A low-cost solution for performing a quantum operation provides for defining a target unitary that rotates a single qubit by a target rotation to place the qubit in a target state. The method further provides for identifying multiple projective rotation circuits that each implement an approximation of the target rotation upon successful measurement and assigning a selection probability to each of the multiple identified projective rotation circuits, the selection probability being defined by a metric that increases in value in proportion to a proximity between a qubit state resulting from the approximation and the target qubit state. The method further provides for probabilistically selecting one of the multiple projective rotation circuits according to the assigned selection probabilities and outputting a circuit definition that includes the selected projective rotation circuit.