  The Kubo-Greenwood (KG) formula is often used in conjunction with Kohn-Sham
(KS) density functional theory (DFT) to compute the optical conductivity,
particularly for warm dense mater. For applying the KG formula, all KS
eigenstates and eigenvalues up to an energy cutoff are required and thus the
approach becomes expensive, especially for high temperatures and large systems,
scaling cubically with both system size and temperature. Here, we develop an
approach to calculate the KS conductivity within the stochastic DFT (sDFT)
framework, which requires knowledge only of the KS Hamiltonian but not its
eigenstates and values. We show that the computational effort associated with
the method scales linearly with system size and reduces in proportion to the
temperature unlike the cubic increase with traditional deterministic
approaches. In addition, we find that the method allows an accurate description
of the entire spectrum, including the high-frequency range, unlike the
deterministic method which is compelled to introduce a high-frequency cut-off
due to memory and computational time constraints. We apply the method to
helium-hydrogen mixtures in the warm dense matter regime at temperatures of
\sim60\text{kK} and find that the system displays two conductivity phases,
where a transition from non-metal to metal occurs when hydrogen atoms
constitute \sim0.3 of the total atoms in the system.
