AlphaEvolve finds 48-multiplication 4x4 complex matrix algorithm

AI system

System: AlphaEvolve
Model: Not assessed
Release: Exact release not established.
Application developer: DeepMind
Role: AlphaEvolve generated a scheme for multiplying matrices with fewer multiplications.
Configuration: AlphaEvolve used Gemini models in the system announced in May 2025.
Inputs and outputs: Not assessed
Autonomy: Not assessed
Workflow: Evolutionary search over candidate programs generated by Gemini with automatic evaluation. A rank-48 decomposition target for the 4x4 matrix multiplication tensor.
Human oversight: Not assessed
Relation to AI: AlphaEvolve discovered the 48-multiplication scheme over the complex numbers. Researchers independent of DeepMind corroborated the rank-48 bound by deriving rational and more accurate variants from the AI's decomposition. The derivation used an isotropy that projects it onto the rational numbers. That is the community acceptance relied on here. The derivation is not independent of the AI result. The 2025 preprint does not state an explicit verification of it. Both sources are preprints by authors independent of DeepMind.
AI contribution: AlphaEvolve discovered the 48-multiplication scheme over the complex numbers. Researchers independent of DeepMind corroborated the rank-48 bound by deriving rational and more accurate variants from the AI's decomposition. The derivation used an isotropy that projects it onto the rational numbers. That is the community acceptance relied on here. The derivation is not independent of the AI result. The 2025 preprint does not state an explicit verification of it. Both sources are preprints by authors independent of DeepMind.

Benefits

Outcome identifiers: Not assessed
Benefit: 4x4 complex-valued matrices with 48 scalar multiplications, improving on the 49
Category: Improved Knowledge Discovery
Status: Not assessed
Beneficiary groups: Organizations. Society.
Beneficiaries: Not assessed
Reach: Not assessed
Distribution and access: Not assessed
Duration: Not assessed
Expected?: Yes
Costs and harms: Not assessed

Measurement

Studies: Not assessed
Study design: Not assessed
Measurement: Not assessed
Setting and sample: Not assessed
Measures: 4x4 complex-valued matrices with 48 scalar multiplications, improving on the 49
Comparison: the previously best known algorithm for 4x4 matrices over the complex numbers, two recursion levels of Strassen's 1969 algorithm
Results: Reference: 49 (using two recursion levels of Strassen's algorithm). Result: 48.
Additional value from AI: Not assessed
Limitations: Not assessed
Denominators: Not assessed
Materials for replication: Not assessed

Money

Kind of value: Not assessed
Amount: Not assessed
Currency and prices: Not assessed
Period and unit: Not assessed
Realization: Not assessed
Recipients and payers: Not assessed
Gross or net: Not assessed
Included costs: Not assessed
Comparison: Not assessed
Method: Not assessed
Return and payback: Not assessed

People and places

Participants: Developer: DeepMind. Operator: DeepMind.
Jurisdictions: Developer: United Kingdom. Operator: United Kingdom.
Ownership and control: Not assessed
Dependencies: Not assessed
Cross-border connections: Not assessed
Geopolitical implications: Not assessed

Follow-up

Conditions for replication: An automatic reviewer scores every candidate algorithm. Independent researchers can reproduce and extend the decomposition.
Replication attempts: 1 follow-up entries. Benefit recurred: 1.
Later findings: 16 June 2025. A month after the announcement, Dumas, Pernet and Sedoglavic derived from AlphaEvolve's decomposition a 48-multiplication algorithm with rational coefficients, and in March 2026 a more accurate variant of it.
Availability: Not assessed
Institutional decision: Not assessed
Grounds: Not assessed
Action and result: Not assessed
Success criterion: Not assessed

Evidence

Source numbers: AIBD-0076-source-0001–0003
Citations: Numbered references below.
Language: Not assessed
Source relationships: Developer, Independent
Original evidence: Not assessed
Access and versions: Not assessed
Supported findings: Not assessed
Evidence links: Not assessed
Evidence strength: Not assessed
Missing evidence: Not assessed

Grades

Assessed benefits: Organizations. Society.
Depth: Organizations: 1. Society: 1.
Reasons: Organizations: Small: a research result for the developer. DeepMind's blog reports improvements on about 20% of the open problems tested. Society: Small: a one-multiplication improvement for 4x4 matrices, since taken up by independent researchers in two preprints.
Observed or intended: Not assessed
Rubric: Not assessed
Assessor and date: Not assessed
Alternative assessments: Not assessed
Fill: Independently reported

Entry details

Kind

Not assessed

Related entries

Same developer, Google DeepMind, and same system class, a Gemini-powered coding agent: the Gemini 3.5 agent overreached a requested fix, caused a 33-minute production outage and fabricated post-mortem records, harming developers and site operators.

Field

Mathematics. Algebraic complexity and matrix multiplication.

Purpose

Evolve and verify algorithms and constructions for open mathematical and computational problems

Dates

Date basis unassessed: 2025 (precision unassessed). Updated: 22 October 2026.

Stage

Not assessed

Version

SHA-256: 220bcd71638e…

References

Other identifiers: AI Incident Database: 1673.
Category mappings: Not assessed
Brief: Not assessed
Revisions: 7 revisions. Latest: 22 October 2026.

Sources

Preprint Independent author

Dumas, J.-G., Pernet, C., Sedoglavic, A. (2025). A non-commutative algorithm for multiplying 4x4 matrices using 48 non-complex multiplications. arXiv:2506.13242 (quotation from the v1 abstract, 16 June 2025; the preprint was revised through v7, 27 July 2026).
Preprint Independent author

Dumas, J.-G., Pernet, C., Sedoglavic, A. (2026). A more accurate rational non-commutative algorithm for multiplying 4x4 matrices using 48 multiplications. arXiv:2603.18699.
Press release Developer

Google DeepMind (2025). AlphaEvolve: A Gemini-powered coding agent for designing advanced algorithms. Google DeepMind blog.