The aim of this study is to construct a finite object (discretization operator) that approximates with any accuracy the solution of the heat equation with an initial condition from  the multidimensional Nikol’skii - Besov   functional class in a certain metric. The research methodology is based on well-known statements of approximation theory. The relevance of the study carried out here is explained by the following circumstances: firstly, in this article, for a solution of the heat equation representable in the form of an absolutely convergent multiple functional series and belonging to the periodic Nikol’skii - Besov   class, a discretization operator approximating it with any accuracy is proposed, constructed from the values of the initial condition at the points of a uniform grid of the unit cube;  secondly, it is proven that any discretization operator constructed from a given finite set of values of linear functionals defined on the functional class under consideration does not improve the order of the error obtained when approximating the solution by the proposed discretization operator.