  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
parametrised rotation by randomly choosing one of three discretised 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 c) the overhead is
remarkably modest even with thousands of parametrised gates and only $7$ 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-NISQ hardware, no more than $9$ bits will be necessary.
