Ensembles of physical states and random quantum circuits on graphs

Abstract

In this paper we continue and extend the investigations of the ensembles of random physical states introduced in A. Hamma et al. [ http://prl.aps.org/abstract/PRL/v109/i4/e040502 Phys. Rev. Lett. 109, 040502 (2012)]. These ensembles are constructed by finite-length random quantum circuits (RQC) acting on the (hyper)edges of an underlying (hyper)graph structure. The latter encodes for the locality structure associated with finite-time quantum evolutions generated by physical i.e., local, Hamiltonians. Our goal is to analyze physical properties of typical states in these ensembles, in particular here we focus on proxies of quantum entanglement as purity and $\alpha$-Renyi entropies. The problem is formulated in terms of matrix elements of superoperators which depend on the graph structure, choice of probability measure over the local unitaries and circuit length. In the $\alpha =2$ case these superoperators act on a restricted multi-qubit space generated by permutation operators associated to the subsets of vertices of the graph. For permutationally invariant interactions the dynamics can be further restricted to an exponentially smaller subspace. We consider different families of RQCs and study their typical entanglement properties for finite-time as well as their asymptotic behavior. We find that area law holds in average and that the volume law is a typical property (that is, it holds in average and the fluctuations around the average are vanishing for the large system) of physical states. The area law arises when the evolution time is O(1) with respect to the size $L$ of the system, while the volume law arises as typical when the evolution time scales like O(L).

updated to published version

References in corpus (22)

• Thermalization and its mechanism for generic isolated quantum systems
• Black holes as mirrors: quantum information in random subsystems
• Lieb-Robinson bounds and the generation of correlations and topological quantum order
• Foundation of Statistical Mechanics under experimentally realistic conditions
• Exact relaxation in a class of non-equilibrium quantum lattice systems
• Evenly distributed unitaries: on the structure of unitary designs
• Randomizing quantum states: Constructions and applications
• Typicality for Generalized Microcanonical Ensembles
• Quantum simulation of time-dependent Hamiltonians and the convenient illusion of Hilbert space
• Ground state entanglement and geometric entropy in the Kitaev's model
• The mother of all protocols: Restructuring quantum information's family tree
• General entanglement scaling laws from time evolution
• Exploring local quantum many-body relaxation by atoms in optical superlattices
• Unitary designs and codes
• Convergence to equilibrium under a random Hamiltonian
• Exact convergence times for generation of random bipartite entanglement
• Emergence of typical entanglement in two-party random processes
• Quantum entanglement in random physical states
• Comment on the paper "Random Quantum Circuits are Approximate 2-designs"
• Optimal two-qubit gate for generation of random bipartite entanglement
• Efficient algorithm for multi-qudit twirling for ensemble quantum computation
• Trapped surfaces and emergent curved space in the Bose-Hubbard model

Cited by in corpus (21)

• Operator Spreading in Random Unitary Circuits
• Solution of a minimal model for many-body quantum chaos
• Quantum Chaos is Quantum
• Random Quantum Batteries
• Spectral statistics in constrained many-body quantum chaotic systems
• Eigenstate thermalization in dual-unitary quantum circuits: Asymptotics of spectral functions
• Necessity of Eigenstate Thermalization
• Energy storage and coherence in closed and open quantum batteries
• Unifying Emergent Hydrodynamics and Lindbladian Low Energy Spectra across Symmetries, Constraints, and Long-Range Interactions
• Transitions in Entanglement Complexity in Random Circuits
• How much entanglement is needed to reduce the energy variance?
• Entanglement of random hypergraph states
• Is macroscopic entanglement a typical trait of many-particle quantum states?
• Entanglement complexity of the Rokhsar-Kivelson-sign wavefunctions
• Generalized quantum microcanonical ensemble from random matrix product states
• Probability density of quantum expectation values
• Pure state thermodynamics with matrix product states
• Entanglement susceptibility: Area laws and beyond
• Stability of topological purity under random local unitaries
• Local random quantum circuits: ensemble CP maps and swap algebras
• Propagation of correlations in Local Random Quantum Circuits