The idea of the out-of-time-order correlator (OTOC) has recently emerged in the study of a variety of research fields, e.g., condensed matter systems, quantum chaos and gravitational systems _{[1][2][3]}. This quantity was suggested by Alexei Kitaev as a quantum generalization of a classical measure of chaotic behaviors _{[}_{4], }see Fig. 1. Despite of the significance of the OTOC revealed by recent theories, experimental measurement of the OTOC remains challenging. First of all, unlike the normal time-ordered correlators, the OTOC cannot be related to conventional spectroscopy measurements, such as ARPES, neutron scattering, through the linear response theory. Secondly, direct simulation of this correlator requires the backward evolution in time, that is, the ability of completely reverse the Hamiltonian, which is extremely challenging.

**Fig. 1.** A classical chaotic system can be diagnosed by the presence of the butterfly effect, in which a small perturbation like the tiny flap of a butterfly’s wing has a huge effect on the system at some later point in time. Analogously, in experiment with a quantum spin system, here described by a wave function. Quantum-control techniques are used to evolve the system forward in time (blue line), to apply a perturbation W, and to evolve the systems backward in time (red line). Then a measurement is performed to diagnose the effect of the perturbation.

**Fig. 2.** Illustration of the physical system, the Ising model and the experimental scheme. (a) The structure of the C2F3I molecule used for the NMR simulation. (b) The four sites Ising spin chain, A and 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.

Recently, Jun Li, a postdoctoral fellow in Chang-pu Sun’s group in CSRC and his collaborators Hui Zhai from Institute for advanced study, Tsinghua University, and Bei Zeng from Institute for quantum computing, University of Waterloo, and Xinhua Peng from University of Science and Technology of China, performed for the first time the measurements of OTOCs on a NMR quantum simulator. The system be simulated is an Ising spin chain model, whose Hamiltonian is written as

The parameter values g = 1, h = 0 correspond to the traverse field Ising model, where the system is integrable. The system is non-integrable whenever both g and h are non-zero. Experiment was done on a four nuclear spins in the iodotrifluroethylene molecule, see Fig. 2. The experiment simulates the dynamics governed by the system Hamiltonian H, and measures the OTOCs of operators that are initially acting on different local sites. The time dynamics of the OTOCs are observed, from which entanglement entropy of the system and butterfly velocities of the chaotic systems are extracted. The results indicate that OTOC provides a faithful reflection of the information scrambling and chaotic behavior of quantum many-body systems.

Measuring the OTOC functions can reveal how quantum entanglement and information scrambles across all of the degrees of freedom in a system. This work here represents a first and encouraging step towards further experimentally observing OTOCs on large-sized quantum systems. The present method can be readily translated to other controllable systems.

**References:**

1. S. H. Shenker and D. Stanford, J. High Energy Phys. 03 067 (2014).

2. Wouter Buijsman, Vladimir Gritsev, Rudolf Sprik, Phys. Rev. Lett. **118**, 080601 (2017).

3. P. Hosur, X.-L. Qi, D. A. Roberts, and B. Yoshida, J. High Energy Phys. 02 004 (2016).

4. A. Kitaev, in Proceedings of the Fundamental Physics Prize Symposium, Vol. 10 (2014).