Phase transitions in sampling and error correction in local Brownian circuits

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.

17+3 pages, 10 figures

References in corpus (4)

• Supplementary information for "Quantum supremacy using a programmable superconducting processor"
• Real time evolution using the density matrix renormalization group
• Strong quantum computational advantage using a superconducting quantum processor
• The mother of all protocols: Restructuring quantum information's family tree

Cited by in corpus (5)

• Noise-induced phase transitions in hybrid quantum circuits
• Hayden-Preskill recovery in chaotic and integrable unitary circuit dynamics
• Efficient circular Dyson Brownian motion algorithm
• Dynamics in the presence of local symmetry-breaking impurities
• Noisy Monitored Quantum Circuits