Quantum computing requires a universal set of gate operations; regarding gates as rotations, any rotation angle must be possible. However a real device may only be capable of B bits of resolution, i.e., it might support only 2^{B} possible variants of a given physical gate. Naive discretization of an algorithm's gates to the nearest available options causes coherent errors, while decomposing an impermissible gate into several allowed operations increases circuit depth. Conversely, demanding higher B can greatly complexify hardware. Here, we explore an alternative: probabilistic angle interpolation (PAI). This effectively implements any desired, continuously parametrized rotation by randomly choosing one of three discretized gate settings and postprocessing individual circuit outputs. The approach is particularly relevant for near-term applications where one would in any case average over many runs of circuit executions to estimate expected values. While PAI increases that sampling cost, we prove that (a) the approach is optimal in the sense that PAI achieves the least possible overhead and (b) the overhead is remarkably modest even with thousands of parametrized gates and only seven bits of resolution available. This is a profound relaxation of engineering requirements for first generation quantum computers where even 5-6 bits of resolution may suffice and, as we demonstrate, the approach is many orders of magnitude more efficient than prior techniques. Moreover we conclude that, even for more mature late noisy intermediate-scale quantum era hardware, no more than nine bits will be necessary.