AlphaEvolve

This repository accompanies the paper

TODO(franrruiz): Add link to paper.

There are the following directories:

• /src contains a minimal implementation of AlphaEvolve with local evaluation.
• /omega contains code and data related to the new upper bound on the matrix multiplication exponent discovered by AlphaEvolve. It includes a Colab notebook for rigorous verification using rational arithmetic, and a level_3/ subdirectory with the gradient-based optimisation code at recursion depth 3:
  • level_3/final_program.py — the optimisation program evolved by AlphaEvolve. It builds a differentiable upper-bound model in JAX/Haiku, runs gradient descent over 32k steps with 32 random seeds, and converts the best solution into a format suitable for verification. Run it directly with python3 final_program.py.
  • level_3/verification.py — a NumPy/SciPy verifier that takes a flat parameter vector produced by the optimiser and evaluates the proven omega upper bound. It uses floating-point arithmetic for fast hill-climbing; the companion Colab notebook uses rational arithmetic for a rigorous proof.

  A requirements.txt listing the pinned Python dependencies and a Dockerfile for containerised execution are also provided in level_3/.

• /maths contains the initial programs, discovered results, and verification code for the mathematical problems on which AlphaEvolve found a solution outperforming the state-of-the-art (see the "Results" section of the paper). There is a subfolder for each of such problems, each containing the files needed to run an AlphaEvolve experiment and a Colab notebook to verify the found construction.
• /matmul_tensor_decompositions contains Colab notebooks with the decompositions of matrix multiplication tensors across the matrix sizes for which the gradient-based optimization method from AlphaEvolve found a solution outperforming the state-of-the-art, along with the code to verify their correctness. Specifically, there are two Colabs: one for the standard tensor rank decomposition, and another one for the support rank.

Installation

For the folders /maths and /matmul_tensor_decompositions, no installation is required. The provided notebooks can be opened and run in Google Colab.

For the folder /src, these are the installation instructions:

1. Install docker (required for program evaluation): https://www.docker.com/get-started/
2. (optional) Create venv:

sudo apt-get install virtualenv python3-venv
virtualenv -p python3.11 alpha_evolve
source alpha_evolve/bin/activate

1. Install pip dependencies:

pip install -r src/requirements.txt

1. Set Gemini API key:

export GEMINI_API_KEY=<your_api_key>

Usage

• /omega: The Colab notebook to verify the new upper bound on the matrix multiplication exponent can be opened via: Open In Colab. Note: To verify our best omega upper bound, you must download the clean_level4.npz.xz file from this repository to your local machine, and then upload it when prompted by the Colab notebook.
• /maths: The Colab notebook to verify the found solution for each problem can be opened via:
  • Circle packing: Open In Colab.
  • Erdös minimum overlap: Open In Colab.
  • First autocorrelation inequality: Open In Colab.
  • Second autocorrelation inequality: Open In Colab.
  • Third autocorrelation inequality: Open In Colab.
  • Heilbronn problem for convex regions: Open In Colab.
  • Heilbronn problem for triangles: Open In Colab.
  • Hexagon packing: Open In Colab.
  • Kissing numbers: Open In Colab.
  • Max to min ratio: Open In Colab.
  • Sums and differences: Open In Colab.
  • Uncertainty principles (Hermite polynomials approach): Open In Colab
  • Uncertainty principles (Laguerre polynomials approach): Open In Colab
  • Extremal girth 5: Open In Colab
  • Shannon capacity of cycle graphs: Open In Colab
  • Cap sets: Open In Colab
• /matmul_tensor_decompositions: There are two Colab notebooks:
  • Standard rank: Open In Colab.
  • Support rank: Open In Colab.
• /src: We provide an example launch command (for the circle packing problem) below:

python run.py --problem_config=./data/problem_config/circle_packing.yaml

DISCLAIMER: AlphaEvolve involves the execution of code generated by Large Language Models (LLMs). LLM-generated code is intrinsically unpredictable and should be handled as untrusted code. In this version of AlphaEvolve, this code is run in a Docker sandbox as an isolation measure. However, despite sandboxing, the potential for unintended behavior, resource misuse, or security vulnerabilities cannot be fully eliminated. All users should be aware of these inherent risks and understand that they use this software at their own discretion. Google DeepMind shall not be liable for any claims, damages, or other liability arising from the execution of LLM-generated code.

Citing this work

If you use the code or data in this package, please cite:

TODO(franrruiz): Add citation.

License and disclaimer

Copyright 2026 Google LLC

All software is licensed under the Apache License, Version 2.0 (Apache 2.0); you may not use this file except in compliance with the Apache 2.0 license. You may obtain a copy of the Apache 2.0 license at: https://www.apache.org/licenses/LICENSE-2.0

All other materials are licensed under the Creative Commons Attribution 4.0 International License (CC-BY). You may obtain a copy of the CC-BY license at: https://creativecommons.org/licenses/by/4.0/legalcode

Unless required by applicable law or agreed to in writing, all software and materials distributed here under the Apache 2.0 or CC-BY licenses are distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the licenses for the specific language governing permissions and limitations under those licenses.

This is not an official Google product.