Phase transitions in sampling and error correction in local Brownian circuits

TL;DR

This paper investigates phase transitions in local Brownian circuits related to anticoncentration, unitary design, and error correction, revealing sharp thresholds and regimes through large-scale numerics and effective Hamiltonian analysis.

Contribution

It introduces a novel framework linking Brownian circuit dynamics to thermodynamic phases and identifies multiple phase transitions relevant to quantum information processing.

Findings

Anticoncentration transition at log N timescale

Hardness transition for classical approximation algorithms

Noise-induced phase transition in linear cross entropy benchmark

Abstract

We study the emergence of anticoncentration and approximate unitary design behavior in local Brownian circuits. The dynamics of circuit averaged moments of the probability distribution and entropies of the output state can be represented as imaginary time evolution with an effective local Hamiltonian in the replica space. This facilitates large scale numerical simulation of the dynamics in $1+1d$ of such circuit-averaged quantities using tensor network tools, as well as identifying the various regimes of the Brownian circuit as distinct thermodynamic phases. In particular, we identify the emergence of anticoncentration as a sharp transition in the collision probability at $\log N$ timescale, where $N$ is the number of qubits. We also show that a specific classical approximation algorithm has a computational hardness transition at the same timescale. In the presence of noise, we show there is a noise-induced first order phase transition in the linear cross entropy benchmark when the noise rate is scaled down as $1/N$. At longer times, the Brownian circuits approximate a unitary 2-design in $O(N)$ time. We directly probe the feasibility of quantum error correction by such circuits, and identify a first order transition at $O(N)$ timescales. The scaling behaviors for all these phase transitions are obtained from the large scale numerics, and corroborated by analyzing the spectrum of the effective replica Hamiltonian.