Automatic post-selection by ancillae thermalization

Tasks such as classification of data and determining the groundstate a Hamiltonian cannot be carried out through purely unitary quantum evolution. Instead, inherent non-unitarity measurement process must harnessed. Post-selection its extensions provide way to do this. However they make inefficient use time resources -- typical computation might require $O(2^m)$ measurements over $m$ qubits reach desired accuracy. We propose method inspired by eigenstate thermalisation hypothesis, that harnesses induced non-linearity on subsystem. ancillae is replaced with tracing $O(\log\epsilon / \log(1-p))$ (where p probability successful measurement) attain same accuracy post-selection circuit. demonstrate this scheme perceptron phase estimation algorithm. This particularly advantageous current computers involving superconducting circuits.