQCs with a simple structure already outperform any classical neural network for generative tasks, unless the polynomial hierarchy collapses. Our proof builds on known results from tensor networks and quantum circuits (in particular, instantaneous quantum polynomial circuits). In addition, PQCs equipped with ancillary qubits for post-selection have even stronger expressive power than those without post-selection. We employ them as an application for Bayesian learning, since it is possible to learn prior probabilities rather than assuming they are known. We expect that it will find many more applications in semi-supervised learning where prior distributions are normally assumed to be unknown. Lastly, we conduct several numerical experiments using the Rigetti Forest platform to demonstrate the performance of the proposed Bayesian quantum circuit. △ Less

Submitted 28 October, 2018; originally announced October 2018.

Comments: Comments welcomed!

Journal ref: Phys. Rev. Research 2, 033125 (2020)

A Grover-search Based Quantum Learning Scheme for Classification

Authors: Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, Dacheng Tao

Abstract: 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 a… ▽ More 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. △ Less

Submitted 29 May, 2022; v1 submitted 17 September, 2018; originally announced September 2018.

Comments: final version

Journal ref: New J. Phys. 23, 023020 (2021 )