Scalar multiplication complexity of 4×4 matrices over ℂ
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What is the bilinear tensor rank of multiplication of two 4×4 matrices over the complex numbers? This counts the scalar multiplications in a bilinear decomposition of the matrix-multiplication tensor. Two recursive levels of Strassen’s algorithm give rank 49; products in other commutative arithmetic models are a different cost measure.
Why it matters and what counts as progress
Why it matters: The fixed-size tensor-rank problem tests how much the multiplication count can be reduced over ℂ, independently of algorithms over finite fields or measurements on particular hardware.
What counts as progress: Give an exact lower-rank bilinear decomposition or prove a stronger lower bound over ℂ. A fixed-dimension improvement alone does not establish a new asymptotic exponent or a universal runtime speedup; Waksman-type products in another model are not a tensor-rank baseline.
How the problem developed
- 1969 · Algorithm [1]
Volker Strassen Strassen’s seven-product 2×2 algorithm yields the standard 49-multiplication recursive upper bound for 4×4 matrix multiplication.
- 14 May 2025 · Algorithm [2]
Google DeepMind Announced an AlphaEvolve-discovered rank-48 decomposition of the 4×4 matrix-multiplication tensor over the complex numbers.
- 16 Jun 2025 · Technical report [3]
AlphaEvolve authors Published the technical report detailing the rank-48 complex decomposition and its verification.
AI contribution
AlphaEvolve: A coding agent for scientific and algorithmic discovery
The AlphaEvolve report gives a rank-48 decomposition of the 4×4 matrix-multiplication tensor over the complex numbers, yielding a bilinear algorithm using 48 scalar multiplications.
Sources
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