  Arikan's Polar codes attracted much attention as the first efficiently
decodable and capacity achieving codes. Furthermore, Polar codes exhibit an
exponentially decreasing block error probability with an asymptotic error
exponent upper bounded by 1/2. Since their discovery, many attempts have been
made to improve the error exponent and the finite block-length performance,
while keeping the bloc-structured kernel. Recently, two of us introduced a new
family of efficiently decodable error-correction codes based on a recently
discovered efficiently-contractible tensor network family in quantum many-body
physics, called branching MERA. These codes, called branching MERA codes,
include Polar codes and also extend them in a non-trivial way by substituting
the bloc-structured kernel by a convolutional structure. Here, we perform an
in-depth study of a particular example that can be thought of as a direct
extension to Arikan's Polar code, which we therefore name Convolutional Polar
codes. We prove that these codes polarize and exponentially suppress the
channel's error probability, with an asymptotic error exponent log_2(3)/2 which
is provably better than for Polar codes under successive cancellation decoding.
We also perform finite block-size numerical simulations which display improved
error-correcting capability with only a minor impact on decoding complexity.
