An enhanced machine learning algorithm for precise adaptive phase estimation

Sequential adaptive measurement is a promising strategy for quantum-enhanced single-shot measurements, but appropriate adaptive procedures are difficult to devise even for ideal cases. Machine learning techniques enable suitable procedures to be found, but these machine learning methods are severely limited by computational time and space restrictions. Thus far the collective intelligence algorithm known as particle swarm optimization has been used to beat the best previous results (obtained by clever guessing) for adaptive phase measurement in quantum interferometry with an input state of an entangled multi-photon pulse. Specifically, particle swarm optimisation has successfully devised procedures with a space cost that is linear in the number N of operations on the multi-photon pulse and a time that scales as N6 [1, 2]. The resultant procedure delivers a power-law scaling for the interferometric phase uncertainty,\\r\\n∆ vs N, but this power law scaling breaks suddenly at an input N 50 due to a failure of the technique for specific computational resources, namely the number of particles (particle swarm optimisation candidate solutions) and number of iterations. We devise a far superior algorithm that is able to deliver much better precision for given N with the restriction of using the same computational resources as the competing algorithm. We show that the differential evolution approach to machine learning delivers this same power law scaling as particle swarm optimisation and shows no sign of breakdown up to one hundred photons. As the true cost scales as N6, this doubling (at least) of the number of photons actually corresponds to a 64 fold increase in the efficiency of devising the algorithm.