Exact analytic toolbox for quantum dynamics with tunable noise strength

TL;DR

This paper develops an exact analytic framework for quantum dynamics under tunable coherent noise, focusing on Gaussian unitary ensembles, enabling precise calculations of spectral form factors and quantum channels for any system size.

Contribution

It introduces a novel method to analytically analyze quantum channels affected by noise modeled via random-matrix ensembles, especially GUE, with exact results for any Hilbert space dimension.

Findings

Exact spectral form factors for GUE at finite N

Analytic expressions for GUE quantum channels and their variance

Identification of nonmonotonic behavior of operator coefficients with noise

Abstract

We introduce a framework that allows for the exact analytic treatment of quantum dynamics subject to coherent noise. The noise is modeled via unitary evolution under a Hamiltonian drawn from a random-matrix ensemble for arbitrary Hilbert-space dimension $N$. While the methods we develop apply to generic such ensembles with a notion of rotation invariance, we focus largely on the Gaussian unitary ensemble (GUE). Averaging over the ensemble of ''noisy'' Hamiltonians produces an effective quantum channel, the properties of which are analytically calculable and determined by the spectral form factors of the relevant ensemble. We compute spectral form factors of the GUE exactly for any finite $N$, along with the corresponding GUE quantum channel, and its variance. Key advantages of our approach include the ability to access exact analytic results for any $N$ and the ability to tune to the noise-free limit (in contrast, e.g., to the Haar ensemble), and analytic access to moments beyond the variance. We also highlight some unusual features of the GUE channel, including the nonmonotonicity of the coefficients of various operators as a function of noise strength and the failure to saturate the Haar-random limit, even with infinite noise strength.