Measurement-induced entanglement and complexity in random constant-depth 2D quantum circuits

TL;DR

This paper proves that random constant-depth 2D quantum circuits generate macroscopic entanglement above a certain depth, impacting classical simulation methods and establishing new theoretical tools free of replica assumptions.

Contribution

The authors develop replica-free techniques from quantum information theory to rigorously analyze entanglement in random quantum circuits, confirming long-standing conjectures.

Findings

Macroscopic entanglement appears above a critical circuit depth.

Standard classical algorithms fail to simulate these circuits beyond certain parameters.

Random shallow circuits cannot be efficiently simulated by sublogarithmic depth classical circuits.

Abstract

We analyse the entanglement structure of states generated by random constant-depth two-dimensional quantum circuits, followed by projective measurements of a subset of sites. By deriving a rigorous lower bound on the average entanglement entropy of such post-measurement states, we prove that macroscopic long-ranged entanglement is generated above some constant critical depth in several natural classes of circuit architectures, which include brickwork circuits and random holographic tensor networks. This behaviour had been conjectured based on previous works, which utilize non-rigorous methods such as replica theory calculations, or work in regimes where the local Hilbert space dimension grows with system size. To establish our lower bound, we develop new replica-free theoretical techniques that leverage tools from multi-user quantum information theory, which are of independent interest, allowing us to map the problem onto a statistical mechanics model of self-avoiding walks without requiring large local Hilbert space dimension. Our findings have consequences for the complexity of classically simulating sampling from random shallow circuits, and of contracting tensor networks: First, we show that standard algorithms based on matrix product states which are used for both these tasks will fail above some constant depth and bond dimension, respectively. In addition, we also prove that these random constant-depth quantum circuits cannot be simulated by any classical circuit of sublogarithmic depth.