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Is anything known about the size of $\Sigma\Pi\Sigma$ (or other constant depth) circuits for the product of $k$ matrices?

Trivial upper bounds (up to small factors) are:

  • if $k=2$, then there are $\Sigma\Pi\Sigma$ circuits of size $n^{\omega}$ (where $\omega$ is the matrix multiplication exponent.)
  • if $k>2$, then there are $\Sigma\Pi\Sigma$ circuits of size $n^{k+1}$.
  • if $k>2$, then there are $O(log(k))$-depth circuits of size $n^{\omega}$.

Is it possible to construct circuits of size $n^{\omega}$ for every $k$, for some fixed depth? Is it even possible to beat the trivial construction?

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I am not sure about specifically depth-three lower bounds, but there has been a lot of depth-4 (and 5) lower bounds, usually assuming other constraints as well. For instance (and without any claim of exhaustivity):

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