arXiv:2611.09135v1 Announce Type: new Abstract: We establish strong convergence rates for a fully discrete method of the one-dimensional stochastic Burgers equation driven by additive rough noise. Using the Cole-Hopf transformation, the equation is converted into a linear stochastic heat equation, which is discretized by a spectral Galerkin method in space and an accelerated exponential Euler method in time. A numerical approximation of the original solution is then reconstructed through a regularized inverse Cole-Hopf transformation. The analysis combines strong Sobolev error estimates with the positivity and inverse-moment bounds for the transformed solution. For additive space-time white noise, the numerical method achieves mean-square convergence rates arbitrarily close to 1/2 in space and 1/4 in time. The rates are slower when the noises are rougher.