of the measurement process must be harnessed. Post-selection and its extensions provide a way to do this. However they make inefficient use of time resources — a typical computation might require O(2m) measurements over m qubits to reach a desired accuracy. We propose a method inspired by the eigenstate thermalisation hypothesis, that harnesses the induced non-linearity of measurement on a subsystem. Post-selection on m ancillae qubits is replaced with tracing out O(logϵ/log(1−p)) (where p is the probability of a successful measurement) to attain the same accuracy as the post-selection circuit. We demonstrate this scheme on the quantum perceptron and phase estimation algorithm. This method is particularly advantageous on current quantum computers involving superconducting circuits.