We introduce an algorithm for sampling many-body quantum states in Fock space. The algorithm efficiently samples states with probability approximately proportional to an arbitrary function of the second-quantized Hamiltonian matrix element connecting the sampled state to the current state. We apply the new sampling algorithm to the recently-developed Semistochastic Full Configuration Interaction Quantum Monte Carlo method (S-FCIQMC), a semistochastic implementation of the power method for projecting out the ground state energy in a basis of Slater determinants. The heat-bath sampling requires modest additional computational time and memory compared to uniform sampling but results in newly-spawned weights that are approximately of the same magnitude, thereby greatly improving the efficiency of projection. A comparison in efficiency between uniform and approximate heat-bath sampling is performed on the all-electron nitrogen dimer at equilibrium in Dunnings cc-pVXZ basis sets with X in {D, T, Q, 5}, demonstrating a large gain in efficiency that increases with basis set size. In addition, a comparison in efficiency is performed on three all-electron first-row dimers, B_2, N_2, and F_2, in a cc-pVQZ basis, demonstrating that the gain in efficiency compared to uniform sampling also increases dramatically with the number of electrons.
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