For a given collection of M operators and a given reference state, the purpose of the AR method is to find a quasi-optimal order from the total number of M ! different orders. The method takes only several times of measurement cost, rather than a global search in all the sequences. The essence of the AR method is simple, reordering the operators {e θp Tp } in a descending manner according to their corresponding optimized amplitudes of {θ opt p } to decrease the scalar part of X defined in Eq.S2 in SI. The reason why this method can generate a quasi-optimal order is shown in Section A of SI.