  Consider the optimization problem $p_{\min, Q} := \min_{\mathbf{x} \in Q}
p(\mathbf{x})$, where $p$ is a degree $m$ multivariate polynomial and $Q := [0,
1]^n$ is the hypercube. We provide explicit degree and error bounds for the
sums of squares approximations of $p_{\min, Q}$ corresponding to the
Positivstellensatz of Putinar. Our approach uses Bernstein multivariate
approximation of polynomials, following the methodology of De Klerk and Laurent
to provide error bounds for Schm\"udgen type positivity certificates over the
hypercube. We give new bounds for Putinar type representations by relating the
quadratic module and the preordering associated with the polynomials $g_i :=
x_i (1 - x_i), \: i=1,\dots,n$, describing the hypercube $Q$.
