  The rough Heston model is a very popular recent model in mathematical
finance; however, the lack of Markov and semimartingale properties poses
significant challenges in both theory and practice. A way to resolve this
problem is to use Markovian approximations of the model. Several previous works
have shown that these approximations can be very accurate even when the number
of additional factors is very low. Existing error analysis is largely based on
the strong error, corresponding to the $L^2$ distance between the kernels.
Extending earlier results by [Abi Jaber and El Euch, SIAM Journal on Financial
Mathematics 10(2):309--349, 2019], we show that the weak error of the Markovian
approximations can be bounded using the $L^1$-error in the kernel approximation
for general classes of payoff functions for European style options. Moreover,
we give specific Markovian approximations which converge super-polynomially in
the number of dimensions, and illustrate their numerical superiority in option
pricing compared to previously existing approximations. The new approximations
also work for the hyper-rough case $H > -1/2$.
