Rui-An Chang, The University of Texas at Austin

"Thermalization of quantum circuits with U(1) symmetry"

Abstract: In isolated quantum systems, thermalization occurs when local subsystems approach universal equilibrium states where observables acquire thermal expectation values. Recently, it was shown that equilibration can happen not only at the level of expectation values, but also of statistical distributions of wavefunctions, a phenomenon called “deep thermalization”. In particular the quantum states in a projected ensemble, obtained on a subsystem after measuring its complement, may also approach a universal form. For systems without conservation laws, this was found to be the unitarilyinvariant (Haar) ensemble. In our work, we consider the projected ensemble constructed from random circuit dynamics with U(1) symmetry and investigate its approach to a universal ensemble. Our numerical results on the second moment of the projected ensemble reveal an interesting dependence on the choice of initial state and measurement basis. We find preliminary evidence of the projected ensemble’s approach to the Haar ensemble (forming a quantum state 2-design) if the initial state and/or measurement basis are off-diagonal in the charge. On the other hand, if the initial state and the measurement basis are both diagonal in the charge, the projected ensemble does not form a quantum state 2-design in the overall Hilbert space, but rather a direct sum of quantum state 2-designs in each charge sector. Our research may give new insights into how U(1)-symmetric quantum circuits thermalize and suggests possible formation of quantum state designs even in systems constrained by a symmetry, with applications in quantum information science.