{ "id": "AIBD-0076", "title": "AlphaEvolve finds a 48-multiplication algorithm for 4x4 complex matrices; Dumas, Pernet and Sedoglavic derive a rational-coefficient variant from its decomposition", "summary": "AlphaEvolve found a 48-multiplication 4x4 complex matrix algorithm (May 2025); Dumas, Pernet and Sedoglavic derived rational variants from its decomposition", "date_observed": "2025-05-14", "date_basis": "unassessed", "date_precision": "unassessed", "benefit_status": null, "source_collection": null, "source_entry": null, "account": "On 14 May 2025 Google DeepMind announced AlphaEvolve, a Gemini-powered evolutionary coding agent. DeepMind reported that it had found an algorithm multiplying two 4x4 complex-valued matrices with 48 scalar multiplications. This improves on the 49 obtained from two recursion levels of Strassen's 1969 algorithm. On 16 June 2025 Jean-Guillaume Dumas, Clément Pernet and Alexandre Sedoglavic (CASC and CRIStAL, France), researchers independent of DeepMind, posted a preprint. It presents a 48-multiplication algorithm with only rational coefficients, removing the need for complex arithmetic. They state that it was derived from AlphaEvolve's algorithm under the action of an isotropy that projects the decomposition onto the field of rational numbers. The derivation is therefore not independent of the AI result. The preprint contains no explicit verification statement. On 19 March 2026 the same authors posted a more accurate variant of that rational algorithm, valid over any ring containing an inverse of 2. These derived variants corroborate and extend the rank-48 bound. Journal publication of the preprints is unconfirmed.", "category": "benefit created", "unexpected": false, "purpose": "Evolve and verify algorithms and constructions for open mathematical and computational problems", "system": "AlphaEvolve", "model_version": "AlphaEvolve (May 2025), Gemini-powered", "developer": "DeepMind", "deployer": "DeepMind", "location": null, "domain": "Mathematics", "segment": "Algebraic complexity and matrix multiplication", "benefit_type": "Improved Knowledge Discovery", "taxonomy_note": null, "registry_ids": {}, "ratings": [ { "group": "organizational", "depth": 1, "reason": "Small: a research result for the developer. DeepMind's blog reports improvements on about 20% of the open problems tested" }, { "group": "societal", "depth": 1, "reason": "Small: a one-multiplication improvement for 4x4 matrices, since taken up by independent researchers in two preprints" } ], "ai_role": { "status": "assessed", "statement": "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." }, "evidence": [ { "kind": "press release", "citation": "Google DeepMind (2025). AlphaEvolve: A Gemini-powered coding agent for designing advanced algorithms. Google DeepMind blog.", "title": "AlphaEvolve: A Gemini-powered coding agent for designing advanced algorithms", "url": "https://deepmind.google/discover/blog/alphaevolve-a-gemini-powered-coding-agent-for-designing-advanced-algorithms/", "date": "2025-05-14", "written_by": "makers", "quote": "AlphaEvolve's procedure found an algorithm to multiply 4x4 complex-valued matrices using 48 scalar multiplications, improving upon Strassen's 1969 algorithm that was previously known as the best in this setting." }, { "kind": "preprint", "citation": "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).", "title": "A non-commutative algorithm for multiplying 4x4 matrices using 48 non-complex multiplications", "url": "https://arxiv.org/abs/2506.13242", "date": "2025-06-16", "written_by": "independent", "quote": "We propose an algorithm in 48 multiplications with only rational coefficients, hence removing the complex number requirement. It was derived from the latter one, under the action of an isotropy which happen to project the algorithm on the field of rational numbers." }, { "kind": "preprint", "citation": "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.", "title": "A more accurate rational non-commutative algorithm for multiplying 4x4 matrices using 48 multiplications", "url": "https://arxiv.org/abs/2603.18699", "date": "2026-03-19", "written_by": "independent", "quote": "We propose a more accurate variant of an algorithm for multiplying 4x4 matrices using 48 multiplications over any ring containing an inverse of 2." } ], "replication": { "produced_by": [ "Evolutionary search over candidate programs generated by Gemini with automatic evaluation", "A rank-48 decomposition target for the 4x4 matrix multiplication tensor" ], "must_hold": [ "An automatic reviewer scores every candidate algorithm", "Independent researchers can reproduce and extend the decomposition" ], "echoes": [ { "date": "2026-03-19", "setting": "More accurate rational variant derived from AlphaEvolve's decomposition by Dumas, Pernet and Sedoglavic", "model_version": null, "outcome": "recurred", "source": "https://arxiv.org/abs/2603.18699" } ] }, "paired_incident": { "database": "AI Incident Database", "identifier": "1673", "title": "Google Gemini 3.5 Coding Agent Reportedly Caused Production Portal Outage and Generated Fabricated Recovery Records", "url": "https://incidentdatabase.ai/cite/1673", "severity": null, "group": "organizational", "basis": "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." }, "headline": "AlphaEvolve finds 48-multiplication 4x4 complex matrix algorithm", "metric": "4x4 complex-valued matrices with 48 scalar multiplications, improving on the 49", "comparator": { "against": "the previously best known algorithm for 4x4 matrices over the complex numbers, two recursion levels of Strassen's 1969 algorithm", "before": "49 (using two recursion levels of Strassen's algorithm)", "after": "48", "basis": "measured", "quote": "Specifically, the number of scalar multiplications required to multiply two 4x4 matrices was reduced in \\cite{Fawzi:2022aa} from 49 (using two recursion levels of Strassen's algorithm) to 47 in characteristic 2, and more recently to 48 in \\cite{alphaevolve} over the complex numbers.", "url": "https://arxiv.org/abs/2506.13242" }, "later": { "status": "persisted", "date": "2025-06-16", "note": "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.", "quote": "We propose an algorithm requiring 48 multiplications that uses only rational coefficients, thereby removing the requirement for complex-number arithmetic, and making this algorithm valid over any ring except those of characteristic 2.", "url": "https://arxiv.org/abs/2506.13242" }, "value": null, "developer_country": "United Kingdom", "deployer_country": "United Kingdom", "unrated_because": null, "brief_notes": null, "brief": null, "revisions": [ { "date": "2026-10-20", "change": "Record drafted from the cited sources.", "source": null, "editor": "AIBD" }, { "date": "2026-10-20", "change": "Paired with AI Incident Database incident 1673 (medium confidence); severity not given by the source.", "source": "https://incidentdatabase.ai/cite/1673", "editor": "AIBD" }, { "date": "2026-10-20", "change": "Reworded the title, summary, account, AI-role statement and echo setting so that the Dumas-Pernet-Sedoglavic rational algorithm is described as derived from AlphaEvolve's decomposition by an isotropy (corroborating, not independently verifying, the rank-48 bound), and replaced the second evidence quotation with the v1 abstract wording of 16 June 2025, because the preprint itself states that its algorithm was derived from AlphaEvolve's and the record had quoted the revised v7 abstract under the v1 date.", "source": "https://arxiv.org/abs/2506.13242v1", "editor": "AIBD" }, { "date": "2026-10-21", "change": "Added what happened since from the cited sources.", "source": "https://arxiv.org/abs/2506.13242", "editor": "AIBD" }, { "date": "2026-10-21", "change": "Country of the developer and the deployer added from the organization's own site or a registry.", "source": "https://en.wikipedia.org/wiki/Google_DeepMind; https://en.wikipedia.org/wiki/Google_DeepMind", "editor": "AIBD" }, { "date": "2026-10-21", "change": "Prose pass: a rating reason, account, the system's part rewritten in shorter sentences with abbreviations expanded; facts, figures and quotations unchanged.", "source": null, "editor": "AIBD" }, { "date": "2026-10-22", "change": "Organization name standardized to DeepMind.", "source": null, "editor": "AIBD" } ], "confidence": "reported", "shade": "medium", "claim_status": "Independently reported", "claim_basis": "Sources: two independent preprints and the makers' press release. 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