Table 3 Leading-order scaling for various properties of PMMs and a fixed-width MLP

Quantity	In terms of…	MLP	AE-PMM	AO-PMM
Non-analytic points, ξ	Architecture hyperparameters	lm	n²	n²
	Inference complexity χ	χ/m	χ/q	χ/q
	Trainable parameters Σ	Σ/m	Σ/p	\(\Sigma /\left(p+q{r}^{2}\right)\)
Inference complexity, χ	Architecture hyperparameters	lm²	qn²	qr²n²
	Non-analytic points ξ	mξ	qξ	qr²ξ
	Trainable parameters Σ	Σ	Σ	Σ
Trainable parameters, Σ	Architecture hyperparameters	lm²	pn²	\(\left(p+q{r}^{2}\right){n}^{2}\)
	Non-analytic points ξ	mξ	pξ	\(\left(p+q{r}^{2}\right)\xi\)
	Inference complexity χ	χ	χ	χ

The two constructed model PMMs considered in this work are shown: the affine eigenvalue PMM (AE-PMM, Methods section “Eigenvalue and Eigenstate Observable Emulation”) and the affine observable PMM (AO-PMM, Methods section “Regression and Classification PMMs”). All models are considered to have p input features and q output values. Each of the l hidden layers of the MLP has m neurons. The size of the matrices in the PMMs is n × n and the number of eigenvectors used in the AO-PMM is denoted by r. We assume that l ≫ p, l ≫ q, and p ~ q.

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