Fig. 2. Phase transitions in the linear cross-entropy.

a–d, At a low number of cycles, XEB grows with the size of the system. In a noiseless device, XEB will converge to 1 with the number of cycles. In the presence of noise, XEB becomes an estimator of the system fidelity. a,b, We experimentally observed a dynamical phase transition at a fixed number of cycles between these two regimes in one (a) and two dimensions (b). The random circuits have Haar random single-qubit gates and an iSWAP-like gate as an entangler. c,d, We experimentally probed a noise-induced phase transition using a weak-link model (see the main text), where the weak link is applied every 12 cycles (discrete gate set; see main text). c, The two different regimes. In the weak-noise regime, XEB converges to the fidelity, whereas in the strong-noise regime, XEB remains higher than predicted by the digital error model. d, We induced errors to scan the transition from one regime to the other.