Kibble-Zurek dynamical scaling hypothesis in the Google analog-digital quantum simulator of the $XX$ model

Abstract

State-of-the-art tensor networks are employed to simulate the Hamiltonian ramp in the analog-digital quantum simulation of the quantum phase transition to the quasi-long-range ordered phase of the two-dimensional square-lattice $XX$ model [T.I. Andersen \textit{et al.}, Nature (London) \textbf{638}, 79 (2025)]. We focus on the quantum Kibble-Zurek (KZ) mechanism near the quantum critical point. Using the infinite projected entangled pair state, we simulate an infinite lattice and demonstrate the KZ scaling hypothesis for the $XX$ correlations across a wide range of ramp times. We use the time-dependent variational principle algorithm to simulate a finite $8\times 8$ lattice, similar to the one in the quantum simulation, and find that adiabatic finite-size effects dominate for longer ramp times, where the correlation length's growth with increasing ramp time saturates and the excitation energy's dependence on the ramp time crosses over to a power-law decay characteristic of adiabatic transitions. This finding contradicts the quantum simulation data where the correlation length seems to obey KZ-like power laws, although with modified exponents.