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

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, 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.

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Plain Language 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.