Fig. 3: Experimental results for single-instance verification.

Root infidelity estimate \(\sqrt{1-F}\) (hexagons), and empirical TVD (stars) for single instances of random MBQC cluster states with recycling (blue) and without (pink). Note that the horizontal axis is labeled with the cluster size n × m and scaled with qubit number n m. The root infidelity upper-bounds the TVD per Eq. (1). Colored error bars represent the 3σ interval of the statistical error. Uncorrelated measurement noise reduces or increases the measured state fidelity compared to the true fidelity asymmetrically depending on its value, such that the shown values are lower bounds to the true state fidelity, see the Methods section for details. The worst-case behavior of the measurement noise is represented by gray error bars. In the non-recycling experiment, the register size is increased between the 2 × 3 and the 3 × 3 instance, leading to a decrease in the local gate fidelities. Modeling the noise as local depolarizing noise after each entangling gate (dotted lines), we obtain effective local Pauli error probabilities after the two-qubit gates of 5.3%, 2.6%, and 1.0%, for the recycling data, for the large-register non-recycling data, and the small register non-recycling data, respectively; see Supplementary Note 5. The shaded green area is the acceptance region corresponding to an infidelity threshold of 8.6% arising in the rigorous hardness argument as sketched in Supplementary Note 4. Since the accuracy of the TVD estimate scales with the system dimension already for cluster sizes of 4 × 3 and 4 × 4 infeasible amount of samples would be required for an accurate estimate, and hence these are not shown. See Supplementary Table 1 for experimental details.