Measuring Out-of-Time-Order Correlators on a Nuclear Magnetic Resonance Quantum Simulator

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, it also diagnoses the chaotic behavior of many-body quantum systems and characterizes information scrambling. Based on OTOCs, three different concepts—quantum chaos, holographic duality, and information scrambling—are found to be intimately related to each other. Despite its theoretical importance, the experimental measurement of the OTOC is quite challenging, and thus 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 nonintegrable 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 nonintgrable 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.

Popular Summary

Chaos is a phenomenon where system dynamics are extremely sensitive to changes in initial conditions. For decades, physicists have attempted to find chaos in quantum mechanics. Unfortunately, because of the linear nature of quantum theory, such sensitivity does not exist. However, along the way, researchers found that chaos appears in quantum systems in other ways such as information scrambling. Surprisingly, this phenomenon shows up naturally in many branches of physics such as condensed-matter physics, high-energy physics, and quantum information science. This raises the question of how to quantitatively measure information scrambling. A mathematical tool known as an out-of-time-ordered correlation (OTOC) function has recently been identified as a candidate, but it is challenging to observe OTOC in experiments. We have used quantum simulations to demonstrate proof-of-concept measurements of OTOC, with high precision and strong robustness against noise, for the first time.

One of the difficulties in measuring OTOC is its alternating time ordering that requires one to “reverse” the system dynamics. The development of small-scale quantum computers provides a way around this hurdle. We use a four-qubit nuclear-magnetic-resonance quantum processor to simulate the dynamics of other quantum systems. We observe how the OTOC behaves in different scenarios and use the measured OTOC to determine how entropy changes over time.

Our method opens up a way to study OTOC with quantum computers built from other physical systems. The rapid development of quantum computing technology will likely reveal more interesting physics through a unifying understanding of quantum chaos and information scrambling.

Figure 1

Illustration of the physical system, the Ising model, and the experimental scheme. (a) The structure of the ${\mathrm{C}}_2{\mathrm{F}}_3\mathrm{I}$ molecule used for the NMR simulation. (b) The four site Ising spin chain. $\mathcal{A}$ and $\mathcal{B}$ label two subsystems for the later discussion of the entanglement entropy. (c) Quantum circuit for measuring the OTOC for the general $N$-site Ising chain when $\beta =0$ (in our case, $N=4$). Here, $\widehat{R}=\mathbf{1}$, ${\widehat{R}}_x(-\pi /2)$, ${\widehat{R}}_y(\pi /2)$ for $\widehat{A}={\widehat{\sigma }}_1^z$, ${\widehat{\sigma }}_1^y$, ${\widehat{\sigma }}_1^x$, respectively.

Figure 2

Experimental results of OTOC measurement for an Ising spin chain. (a) $\widehat{A}={\widehat{\sigma }}_1^z$ at the first site and $\widehat{B}={\widehat{\sigma }}_4^x$ at the fourth site. (b) $\widehat{A}={\widehat{\sigma }}_1^x$ at the first site and $\widehat{B}={\widehat{\sigma }}_4^y$ 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. (2), respectively. The red points are experimental data, the blue curves are theoretical calculation of OTOC with model Eq. (2) for four sites.

Figure 3

The second Rényi entropy $S_{\mathcal{A}}^{(2)}$ after a quench. A quench operator $(\mathbf{1}+{\widehat{\sigma }}_1^x)$ (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) The OTOCs for $\widehat{A}={\widehat{\sigma }}_1^z$ and $\widehat{B}={\widehat{\sigma }}_i^x$, with $i=4$ (blue), $i=3$ (green), and $i=2$ (red). (b) The OTOCs for $\widehat{A}={\widehat{\sigma }}_1^y$ and $\widehat{B}={\widehat{\sigma }}_i^z$, with $i=4$ (blue), $i=3$ (green), and $i=2$ (red). The insets of (a) and (b) show the time for the onset of chaos $t_d$ for the OTOCs versus the distance between two operators. The slope gives $1/v_B$. 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 ${}^{13}\mathrm{C}$ or ${}^{19}\mathrm{F}$ transmitters on the 400-MHz spectrometer that we use.

Figure 6

(a) Quantum circuit that measures the OTOCs. The first part aims to reset an arbitrary state to the desired initial state. Here, the time interval between the $\pi$ pulses is 25 ms, the number of cycles is $l=500$, and $G_z$ denotes $z$ axis gradient pulse. (b) Sequences for implementing the dynamics of $e^{-i{\widehat{H}}_{zz}\tau }$ (left) and $e^{i{\widehat{H}}_{zz}\tau }$ (right). The refocusing circuits are designed to generate the right amount of coupled evolution. (c) ${}^{13}\mathrm{C}$ experimental spectrum for equilibrium state (blue) and state ${\widehat{\rho }}_0$ (red) after a readout pulse ${\widehat{R}}_y^1(\pi /2)$. They are shown at the same scale for comparison.

Figure 7

Experimental results for measuring OTOCs for different Ising model parameters and different pairs of $\widehat{A}$ and $\widehat{B}$. The red points are experimental data, the blue curves are theoretical calculation of OTOCs with the model, and the blue points are theoretical values displayed for comparison.

Figure 8

Robustness of the used GRAPE pulses against rf field inhomogeneity. Here, the transverse axis denotes relative error of output field power of the ${}^{13}\mathrm{C}$ channel and the longitudinal axis is that of the ${}^{19}\mathrm{F}$ channel. $U_{\mathrm{sim}}$ denotes the corresponding propagator and $f$ is the fidelity function.

Figure 9

Numerical results of OTOCs for the integrable case. The arrows denote the revival time, which approximately linearly increases with respect to the distance between operators. Here, we choose $\widehat{A}={\widehat{\sigma }}_z^1$ on the first site and $\widehat{B}={\widehat{\sigma }}_z^n$ on the final site. The parameters are $g=1$, $h=0$.