Fig. 2: Relationships between the key classical and quantum circuit (complexity) classes considered.

Notably, \({\mathsf{Majority}}\) is in \({{\mathsf{TC}}}^{0}\) but not \({{\mathsf{AC}}}^{0}\). The ISMR family, introduced here, separates constant-depth quantum circuits from biased polynomial threshold circuits (bias \(k=\omega (\log n)\)). The latter class contains \({{\mathsf{AC}}}^{0}\), and can solve \({{\mathsf{NC}}}^{1}\)-complete problems for super-logarithmic biases and input lengths, suggesting non-trivial overlap with \({{\mathsf{TC}}}^{0}\). We also note computational tasks and models of significant practical value, such as neural networks. For example, Large Language Models (LLMs) that use the transformer architecture can be simulated by \({{\mathsf{AC}}}^{0}\) circuits when the attention mechanism is limited in certain natural ways^24,77. Meanwhile, \({{\mathsf{TC}}}^{0}\), the standard for modeling discretized neural networks^20,78, can simulate LLMs with realistic constraints on variable precision and autoregression^25,79,80, in the absence of more complicated elements such as feedback loops. The biased polynomial threshold circuits we study can simulate neural network variants and approximate activation functions controlled by the bias parameter k.