Universal Non-stabilizerness Dynamics Across Quantum Phase Transitions

Abstract

Quantum magic and non-stabilizerness are important quantum resources that characterize computational power beyond classically simulable Clifford operations and are therefore essential for achieving quantum advantage. While non-stabilizerness has so far been investigated only at equilibrium, here we extend its dynamics to time-dependent drivings across quantum phase transitions. In particular, we show that the stabilizer Rényi entropies and the cumulants of the Pauli spectrum exhibit universal power-law scaling with the driving rate in slow processes. Moreover, we show that the logarithmic Pauli spectrum is asymptotically Gaussian, implying a lognormal distribution for the Pauli spectrum values. Our results are explicitly demonstrated by exact results in the transverse-field Ising model and by analytical approximations in long-range Kitaev models.

Figures (13)

• Figure 1: Stabilizer Rényi entropies versus $\tau_Q$ for various $\alpha$ in the TFIM. All curves show the predicted universal power-law decay with weak superimposed oscillations, captured by the analytical approximations (red dashed, $L=1600$).
• Figure 2: First three cumulants of the logarithmic Pauli spectrum for the TFIM, showing precise agreement with the predicted KZ power-law $(L=200)$.
• Figure 3: Universal time evolution of quantum magic relative to the instantaneous ground state for different driving rates in the TFIM. The time of evolution is measured relative to the instant $t_c$ at which the critical point is reached and scaled by the freeze-out time $\hat{t}$. Near the critical point, the curves exhibit a sudden increase and collapse onto a universal scaling form, followed by an oscillatory intermediate regime ($L=1600$).
• Figure 4: Quantum magic, $\alpha=1/2$, and higher order SREs relative to the final ground state in the LRKM following precisely the predicted dynamical scaling laws for $\gamma=1.4,\,\beta=1.6$, in close agreement with the analytical approximations for $\alpha=1/2,\,\alpha=2$ ($L=1000$).
• Figure 5: Pauli spectrum statistics in the LRKM ($\gamma=1.4,\,\beta=1.6$). The approximately Gaussian statistics of the logarithmic Pauli spectrum converges to the ground-state distribution with increasing driving times. Inset: Pauli spectrum distributions following the corresponding lognormal distributions.