$L^2$-estimates of error in homogenization of parabolic equations with correctors taken into account

We consider second-order parabolic equations with bounded measurable $\varepsilon$-periodic coefficients. To solve the Cauchy problem in the layer $\mathbb{R}^d\times(0,T)$ with the nonhomogeneous equation, we obtain approximations in the norm $\|\cdot\|_{L^2(\mathbb{R}^d\times(0,T))}$ with remainder of order $\varepsilon^2$ as $\varepsilon\to 0.$