arXiv subjects

Haigang Zhou

Publications and source records attributed to Haigang Zhou.

6 recordsLinked to original sources

arXiv2026

Dimension-Free Approximate Tensorization of Quantum Hypercontractivity for Qudit Depolarizing Semigroups

We prove approximate tensorization for hypercontractivity and logarithmic-Sobolev constants for a class of primitive reversible quantum Markov semigroups satisfying the positive off-diagonal scaling (PODS) condition. This class includes qubit examples and generalized depolarizing semigroups with respect to full-rank states in arbitrary finite dimensions. For any such semigroup $(\Phi_t)_{t\ge 0}$ and every tensor power $n$, we show that the log-Sobolev constant of the product semigroup $\Phi_t^{\otimes n}$ is at least $2/(3\ln 2)\approx 0.96$ times the log-Sobolev constant of the single-site semigroup $\Phi_t$, independently of $n$ and the local dimension $d$. The proof first establishes an exact tensorization of the $(q,2)$-hypercontractive inequality for integer $q$, in particular $q=3$, and then extends the estimate to all real $q>2$ by complex interpolation; the standard implication from hypercontractivity to logarithmic-Sobolev inequalities yields the stated almost tensorization result. In the qubit case, we further prove exact tensorization for primitive reversible PODS semigroups and obtain sharp $(q,2)$-hypercontractivity estimates for generalized qubit depolarizing channels.

quant-ph↗

arXiv2026

The MiniMax-M2 Series: Mini Activations Unleashing Max Real-World Intelligence

We introduce the MiniMax-M2 series, a family of Mixture-of-Experts language models built around the principle that mini activations can unleash maximum real-world intelligence. The flagship M2 contains 229.9B total parameters with only 9.8B activated per token. Designed end-to-end for agentic deployment, the M2 series rests on three components: (i) agent-driven data pipelines producing large-scale, verifiable trajectories across agentic coding and agentic cowork, each grounded in an executable workspace and an artifact-aligned reward; (ii) Forge, a scalable agent-native RL system that adapts to long-horizon agent trajectories, paired with windowed-FIFO scheduling, prefix-tree merging, inference optimization, and a clean training-inference-agent decoupling that supports both white-box and black-box agents; (iii) the latest M2.7 checkpoint takes an early step toward self-evolution -- autonomously debugging training runs and modifying its own scaffold. Across M2 through M2.7, this combination translates a mini-activation footprint into frontier-tier performance on agentic coding, deep search, office-task, and reasoning benchmarks.

cs.AI↗

arXiv2025

MiniMax-M1: Scaling Test-Time Compute Efficiently with Lightning Attention

We introduce MiniMax-M1, the world's first open-weight, large-scale hybrid-attention reasoning model. MiniMax-M1 is powered by a hybrid Mixture-of-Experts (MoE) architecture combined with a lightning attention mechanism. The model is developed based on our previous MiniMax-Text-01 model, which contains a total of 456 billion parameters with 45.9 billion parameters activated per token. The M1 model natively supports a context length of 1 million tokens, 8x the context size of DeepSeek R1. Furthermore, the lightning attention mechanism in MiniMax-M1 enables efficient scaling of test-time compute. These properties make M1 particularly suitable for complex tasks that require processing long inputs and thinking extensively. MiniMax-M1 is trained using large-scale reinforcement learning (RL) on diverse problems including sandbox-based, real-world software engineering environments. In addition to M1's inherent efficiency advantage for RL training, we propose CISPO, a novel RL algorithm to further enhance RL efficiency. CISPO clips importance sampling weights rather than token updates, outperforming other competitive RL variants. Combining hybrid-attention and CISPO enables MiniMax-M1's full RL training on 512 H800 GPUs to complete in only three weeks, with a rental cost of just $534,700. We release two versions of MiniMax-M1 models with 40K and 80K thinking budgets respectively, where the 40K model represents an intermediate phase of the 80K training. Experiments on standard benchmarks show that our models are comparable or superior to strong open-weight models such as the original DeepSeek-R1 and Qwen3-235B, with particular strengths in complex software engineering, tool utilization, and long-context tasks. We publicly release MiniMax-M1 at https://github.com/MiniMax-AI/MiniMax-M1.

cs.CL↗

arXiv2024

Type number for orders of level (N_1,N_2)

We establish an explicit formula for the type number of quaternion orders of level $(N_1, N_2)$, where $N_1 = p_1^{2u_1+1} \cdots p_w^{2u_w+1}$ (with $u_i \geq 0$ and $w$ odd) and $\gcd(N_1, N_2) = 1$. Our main result generalizes Pizer's work on Eichler orders (where $N_1$ is squarefree) and Boyd's formula (where $N_1 = p^{2u+1}$) to the general case with arbitrary prime powers in $N_1$. The proof introduces a generalization of the modified Hurwitz class number $H^{(N_1,N_2)}(D)$, originally defined by Li, Skoruppa and the second author for squarefree levels. Through a bijection between quaternion orders and ternary quadratic forms, we express the type number as a weighted sum of representation numbers, which we evaluate explicitly via the Siegel-Weil formula and local density computations. We compute type numbers for all levels with $N_1 N_2 \leq 100$ and correct four entries in Boyd's 1994 table. As a further application, we classify all 27 pairs $(N_1, N_2)$ having type number $1$, extending the list of 9 squarefree pairs found by Boylan, Skoruppa and the second author.

math.NT↗

arXiv2024

The classification and representations of positive definite ternary quadratic forms of level 4N

Classifications and representations are two main topics in the theory of quadratic forms. In this paper, we consider these topics of ternary quadratic forms. For a given squarefree integer $N$, first we give the classification of positive definite ternary quadratic forms of level $4N$ explicitly. Second, we give explicit formulas of the weighted sum of representations over each class in every genus of ternary quadratic forms of level $4N$, which are involved with modified Hurwitz class number. In the proof of the main results, we use the relations among ternary quadratic forms, quaternion algebras, and Jacobi forms. As a corollary, we get the formula for the class number of positive ternary quadratic forms of level $4N$. As applications, we derive an explicit base of Eisenstein series space of modular forms of weight $3/2$ and level $4N$, and give new proofs of some interesting identities involving representation number of ternary quadratic forms.

math.NT↗

arXiv2022

The rise of digital finance: Financial inclusion or debt trap

This study focuses on the impact of digital finance on households. While digital finance has brought financial inclusion, it has also increased the risk of households falling into a debt trap. We provide evidence that supports this notion and explain the channel through which digital finance increases the likelihood of financial distress. Our results show that the widespread use of digital finance increases credit market participation. The broadened access to credit markets increases household consumption by changing the marginal propensity to consume. However, the easier access to credit markets also increases the risk of households falling into a debt trap.

econ.GN↗