  Tasks such as classification of data and determining the groundstate of a
Hamiltonian cannot be carried out through purely unitary quantum evolution.
Instead, the inherent non-unitarity 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 $O(2^m)$ 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
$O(\log\epsilon / \log(1-p))$ (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.
