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Out-of-Time-Order Correlator Spectroscopy

Keisuke Fujii

TL;DR

Out-of-Time-Order Correlators (OTOCs) are key probes of quantum scrambling but lacked a unified operational interpretation. This work embeds higher-order OTOCs in the quantum signal processing (QSP) framework, showing that $\mathrm{OTOC}^{(k)}$ equals the $2k$-th Fourier component of the phase distribution of the singular values of the spatially resolved truncated propagator $A_{i,j}(t)=\langle 0_i|U(t)|0_j\rangle$, i.e., a Chebyshev transform of the spectrum. By generalizing to phase-rotated QSP sequences, the authors define OTOC spectroscopy, enabling degree-$2d$ polynomial transformations and frequency-selective filters on the singular-value spectrum, which probe scrambling and spectral structure in a mode-resolved fashion. Numerical experiments across chaotic, integrable, and many-body localized dynamics validate the Fourier-mode interpretation and demonstrate sharp detection of causal-cone edges with higher-order OTOCs, providing a scalable, verifiable diagnostic toolkit for quantum dynamics. This framework offers a principled route to design spectrally selective probes and to learn Hamiltonian structure from dynamical data.

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.

Out-of-Time-Order Correlator Spectroscopy

TL;DR

Out-of-Time-Order Correlators (OTOCs) are key probes of quantum scrambling but lacked a unified operational interpretation. This work embeds higher-order OTOCs in the quantum signal processing (QSP) framework, showing that equals the -th Fourier component of the phase distribution of the singular values of the spatially resolved truncated propagator , i.e., a Chebyshev transform of the spectrum. By generalizing to phase-rotated QSP sequences, the authors define OTOC spectroscopy, enabling degree- polynomial transformations and frequency-selective filters on the singular-value spectrum, which probe scrambling and spectral structure in a mode-resolved fashion. Numerical experiments across chaotic, integrable, and many-body localized dynamics validate the Fourier-mode interpretation and demonstrate sharp detection of causal-cone edges with higher-order OTOCs, providing a scalable, verifiable diagnostic toolkit for quantum dynamics. This framework offers a principled route to design spectrally selective probes and to learn Hamiltonian structure from dynamical data.

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 measures the -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.

Paper Structure

This paper contains 17 sections, 1 theorem, 49 equations, 2 figures.

Table of Contents

  1. Introduction
  2. QSP formulation of OTOC$^{(k)}$
  3. Higher order OTOC
  4. QSP formulation
  5. Behavior of OTOC$^{(k)}$ from the viewpoint of Fourier analysis
  6. Phase distribution of singular values of spatially resolved truncated propagator
  7. Haar random unitary
  8. Outside the causal-cone
  9. Algorithmic meanings of higher order OTOC
  10. Structural signature of underlying Hamiltonian
  11. Free Fermion example
  12. Implications of Fourier mode analysis
  13. OTOC spectroscopy via QSP
  14. Numerical Experiments
  15. Models
  16. ...and 2 more sections

Key Result

Theorem 1

Let $U$ be the given dynamics and take $B_i=Z_i$ and $M_j=Z_j$ on the probe qubit. Then the $k$th-order OTOC associated with this echo sequence satisfies where $\lambda_l$ are the singular values of $A=\bra{0_i} \tilde{U}\ket{0_i}$ with $\tilde{U} = U S_{j\leftrightarrow i}$ (or equivalently $A_{i,j}= \bra{0_i} U\ket{0_j}$) and $\ket{\psi_{\rm ref}}=\ket{0_i} \otimes\sum_l\alpha_l\ket{\psi_l}$.

Figures (2)

  • Figure 1: Phase distributions $\tilde{p}_{0,9}(\theta,t)$ of the singular values of the truncated propagator $A_{0,9}(t)$ for chaotic (left top), integrable (Bethe ansatz) (right top), integrable (free-fermion) (left bottom) and MBL (right bottom) dynamics. The horizontal axes show time $t$ ($0\leq t \leq 7$ for integrable (free-fermion) and $0\leq t \leq 5$ for others) and the phase variable $\theta$ with $\lambda=\cos(\theta/2)$, and the vertical axis and color scale indicate the density.
  • Figure 2: Time evolution of the second-, fourth-, eighth-order, and twelfth-order Chebyshev moments $\langle T_{4k}(\lambda)\rangle$ of the truncated propagator $A_{i,j}(t)$ for the chaotic Hamiltonian, corresponding respectively to the TOC ($k=1/2$), OTOC$^{(1)}$ ($k=1$), OTOC$^{(2)}$ ($k=2$), and OTOC$^{(3)}$ ($k=3$) from left to right. Each curve shows the moment for a different separation $j=0,...,9$ where $i=0$.

Theorems & Definitions (1)

  • Theorem 1: OTOC as a Chebyshev polynomial transform