Zhan Yu

Tencent Hunyuan Team, Ao Liu, Botong Zhou et al. · tencent-ai

As Large Language Models (LLMs) rapidly advance, we introduce Hunyuan-TurboS, a novel large hybrid Transformer-Mamba Mixture of Experts (MoE) model. It synergistically combines Mamba's long-sequence processing efficiency with Transformer's superior contextual understanding. Hunyuan-TurboS features an adaptive long-short chain-of-thought (CoT) mechanism, dynamically switching between rapid responses for simple queries and deep "thinking" modes for complex problems, optimizing computational resources. Architecturally, this 56B activated (560B total) parameter model employs 128 layers (Mamba2, Attention, FFN) with an innovative AMF/MF block pattern. Faster Mamba2 ensures linear complexity, Grouped-Query Attention minimizes KV cache, and FFNs use an MoE structure. Pre-trained on 16T high-quality tokens, it supports a 256K context length and is the first industry-deployed large-scale Mamba model. Our comprehensive post-training strategy enhances capabilities via Supervised Fine-Tuning (3M instructions), a novel Adaptive Long-short CoT Fusion method, Multi-round Deliberation Learning for iterative improvement, and a two-stage Large-scale Reinforcement Learning process targeting STEM and general instruction-following. Evaluations show strong performance: overall top 7 rank on LMSYS Chatbot Arena with a score of 1356, outperforming leading models like Gemini-2.0-Flash-001 (1352) and o4-mini-2025-04-16 (1345). TurboS also achieves an average of 77.9% across 23 automated benchmarks. Hunyuan-TurboS balances high performance and efficiency, offering substantial capabilities at lower inference costs than many reasoning models, establishing a new paradigm for efficient large-scale pre-trained models.

Zhongjie Shi, Zhan Yu, Ding-Xuan Zhou

In this paper, we aim at establishing an approximation theory and a learning theory of distribution regression via a fully connected neural network (FNN). In contrast to the classical regression methods, the input variables of distribution regression are probability measures. Then we often need to perform a second-stage sampling process to approximate the actual information of the distribution. On the other hand, the classical neural network structure requires the input variable to be a vector. When the input samples are probability distributions, the traditional deep neural network method cannot be directly used and the difficulty arises for distribution regression. A well-defined neural network structure for distribution inputs is intensively desirable. There is no mathematical model and theoretical analysis on neural network realization of distribution regression. To overcome technical difficulties and address this issue, we establish a novel fully connected neural network framework to realize an approximation theory of functionals defined on the space of Borel probability measures. Furthermore, based on the established functional approximation results, in the hypothesis space induced by the novel FNN structure with distribution inputs, almost optimal learning rates for the proposed distribution regression model up to logarithmic terms are derived via a novel two-stage error decomposition technique.

Zhan Yu, Xuanqiang Zhao, Benchi Zhao et al.

Unitary transformations formulate the time evolution of quantum states. How to learn a unitary transformation efficiently is a fundamental problem in quantum machine learning. The most natural and leading strategy is to train a quantum machine learning model based on a quantum dataset. Although the presence of more training data results in better models, using too much data reduces the efficiency of training. In this work, we solve the problem on the minimum size of sufficient quantum datasets for learning a unitary transformation exactly, which reveals the power and limitation of quantum data. First, we prove that the minimum size of a dataset with pure states is $2^n$ for learning an $n$-qubit unitary transformation. To fully explore the capability of quantum data, we introduce a practical quantum dataset consisting of $n+1$ elementary tensor product states that are sufficient for exact training. The main idea is to simplify the structure utilizing decoupling, which leads to an exponential improvement in the size of the datasets with pure states. Furthermore, we show that the size of the quantum dataset with mixed states can be reduced to a constant, which yields an optimal quantum dataset for learning a unitary. We showcase the applications of our results in oracle compiling and Hamiltonian simulation. Notably, to accurately simulate a 3-qubit one-dimensional nearest-neighbor Heisenberg model, our circuit only uses $96$ elementary quantum gates, which is significantly less than $4080$ gates in the circuit constructed by the Trotter-Suzuki product formula.

Zhan Yu, Hongshun Yao, Mujin Li et al.

Quantum neural networks (QNNs) have emerged as a leading strategy to establish applications in machine learning, chemistry, and optimization. While the applications of QNN have been widely investigated, its theoretical foundation remains less understood. In this paper, we formulate a theoretical framework for the expressive ability of data re-uploading quantum neural networks that consist of interleaved encoding circuit blocks and trainable circuit blocks. First, we prove that single-qubit quantum neural networks can approximate any univariate function by mapping the model to a partial Fourier series. We in particular establish the exact correlations between the parameters of the trainable gates and the Fourier coefficients, resolving an open problem on the universal approximation property of QNN. Second, we discuss the limitations of single-qubit native QNNs on approximating multivariate functions by analyzing the frequency spectrum and the flexibility of Fourier coefficients. We further demonstrate the expressivity and limitations of single-qubit native QNNs via numerical experiments. We believe these results would improve our understanding of QNNs and provide a helpful guideline for designing powerful QNNs for machine learning tasks.

Zhan Yu, Qiuhao Chen, Yuling Jiao et al.

Parameterized quantum circuits (PQCs) have emerged as a promising approach for quantum neural networks. However, understanding their expressive power in accomplishing machine learning tasks remains a crucial question. This paper investigates the expressivity of PQCs for approximating general multivariate function classes. Unlike previous Universal Approximation Theorems for PQCs, which are either nonconstructive or rely on parameterized classical data processing, we explicitly construct data re-uploading PQCs for approximating multivariate polynomials and smooth functions. We establish the first non-asymptotic approximation error bounds for these functions in terms of the number of qubits, quantum circuit depth, and number of trainable parameters. Notably, we demonstrate that for approximating functions that satisfy specific smoothness criteria, the quantum circuit size and number of trainable parameters of our proposed PQCs can be smaller than those of deep ReLU neural networks. We further validate the approximation capability of PQCs through numerical experiments. Our results provide a theoretical foundation for designing practical PQCs and quantum neural networks for machine learning tasks that can be implemented on near-term quantum devices, paving the way for the advancement of quantum machine learning.

Naixu Guo, Zhan Yu, Matthew Choi et al.

Powerful generative artificial intelligence from large language models (LLMs) harnesses extensive computational resources for inference. In this work, we investigate the transformer architecture, a key component of these models, under the lens of fault-tolerant quantum computing. We develop quantum subroutines to construct the building blocks in the transformer, including the self-attention, residual connection with layer normalization, and feed-forward network. As an important subroutine, we show how to efficiently implement the Hadamard product and element-wise functions of matrices on quantum computers. Our algorithm prepares an amplitude encoding of the transformer output, which can be measured for prediction or use in the next layer. We find that the matrix norm of the input sequence plays a dominant role in the quantum complexity. With numerical experiments on open-source LLMs, including for bio-informatics applications, we demonstrate the potential of a quantum speedup for transformer inference in practical regimes.

Yuxuan Du, Xinbiao Wang, Naixu Guo et al.

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.

Zhan Yu, Zhongjie Shi, Ding-Xuan Zhou

We propose a new decentralized robust kernel-based learning algorithm within the framework of reproducing kernel Hilbert spaces (RKHSs) by utilizing a networked system that can be represented as a connected graph. The robust loss function $\huaL_σ$ induced by a windowing function $W$ and a robustness scaling parameter $σ>0$ can encompass a broad spectrum of robust losses. Consequently, the proposed algorithm effectively provides a unified decentralized learning framework for robust regression, which fundamentally differs from the existing distributed robust kernel-based learning schemes, all of which are divide-and-conquer based. We rigorously establish a learning theory and offer comprehensive convergence analysis for the algorithm. We show each local robust estimator generated from the decentralized algorithm can be utilized to approximate the regression function. Based on kernel-based integral operator techniques, we derive general high confidence convergence bounds for the local approximating sequence in terms of the mean square distance, RKHS norm, and generalization error, respectively. Moreover, we provide rigorous selection rules for local sample size and show that, under properly selected step size and scaling parameter $σ$, the decentralized robust algorithm can achieve optimal learning rates (up to logarithmic factors) in both norms. The parameter $σ$ is shown to be essential for enhancing robustness and ensuring favorable convergence behavior. The intrinsic connection among decentralization, sample selection, robustness of the algorithm, and its convergence is clearly reflected.

Zhan Yu, Jun Fan, Zhongjie Shi et al.

In recent years, different types of distributed and parallel learning schemes have received increasing attention for their strong advantages in handling large-scale data information. In the information era, to face the big data challenges {that} stem from functional data analysis very recently, we propose a novel distributed gradient descent functional learning (DGDFL) algorithm to tackle functional data across numerous local machines (processors) in the framework of reproducing kernel Hilbert space. Based on integral operator approaches, we provide the first theoretical understanding of the DGDFL algorithm in many different aspects of the literature. On the way of understanding DGDFL, firstly, a data-based gradient descent functional learning (GDFL) algorithm associated with a single-machine model is proposed and comprehensively studied. Under mild conditions, confidence-based optimal learning rates of DGDFL are obtained without the saturation boundary on the regularity index suffered in previous works in functional regression. We further provide a semi-supervised DGDFL approach to weaken the restriction on the maximal number of local machines to ensure optimal rates. To our best knowledge, the DGDFL provides the first divide-and-conquer iterative training approach to functional learning based on data samples of intrinsically infinite-dimensional random functions (functional covariates) and enriches the methodologies for functional data analysis.

Yifei Chen, Zhan Yu, Chenghong Zhu et al.

The rapid advancement of quantum computing has led to an extensive demand for effective techniques to extract classical information from quantum systems, particularly in fields like quantum machine learning and quantum chemistry. However, quantum systems are inherently susceptible to noises, which adversely corrupt the information encoded in quantum systems. In this work, we introduce an efficient algorithm that can recover information from quantum states under Pauli noise. The core idea is to learn the necessary information of the unknown Pauli channel by post-processing the classical shadows of the channel. For a local and bounded-degree observable, only partial knowledge of the channel is required rather than its complete classical description to recover the ideal information, resulting in a polynomial-time algorithm. This contrasts with conventional methods such as probabilistic error cancellation, which requires the full information of the channel and exhibits exponential scaling with the number of qubits. We also prove that this scalable method is optimal on the sample complexity and generalise the algorithm to the weight contracting channel. Furthermore, we demonstrate the validity of the algorithm on the 1D anisotropic Heisenberg-type model via numerical simulations. As a notable application, our method can be severed as a sample-efficient error mitigation scheme for Clifford circuits.

Zhan Yu, Daniel W. C. Ho, Ding-Xuan Zhou

Regularization schemes for regression have been widely studied in learning theory and inverse problems. In this paper, we study distribution regression (DR) which involves two stages of sampling, and aims at regressing from probability measures to real-valued responses over a reproducing kernel Hilbert space (RKHS). Recently, theoretical analysis on DR has been carried out via kernel ridge regression and several learning behaviors have been observed. However, the topic has not been explored and understood beyond the least square based DR. By introducing a robust loss function $l_σ$ for two-stage sampling problems, we present a novel robust distribution regression (RDR) scheme. With a windowing function $V$ and a scaling parameter $σ$ which can be appropriately chosen, $l_σ$ can include a wide range of popular used loss functions that enrich the theme of DR. Moreover, the loss $l_σ$ is not necessarily convex, hence largely improving the former regression class (least square) in the literature of DR. The learning rates under different regularity ranges of the regression function $f_ρ$ are comprehensively studied and derived via integral operator techniques. The scaling parameter $σ$ is shown to be crucial in providing robustness and satisfactory learning rates of RDR.

Zhan Yu, Daniel W. C. Ho

This paper is concerned with functional learning by utilizing two-stage sampled distribution regression. We study a multi-penalty regularization algorithm for distribution regression under the framework of learning theory. The algorithm aims at regressing to real valued outputs from probability measures. The theoretical analysis on distribution regression is far from maturity and quite challenging, since only second stage samples are observable in practical setting. In the algorithm, to transform information from samples, we embed the distributions to a reproducing kernel Hilbert space $\mathcal{H}_K$ associated with Mercer kernel $K$ via mean embedding technique. The main contribution of the paper is to present a novel multi-penalty regularization algorithm to capture more features of distribution regression and derive optimal learning rates for the algorithm. The work also derives learning rates for distribution regression in the nonstandard setting $f_ρ\notin\mathcal{H}_K$, which is not explored in existing literature. Moreover, we propose a distribution regression-based distributed learning algorithm to face large-scale data or information challenge. The optimal learning rates are derived for the distributed learning algorithm. By providing new algorithms and showing their learning rates, we improve the existing work in different aspects in the literature.

Christopher Thorpe, Feng Li, Zijia Li et al.

This paper presents a novel Coprime Blurred Pair (CBP) model for visual data-hiding for security in camera surveillance. While most previous approaches have focused on completely encrypting the video stream, we introduce a spatial encryption scheme by blurring the image/video contents to create a CBP. Our goal is to obscure detail in public video streams by blurring while allowing behavior to be recognized and to quickly deblur the stream so that details are available if behavior is recognized as suspicious. We create a CBP by blurring the same latent image with two unknown kernels. The two kernels are coprime when mapped to bivariate polynomials in the z domain. To deblur the CBP we first use the coprime constraint to approximate the kernels and sample the bivariate CBP polynomials in one dimension on the unit circle. At each sample point, we factor the 1D polynomial pair and compose the results into a 2D kernel matrix. Finally, we compute the inverse Fast Fourier Transform (FFT) of the kernel matrices to recover the coprime kernels and then the latent video stream. It is therefore only possible to deblur the video stream if a user has access to both streams. To improve the practicability of our algorithm, we implement our algorithm using a graphics processing unit (GPU) to decrypt the blurred video streams in real-time, and extensive experimental results demonstrate that our new scheme can effectively protect sensitive identity information in surveillance videos and faithfully reconstruct the unblurred video stream when two blurred sequences are available.