Out-of-Time-Order Correlator Spectroscopy

TL;DR

This paper unifies the understanding of out-of-time-order correlators (OTOCs) within quantum signal processing, introducing OTOC spectroscopy as a mode-resolved method to analyze quantum scrambling and spectral properties of many-body dynamics.

Contribution

It provides a unified algorithmic interpretation of higher-order OTOCs using quantum signal processing and introduces OTOC spectroscopy for mode-resolved analysis.

Findings

Higher-order OTOCs measure Fourier components of phase distributions.

OTOC spectroscopy enables frequency-selective probing of quantum dynamics.

The framework explains sensitivities of correlators to different dynamical regimes.

Abstract

Out-of-time-order correlators (OTOCs) are central probes of quantum scrambling, and their generalizations have recently become key primitives for both benchmarking quantum advantage and learning the structure of Hamiltonians. Yet their behavior has lacked a unified algorithmic interpretation. We show that higher-order OTOCs naturally fit within the framework of quantum signal processing (QSP): each $\mathrm{OTOC}^{(k)}$ measures the $2k$-th Fourier component of the phase distribution associated with the singular values of a spatially resolved truncated propagator. This explains the contrasting sensitivities of time-ordered correlators (TOCs) and higher-order OTOCs to causal-cone structure and to chaotic, integrable, or localized dynamics. Based on this understanding, we further generalize higher-order OTOCs by polynomial transformation of the singular values of the spatially resolved truncated propagator. The resultant signal allows us to construct frequency-selective filters, which we call \emph{OTOC spectroscopy}. This extends conventional OTOCs into a mode-resolved tool for probing scrambling and spectral structure of quantum many-body dynamics.