Consider a holomorphic vector bundle \(E\) of rank \(r\) over a projective manifold of complex dimension \(n\) polarized by an ample line bundle \(L\). Fix Hermitian metrics \(h\) on \(E\) and \(\sigma \) on \(L\) whose curvature defines a Kähler form \(\omega \) on \(X\). Given these data, one can consider the induced Bochner Laplacian over the smooth endomorphisms of \(E\): \(\Delta ^E:C^{\infty }(X, \mathrm{End}(E))\rightarrow C^{\infty }(X, \mathrm{End}(E))\). In this paper, it is proved that certain operators of the form \(P^{\ast}_kP_k\) provide a quantization of \(\Delta ^E\). It is important to note that \(P^{\ast}_kP_k\) lives as an operator on a finite-dimensional space and is defined in a purely algebraic way.