Quantum Entanglement Phase Transitions and Computational Complexity: Insights from Ising Models

TL;DR

This paper investigates how entanglement transitions in 2D bipartite cluster states relate to measurement-induced phase transitions and their implications for the computational complexity of Ising models, revealing conditions for efficient sampling.

Contribution

It establishes a connection between boundary entanglement transitions and the complexity of sampling and computing the Ising partition function with complex parameters.

Findings

Boundary state can transition from volume-law to area-law entanglement.

Identifies parameter regimes for efficient sampling of 2D quantum states.

Links entanglement transitions to the computational complexity of Ising models.

Abstract

In this paper, we construct 2-dimensional bipartite cluster states and perform single-qubit measurements on the bulk qubits. We explore the entanglement scaling of the unmeasured 1-dimensional boundary state and show that under certain conditions, the boundary state can undergo a volume-law to an area-law entanglement transition driven by variations in the measurement angle. We bridge this boundary state entanglement transition and the measurement-induced phase transition in the non-unitary 1+1-dimensional circuit via the transfer matrix method. We also explore the application of this entanglement transition on the computational complexity problems. Specifically, we establish a relation between the boundary state entanglement transition and the sampling complexity of the bipartite $2$d cluster state, which is directly related to the computational complexity of the corresponding Ising partition function with complex parameters. By examining the boundary state entanglement scaling, we numerically identify the parameter regime for which the $2$d quantum state can be efficiently sampled, which indicates that the Ising partition function can be evaluated efficiently in such a region.