Noise-induced contraction of MPO truncation errors in noisy random circuits and Lindbladian dynamics

TL;DR

This paper demonstrates that MPO truncation errors decay exponentially in noisy 1D quantum circuits and Lindbladian dynamics, suggesting efficient simulation of these systems at large sizes and long times.

Contribution

It provides empirical bounds showing exponential contraction of MPO truncation errors in noisy quantum dynamics, improving understanding of simulation efficiency.

Findings

Truncation errors contract exponentially with system size and time.

Average purity relaxes to a steady state inversely proportional to noise rate.

MPO algorithms can efficiently sample from noisy circuit outputs and steady states.

Abstract

We study how matrix-product-operator (MPO) truncation errors evolve when simulating two setups: (1) 1D Haar-random circuits under either depolarizing noise or amplitude-damping noise, and (2) 1D Lindbladian dynamics of a non-integrable quantum Ising model under either depolarizing or amplitude-damping noise. We first show that the average purity of the system density matrix relaxes to a steady value on a timescale that scales inversely with the noise rate. We then show that truncation errors contract exponentially in both system size $N$ and the evolution time $t$, as the noisy dynamics maps different density matrices toward the same steady state. This yields an empirical bound on the $L_1$ truncation error that is exponentially tighter in $N$ than the existing bound. Together, these results provide empirical evidence that MPO simulation algorithms may efficiently sample from the output of 1D noisy random circuits [setup (1)] at arbitrary circuit depth, and from the steady state of 1D Lindbladian dynamics [setup (2)].