Fig. 2: Surpassing single-copy limits through collective measurements.

In all figures the dashed pink, purple and green lines correspond to the single-copy Nagaoka, two-copy Nagaoka and Holevo bounds, respectively. The orange-shaded region corresponds to the m.s.e. attainable with separable measurements. m.s.e.s below the dashed green line are forbidden by quantum mechanics. Error bars are obtained using the bootstrapping technique⁵² and correspond to one standard deviation. All experimental points have error bars but some are smaller than the marker size. Each data point corresponds to the average of 400 individual experimental runs, each using 512 shots, as shown in the inset of a (see Methods for details). a,c, Single-copy (a) and two-copy (c) estimates of θ_x, both with and without error mitigation. Results for estimating θ_y are similar (Extended Data Fig. 1). b,d, The corresponding m.s.e.: single-copy (b) and two-copy (d). The distribution of m.s.e. values follows the expected chi-squared distribution, shown in the inset of d. The black circle in the inset corresponds to the mean m.s.e. value. The results shown in a–d are for decoherence parameter ϵ = 0.5 and are obtained on the F-IBM QS1 device. e, Optimal single-, two- and three-copy measurements at different decoherence strengths, ϵ. The pink, purple and blue markers correspond to experimental single-, two- and three-copy measurements, respectively. For the superconducting devices, all markers correspond to the precision after using error mitigation. The results of the AQTION trapped-ion processor for ϵ = 0.5 are shown in the inset for clarity. f, Bars are one minus the ratio of the Holevo bound to the m-copy Nagaoka bound, for m up to and including 7, calculated theoretically at ϵ = 0.5. Experimental points are one minus the ratio of the Holevo bound to the m.s.e. obtained experimentally. Unfilled black diamonds correspond to the precision that our three- and four-copy projective measurements can obtain in theory. The upper and lower filled black diamonds are simulations based on a depolarizing noise model with gate error rates of 5 × 10⁻³ and 1 × 10⁻³, respectively. The legend is the same as in e.