notesum.ai

Published at November 29

Efficient quantum-enhanced classical simulation for patches of quantum landscapes

quant-ph

cs.LG

stat.ML

Released Date: November 29, 2024

Authors: Sacha Lerch¹, Ricard Puig¹, Manuel S. Rudolph¹, Armando Angrisani¹, Tyson Jones¹, M. Cerezo², Supanut Thanasilp³, Zoë Holmes¹

Aff.: ¹Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland; ²Los Alamos National Laboratory, Los Alamos, NM 87545, USA; ³Chula Intelligent and Complex Systems, Department of Physics, Faculty of Science, Chulalongkorn University, Bangkok, Thailand, 10330

Arxiv: http://arxiv.org/pdf/2411.19896v1

General surrogate: notation table
Symbol	Definition
$\gamma$	Constant such that the $k$ -th partial derivatives are bounded as $\|\partial_{i_{1},i_{2},...i_{k}}f(\boldsymbol{\alpha})\|\leqslant\gamma^{k}% \norm{O}_{\infty}$
$\mathcal{E}_{\boldsymbol{\alpha}}$	Parametrised quantum channel.
$\partial_{i_{1},i_{2},...,i_{k}}$	Partial derivatives with respect to parameters $\{\alpha_{i_{1}},\alpha_{i_{2}},...,\alpha_{i_{k}}\}$ , i.e. $\partial_{i_{1},i_{2},...,i_{k}}:=\frac{\partial^{k}}{\prod_{j=1}^{k}\partial% \alpha_{i_{j}}}$ .
$\partial^{\boldsymbol{k}}$	Partial derivatives of order $k_{l}$ with respect to each parameter $\alpha_{l}$ where $\boldsymbol{k}=(k_{1},k_{2},...,k_{m})$ , i.e. $\partial^{\boldsymbol{k}}:=\prod_{l=1}^{m}\frac{\partial^{k_{l}}}{\partial% \alpha_{l}^{k_{l}}}$ .
$b_{j}$	The set $\{b_{j}\}$ is defined as $\partial_{i_{1},i_{2},...,i_{k}}f(\boldsymbol{\alpha})=\sum_{j=1}^{N_{d}}b_{j}% f(\boldsymbol{\alpha}_{j})$ .
$N_{d}$	Is the number of coefficients $b_{j}$ .
$b_{0}$	$b_{0}:=\max_{j}\|b_{j}\|$ .
$\delta_{i}$	$\delta_{i}=\alpha_{i}-\alpha_{i}^{*}$
$\Delta_{i_{1},...,i_{k}}(\bm{x},\bm{x}^{*})$	$\Delta_{i_{1},i_{2},...,i_{k}}(\bm{x},\bm{x}^{})=(x_{i_{1}}-x^{}_{i_{1}})(x_% {i_{2}}-x^{}_{i_{2}})...(x_{i_{k}}-x^{}_{i_{k}})$