Hilbert space delocalization under random unitary circuits

Abstract

Unitary dynamics of a quantum system initialized in a selected basis state yields, generically, a state that is a superposition of all the basis states. This process, associated with the quantum information scrambling and intimately tied to the resource theory of coherence, may be viewed as a gradual delocalization of the system's state in the Hilbert space. This work analyzes the Hilbert space delocalization under dynamics of random quantum circuits, which serve as a minimal model of chaotic dynamics of quantum many-body systems. We employ analytical methods based on the replica trick and Weingarten calculus to investigate the time evolution of the participation entropies which quantify the Hilbert space delocalization. We demonstrate that the participation entropies approach, up to a fixed accuracy, their long-time saturation value in times that scale logarithmically with the system size. Exact numerical simulations and tensor network techniques corroborate our findings.

17 pages, 3 figures. Invited Feature Paper for Entropy. Comments are welcome!

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• Anticoncentration and state design of random tensor networks
• Quantum Algorithms for Inverse Participation Ratio Estimation in multi-qubit and multi-qudit systems
• Nonstabilizerness in the unitary and monitored quantum dynamics of XXZ-staggered and SYK models
• Exactly solvable many-body dynamics from space-time duality
• The non-stabilizerness of fermionic Gaussian states
• Fermionic Magic Resources of Quantum Many-Body Systems
• Approximate inverse measurement channel for shallow shadows
• Partial projected ensembles and spatiotemporal structure of information scrambling
• Many-body critical phase in a quasiperiodic chain and dynamical Widom lines in Fock space properties
• Anticoncentration in Clifford Circuits and Beyond: From Random Tensor Networks to Pseudo-Magic States