We consider a quantum gas consisting of N hard spheres with radius \frak{a} > 0, obeying bosonic statistics and moving in the box \Lambda = [0;L]^3 with periodic boundary conditions. We are interested in the ground state energy per unit volume in the thermodynamic limit, with N, L \to \infty at fixed density \rho = N / L^3. We derive an upper bound for the ground state energy density, matching the famous Lee-Huang-Yang formula, up to lower order terms, in the dilute limit \rho \frak{a}^3 \ll 1.
