  Recent cosmic shear studies have shown that higher-order statistics (HOS)
developed by independent teams now outperform standard two-point estimators in
terms of statistical precision thanks to their sensitivity to the non-Gaussian
features of large-scale structure. The aim of the Higher-Order Weak Lensing
Statistics (HOWLS) project is to assess, compare, and combine the constraining
power of ten different HOS on a common set of $Euclid$-like mocks, derived from
N-body simulations. In this first paper of the HOWLS series, we computed the
nontomographic ($\Omega_{\rm m}$, $\sigma_8$) Fisher information for the
one-point probability distribution function, peak counts, Minkowski
functionals, Betti numbers, persistent homology Betti numbers and heatmap, and
scattering transform coefficients, and we compare them to the shear and
convergence two-point correlation functions in the absence of any systematic
bias. We also include forecasts for three implementations of higher-order
moments, but these cannot be robustly interpreted as the Gaussian likelihood
assumption breaks down for these statistics. Taken individually, we find that
each HOS outperforms the two-point statistics by a factor of around two in the
precision of the forecasts with some variations across statistics and
cosmological parameters. When combining all the HOS, this increases to a $4.5$
times improvement, highlighting the immense potential of HOS for cosmic shear
cosmological analyses with $Euclid$. The data used in this analysis are
publicly released with the paper.
