Practical learning method for multi-scale entangled states

Abstract

We describe two related methods for reconstructing multi-scale entangled states from a small number of efficiently-implementable measurements and fast post-processing. Both methods only require single-particle measurements and the total number of measurements is polynomial in the number of particles. Data post-processing for state reconstruction uses standard tools, namely matrix diagonalization and conjugate gradient method, and scales polynomially with the number of particles. Both methods prevent the build-up of errors from both numerical and experimental imperfections. The first method is conceptually simpler but requires unitary control. The second method circumvents the need for unitary control but requires more measurements and produces an estimated state of lower fidelity.