AI Mathematician, this time it's not just for solving practice problems.
Previously, the team led by Professor Liu Yang, Dean of the Tsinghua Institute for AI Industry Research (AIR), released an intelligent agent system for mathematical research—
AIM.
Unlike many previous agents focused on problem-solving, AIM does not merely answer math problems but also attempts to participate in the earlier stages of research work:
It can help researchers explore ideas, organize theorems, generate proof drafts, and hand these over to humans for further review.
Recently, centered on AIM, AIR-Truth Academy joint student Wang Yanqiao and Truth Academy Assistant Professor Liu Jinpeng completed a cutting-edge quantum algorithm research with deep AI involvement—
Sign Embedding Quantum Algorithms for Matrix Equations and Matrix Functions.
This research started from a vague intuition: Could rational approximation become a quantum algorithm design principle?
During the research process, AI first helped human researchers outline candidate paths, from which humans then selected directions, audited assumptions, and fixed derivations. AIM participated in later stages, assisting with theorem organization, proof draft generation, and complexity analysis.
Ultimately, the research team proposed Sign Embedding Quantum Algorithms, resulting in an 84-page quantum algorithm paper.
It can be said that, compared to mainly solving open mathematical problems given by researchers previously, this time, AIM began to participate in the proposal and exploration of research directions.
How was this achieved?
AI's Mathematical Capabilities Are Evolving from "Problem-Solving" to "Research"
In recent years, AI has made continuous progress in mathematical reasoning, algorithm search, conjecture testing, and proof assistance.
Many existing cases primarily address relatively clear-cut tasks: given a proposition to prove or refute, an objective function to optimize, or a search space executable and scorable by a program.
However, in real frontier mathematical research, significant advances often occur before the formal appearance of a theorem.
Researchers might first have a vague intuition, a cross-domain analogy, or a technical preference not yet solidified, then gradually decide what problem it should transform into, what assumptions to adopt, which path to pursue, and ultimately what family of theorems to form.
This stage is often difficult to evaluate with standard answers or a single numerical metric, yet it directly influences the value and direction of the research.
Addressing the question of "can AI assist in problem formation," this research provides a relatively complete observational sample:
AI and AIM were placed in a research loop overseen by human researchers, participating in both exploration and derivation, while also undergoing continuous auditing, revision, and integration.
From a Meta-Idea to an Auditable Theorem Family
Notably, the research did not start from a precisely defined quantum algorithm theorem, but from a macroscopic intuition proposed by a human researcher:
Rational approximation has advantages when dealing with step-type functions, especially the sign function. Could this idea serve as a quantum algorithm design principle?
In the early exploration phase, the researchers expanded this intuition into a set of candidate research directions and comparative dimensions through interaction with a general-purpose AI model.
Subsequently, the human researchers filtered and focused on the "Sign-Embedding" route based on mathematical taste, technical feasibility, and potential contributions.
AIM, as part of the human-AI collaborative research system in later stages, helped organize the selected route into auditable theorem objectives and derivation materials.
The final quantum algorithm paper is 84 pages long. The figure below illustrates the roles played by AI/AIM in the formation of this paper.
It should be noted that the capabilities for route divergence, candidate direction organization, and comparison achieved through early reliance on general AI dialogue have been further systematized into capabilities in the subsequent AIM v2.
In other words, this case not only showcases a specific research process but also reflects AIM's evolution from interactive assistance toward supporting a more complete research workflow.
Human-AI Collaborative Workflow: AI High-Throughput Exploration under Human Value Gates
From an AI research perspective, the focus of this study is not on demonstrating "fully automated mathematical discovery," but on presenting a traceable, auditable, reusable human-AI collaborative process.
The entire process can be summarized in five stages.
Divergent Route Expansion: Human researchers provide a core meta-idea or macroscopic research intuition; AI expands this into multiple candidate problems, technical routes, and cross-domain connections, helping researchers quickly see the surrounding research space.
Human Value Gate: Facing candidate branches generated by AI, human researchers filter and focus based on academic judgment, problem value, and technical feasibility, deciding which directions are worth further investment.
Theorem Formation and Derivation: Once the main route is determined, AIM helps translate high-level ideas into auditable materials such as theorem statements, lemma decomposition, proof drafts, and complexity expressions.
Complexity Audit and Repair: In quantum algorithm research, proving correctness does not automatically ensure sufficient algorithmic contribution; assumptions need to be natural, access models reasonable, and complexity bounds not too loose, all requiring repeated checking. The process of repair, optimization, or reconstruction can continue to leverage AI/AIM's derivation, comparison, and rewriting abilities, but critical judgments and final confirmation must be undertaken by human researchers.
Validation and Integration: All mathematical statements, proofs, assumptions, complexity estimates, and contribution descriptions ultimately need to be verified, selected, rewritten, and integrated by human researchers before entering the final published paper.
Connecting Discovery, Derivation Generation, and Prudent Review
In summary, the significance of AIM is not to replace human mathematicians in independently completing research, but to increase the density of exploration and efficiency of derivation within a human-supervised loop.
AI/AIM can rapidly expand candidate routes, organize connections between related concepts, and generate proofs and complexity drafts for review;
Human researchers are responsible for deciding which routes have research value, which assumptions are acceptable, and which derivations need repair.
This collaborative model makes the research process closer to "high-throughput candidate generation + human value gating + AI-assisted audit/repair + human final integration."
Its advantage lies not in making AI output the final conclusion directly, but in transforming the originally hard-to-exhaust route exploration, connection organization, and local derivation into intermediate materials that are checkable, comparable, and revisable step-by-step.
For AI4Math and AI Scientist research, this also suggests: the feedback signal in theoretical research is often not an experimental score, but mathematical judgment.
The system needs to support long-range memory, route management, assumption tracking, complexity auditing, and counterfactual checking, enabling human researchers to more effectively control direction, detect errors, and solidify final outcomes.
Sign Embedding Quantum Algorithms
As the technical outcome formed through this collaborative process, Wang Yanqiao and Liu Jinpeng proposed "Sign Embedding Quantum Algorithms" for a class of matrix equation and matrix function problems, including Sylvester, Lyapunov, Riccati equations, as well as objects like matrix square roots, inverse square roots, and geometric means.
These problems hold fundamental importance in numerical linear algebra, control theory, dynamical systems, and scientific computing.
For readers not in quantum fields, the core idea of the paper can be understood as: first compressing multiple structured matrix problems into the sign function or sign projection of an extended matrix,
then realizing the corresponding objects through quantum algorithm primitives like rational approximation and shifted inverses. This "embed first, then approximate" route provides a unified organizing framework for several seemingly different problems.
The technical contributions of this quantum paper include: establishing usable assumptions and complexity formulations under more general input conditions (non-normal, non-diagonalizable, etc.);
advancing the output from a single vector state to a matrix block encoding accessible to downstream quantum circuits; and forming a relatively systematic operator-output quantum linear algebra framework through scaling, rebalancing, and complexity auditing of the shifted-inverse implementation layer.
Human Judgment and AI Productivity in Theoretical Research
In summary, this research presents a more realistic way for AI to participate in mathematical research:
AI can help researchers expand routes, organize connections, draft proofs, and conduct preliminary complexity analysis faster, thereby reducing the explicit cost of some foundational derivations and local exploration in theoretical research.
Simultaneously, however, professional judgment and continuous review from researchers remain essential for deciding whether a research direction is worth pursuing deeply, whether assumptions are natural and reasonable, and whether results possess sufficient theoretical value.
As agents can quickly generate large numbers of candidate routes, proof drafts, and technical formulations, the focus of theoretical scientists' work may also shift.
As the cost of tedious derivations is partially reduced, researchers can allocate more energy to direction selection, problem definition, assumption gating, and result auditing.
In other words, discerning "which problems are truly worth researching," and identifying routes that appear plausible but contain hidden conditions, technical flaws, or insufficient contributions will become even more critical skills.
This also provides important implications for AIM's future development. What warrants further strengthening is not just single-point proof or local derivation capabilities, but also systemic capabilities supporting the entire research process:
For example, recording and comparing different research routes, explicitly managing key assumptions, preserving auditable derivation traces, discovering hidden conditions and complexity pitfalls, and supporting researchers in subsequent repair, optimization, and reconstruction with AI assistance.
This case demonstrates that AI's value in frontier theoretical research is gradually extending from local task assistance to a more complete research workflow.
By organizing capabilities like route expansion, connection discovery, proof drafting, and audit feedback, AIM enables AI's generative and deductive abilities to better serve human researchers' directional judgment and mathematical oversight.
Such a collaborative approach offers new possibilities for improving the efficiency of theoretical research and expanding research horizons.
Related Links
AIM System Application Report: From Meta Idea to Advanced Mathematical Discovery: Human-AI Co-Discovery of Sign-Embedding Quantum Algorithms (https://arxiv.org/abs/2606.24899)
Quantum Algorithm Paper: Sign Embedding Quantum Algorithms for Matrix Equations and Matrix Functions (http://arxiv.org/abs/2604.25333)
AIM repo: https://github.com/TheoryFoundry/AIMv2AIM
blog: https://ai-mathematician.net
This article is from WeChat Official Account "QbitAI," author: Tsinghua AIR Team
