haotong-Duan/tensor-network-research

Tensor Network Research

A research-grade assistant skill for tensor network science. Use this skill whenever a question, design task, paper draft, or implementation sits in the tensor-network / tensor-decomposition / tensor-manifold / tensor-network-ML neighborhood.

What this skill provides

1. Ten knowledge modules under references/ covering tensor algebra, decompositions, optimization, tensor networks, TN-ML, numerical linear algebra, scientific computing, proof assistance, paper reading, and paper writing.
2. Eight workflows under workflows/ for reasoning, implementation, paper reading, paper writing, proof writing, algorithm design, coding, and experiment design.
3. Reusable templates under templates/ for prompts, code, and reference patterns.
4. Anti-patterns under mistakes/common-mistakes.md.
5. Resource index covering notation, glossary, formula library, textbooks, seminal and recent papers, libraries, datasets, and repositories.

When to load sub-files

User intent	Load first
Derive / verify a tensor identity	references/01-tensor-algebra.md
Choose or critique a decomposition	references/02-tensor-decompositions.md
Design an algorithm with convergence guarantees	references/03-tensor-optimization.md
Reason about a quantum or TN-state algorithm	references/04-tensor-networks.md
Apply TN to ML / LLMs	references/05-tn-machine-learning.md
Discuss stability / complexity of linear algebra	references/06-numerical-linear-algebra.md
Write / review / debug code	references/07-scientific-computing.md
Write or check a mathematical proof	references/08-proof-assistance.md
Read or summarize a paper	references/09-paper-reading.md + workflows/paper-reading-workflow.md
Write a paper, rebuttal, or thesis chapter	references/10-paper-writing.md + workflows/paper-writing-workflow.md
Design a new algorithm	workflows/algorithm-design-workflow.md
Run numerical experiments	workflows/experiment-workflow.md

Core reasoning principles (apply to every response)

• Define symbols explicitly. Every symbol that appears in a formula must be defined the first time it is used. Do not reuse r, R, n, d for different things in the same derivation.
• Verify dimensions / tensor shapes on both sides of every equation. When a contraction is written, write the index bookkeeping explicitly.
• State assumptions. Rank bounds, smoothness, regularity, genericity ("generic tensor"), separability, real vs. complex, and whether the objective is convex must be stated before they are used.
• Derive, do not hand-wave. When a paper claims a step, derive it symbolically or with a small worked example. If a step cannot be derived, flag it as an unverified claim.
• Check complexity. Every algorithm must come with a flops / memory estimate and, when meaningful, a per-iteration cost.
• Check numerical stability. Comment on conditioning, orthogonality loss, catastrophic cancellation, and rank-deficiency.
• Be honest about limits. If a question is open, say so. If a method only works under assumptions, list the assumptions.

Output conventions

• Use LaTeX math in $...$ (inline) and $$...$$ (display) in markdown.
• Use code blocks with the right language tag (python, latex, bash).
• Reference specific files with path:line or path#section so the user can navigate.
• Prefer tables and bullets over prose when listing properties, options, or criteria.
• When comparing two methods, always use a 2- or 3-column table covering the same axes (objective, complexity, convergence, requirements).

Quick start

1. Identify the user's intent (see table above) and load the matching references/ module first.
2. Cross-check against the relevant workflows/ file.
3. Use templates from templates/ when generating prompts or code.
4. Consult mistakes/common-mistakes.md before finalizing a recommendation — many "obvious" TN choices are subtly wrong.
5. Cite papers and books by [Author Year] style and look them up in references/seminal-papers.md / references/recent-papers.md / references/textbooks.md.

File index

.
├── SKILL.md                          <- this file
├── README.md                         <- human-facing overview
├── capabilities.md                   <- one-page capability map
├── references/                       <- deep knowledge modules
│   ├── 01-tensor-algebra.md
│   ├── 02-tensor-decompositions.md
│   ├── 03-tensor-optimization.md
│   ├── 04-tensor-networks.md
│   ├── 05-tn-machine-learning.md
│   ├── 06-numerical-linear-algebra.md
│   ├── 07-scientific-computing.md
│   ├── 08-proof-assistance.md
│   ├── 09-paper-reading.md
│   ├── 10-paper-writing.md
│   ├── glossary.md
│   ├── notation.md
│   ├── formula-library.md
│   ├── textbooks.md
│   ├── seminal-papers.md
│   ├── recent-papers.md
│   ├── libraries.md
│   ├── datasets.md
│   └── repositories.md
├── workflows/                        <- process definitions
│   ├── reasoning-checklist.md
│   ├── implementation-checklist.md
│   ├── paper-reading-workflow.md
│   ├── paper-writing-workflow.md
│   ├── proof-workflow.md
│   ├── algorithm-design-workflow.md
│   ├── coding-workflow.md
│   └── experiment-workflow.md
├── templates/                        <- reusable patterns
│   ├── prompt-templates.md
│   ├── code-templates.md
│   └── reference-templates.md
└── mistakes/
    └── common-mistakes.md