Matrix rigidity is a notion put forth by Valiant [Val77] as a means for proving arithmetic circuit lower bounds. A matrix is rigid if it is far, in Hamming distance, from any low rank matrix. Despite decades of efforts, no explicit matrix rigid enough to carry out Valiant’s plan has been found. Recently, Alman and Williams [AW17] showed, contrary to common belief, that the 2 × 2 Hadamard matrix...

The vast majority of quantum states and unitaries have circuit complexity exponential in the number qubits. In a similar vein, most them also minimum description length, which makes it difficult to pinpoint examples complexity. this work, we construct constant length but We provide infinite families such that each element requires an two-qubit gates be generated exactly from product where same ...

In the present note we show that for any constant k ∈ N an arbitrary Boolean circulant matrix can be implemented via modulo 2 rectifier circuit of depth 2k − 1 and complexity O ( n1+1/k ) , and also via circuit of depth 2k and complexity O (

Abstract This paper analyzes three formal models of Transformer encoders that differ in the form their self-attention mechanism: unique hard attention (UHAT); generalized (GUHAT), which generalizes UHAT; and averaging (AHAT). We show UHAT GUHAT Transformers, viewed as string acceptors, can only recognize languages complexity class AC0, recognizable by families Boolean circuits constant depth po...

Model reduction is a popular approach for incorporating detailed physical effects into high level simulations. In this paper we present a simple method for automatically extracting macromodels of nonlinear circuit with time-varying operating points. The models we generate are truly "reduced", meaning that the complexity of macromodel evaluation is not strongly dependent on the size or complexit...

Semiring complexity is the version of arithmetic circuit complexity that allows only two operations: addition and multiplication. We show that semiring complexity of a Schur polynomial sλ(x1, . . . , xk) labeled by a partition λ = (λ1 ≥ λ2 ≥ · · · ) is bounded by O(log(λ1)) provided the number of variables k is fixed.

Proving super-polynomial lower bounds against depth-2 threshold circuits of the form THR ◦THR is a well-known open problem that represents a frontier of our understanding in boolean circuit complexity. By contrast, exponential lower bounds on the size of THR◦MAJ circuits were shown by Razborov and Sherstov [31] even for computing functions in depth-3 AC. Yet, no separation among the two depth-2...