Purity decay rate in random circuits with different configurations of gates

TL;DR

This paper analyzes how the purity, a measure of entanglement, decays over time in a chain of qubits with various configurations of random two-site gates, revealing a two-stage decay process influenced by circuit geometry.

Contribution

It introduces a Markov chain approach to model purity decay in different qubit circuit geometries, including a reduction to a polynomial transfer matrix for analysis.

Findings

Purity decays in two stages: initial decay with rate λ_eff and asymptotic decay with second largest eigenvalue λ_2.

The effective decay rate λ_eff depends on bipartition location and gate geometry.

Most configurations exhibit a two-stage decay, with brick-wall being an exception.

Abstract

We study purity decay -- a measure of bipartite entanglement -- in a chain of $n$ qubits under the action of various geometries of nearest-neighbor random two-site unitary gates. We use a Markov chain description of average purity evolution, using further reduction to obtain a transfer matrix of only polynomial dimension in $n$. In most circuits, an exception being the brick-wall configuration, purity decays to its asymptotic value in two stages: the initial thermodynamically relevant decay persisting up to extensive times is $\sim \lambda_{\mathrm{eff}}^t$, with $\lambda_{\mathrm{eff}}$ not necessarily being in the spectrum of the transfer matrix, while the ultimate asymptotic decay is given by the second largest eigenvalue $\lambda_2$ of the transfer matrix. The effective rate $\lambda_{\mathrm{eff}}$ depends on the location of bipartition boundaries as well as on the geometry of applied gates.