Measuring out-of-time-order correlators on a nuclear magnetic resonance quantum simulator

TL;DR

This work demonstrates the first experimental measurement of local-operator OTOCs on a nuclear magnetic resonance quantum simulator using a four-qubit Ising spin chain, enabling direct observation of chaos- and scrambling-related dynamics. By combining Trotterized time evolution with local control, the authors compare integrable and non-integrable regimes, extract entanglement growth via a relation between OTOCs and the second Rényi entropy, and determine the butterfly velocity from the spreading of correlations. The results show revival behavior in the integrable case and scrambling in the non-integrable cases, with entanglement entropy growing linearly before saturating in non-integrable dynamics. The study provides a practical pathway toward probing quantum chaos, information scrambling, and holographic connections in larger quantum simulators across multiple platforms.

Abstract

The idea of the out-of-time-order correlator (OTOC) has recently emerged in the study of both condensed matter systems and gravitational systems. It not only plays a key role in investigating the holographic duality between a strongly interacting quantum system and a gravitational system, but also diagnoses the chaotic behavior of many-body quantum systems and characterizes the information scrambling. Based on the OTOCs, three different concepts -- quantum chaos, holographic duality, and information scrambling -- are found to be intimately related to each other. Despite of its theoretical importance, the experimental measurement of the OTOC is quite challenging and so far there is no experimental measurement of the OTOC for local operators. Here we report the measurement of OTOCs of local operators for an Ising spin chain on a nuclear magnetic resonance quantum simulator. We observe that the OTOC behaves differently in the integrable and non-integrable cases. Based on the recent discovered relationship between OTOCs and the growth of entanglement entropy in the many-body system, we extract the entanglement entropy from the measured OTOCs, which clearly shows that the information entropy oscillates in time for integrable models and scrambles for non-intgrable models. With the measured OTOCs, we also obtain the experimental result of the butterfly velocity, which measures the speed of correlation propagation. Our experiment paves a way for experimentally studying quantum chaos, holographic duality, and information scrambling in many-body quantum systems with quantum simulators.

Figures (9)

• Figure 1: Illustration of the physical system, the Ising model and the experimental scheme. (a) The structure of the $\text{C}_2\text{F}_3\text{I}$ molecule used for the NMR simulation. (b) The four sites Ising spin chain, $\mathcal{A}$ and $\mathcal{B}$ label dividing the entire system into two subsystems in the later discussion of entanglement entropy. (c) Quantum circuit for measuring the OTOC for general $N$-site Ising chain when $\beta = 0$ (in our case $N=4$). Here $\hat{R} = \mathbf{1}, \hat{R}_x(-\pi/2), \hat{R}_y(\pi/2)$ for $\hat{A} = \hat{\sigma}_1^z, \hat{\sigma}_1^y, \hat{\sigma}_1^x$, respectively.
• Figure 2: Experimental results of OTOC measurement for an Ising spin chain: (a) $\hat{A}=\hat{\sigma}^z_1$ at the first site, and $\hat{B}=\hat{\sigma}^x_4$ at the fourth site. (b) $\hat{A}=\hat{\sigma}^x_1$ at the first site, and $\hat{B}=\hat{\sigma}^y_4$ at the fourth site. The three columns correspond to $g=1$, $h=0$; $g=1.05$, $h=0.5$; and $g=1$, $h=1$ of model Eq. \ref{['Ising']}, respectively. The red points are experimental data, the blue curves are theoretical calculation of OTOC with model Eq. \ref{['Ising']} for four sites.
• Figure 3: The 2nd Rényi entropy $S^{(2)}_\mathcal{A}$ after a quench. A quench operator $(\mathbf{1}+\hat{\sigma}^x_1)$ (up to a normalization factor) is applied to the system at $t=0$, and the entropy is measured by tracing out the fourth site as the subsystem $\mathcal{B}$. Different colors correspond to different parameters of $g$ and $h$ in the Ising spin model. The points are experimental data, the curves are theoretical calculations.
• Figure 4: Measurement of the butterfly velocity: (a) shows the OTOCs for $\hat{A}=\hat{\sigma}^z_1$ and $\hat{B}=\hat{\sigma}^x_i$ with $i=4$ (blue), $i=3$ (green) and $i=2$ (red); (b) shows the OTOCs for $\hat{A}=\hat{\sigma}^y_1$ and $\hat{B}=\hat{\sigma}^z_i$ with $i=4$ (blue), $i=3$ (green) and $i=2$ (red). The insets of (a) and (b) shows the time for the onset of chaos $t_d$ for the OTOCs v.s. the distance between two operators. The slope gives $1/v_\text{B}$. Here $g=1.05$ and $h=0.5$.
• Figure 5: Characteristics of Iodotrifluroethylene. Molecular structure together with a table of the chemical shifts (on the diagonal) and $J$-coupling strengths (lower off-diagonal), all in Hz. The chemical shifts are given with respect to base frequency for $^{13}$C or $^{19}$F transmitters on the 400 MHz spectrometer that we used.