Out-of-time-ordered correlators (OTOCs) are a key observable in a wide range of interconnected fields including many-body physics, quantum information science, and quantum gravity. Measuring OTOCs using near-term quantum simulators will extend our ability to explore fundamental aspects of these fields and the subtle connections between them. Here, we demonstrate an experimental method to measure OTOCs at finite temperatures and use the method to study their temperature dependence. These measurements are performed on a digital quantum computer running a simulation of the transverse field Ising model. Our flexible method, based on the creation of a thermofield double state, can be extended to other models and enables us to probe the OTOC's temperature-dependent decay rate. Measuring this decay rate opens up the possibility of testing the fundamental temperature-dependent bounds on quantum information scrambling.
Out-of-time-ordered correlators (OTOCs) are a key observable in a wide range of interconnected fields including many-body physics, quantum information science, and quantum gravity. Measuring OTOCs using near-term quantum simulators will extend our ability to explore fundamental aspects of these fields and the subtle connections between them. Here, we demonstrate an experimental method to measure OTOCs at finite temperatures and use the method to study their temperature dependence. These measurements are performed on a digital quantum computer running a simulation of the transverse field Ising model. Our flexible method, based on the creation of a thermofield double state, can be extended to other models and enables us to probe the OTOC's temperature-dependent decay rate. Measuring this decay rate opens up the possibility of testing the fundamental temperature-dependent bounds on quantum information scrambling.
Abstract Information scrambling, which is the spread of local information through a system’s many-body degrees of freedom, is an intrinsic feature of many-body dynamics. In quantum systems, the out-of-time-ordered correlator (OTOC) quantifies information scrambling. Motivated by experiments that have measured the OTOC at infinite temperature and a theory proposal to measure the OTOC at finite temperature using the thermofield double state, we describe a protocol to measure the OTOC in a finite temperature spin chain that is realized approximately as one half of the ground state of two moderately-sized coupled spin chains. We consider a spin Hamiltonian with particle–hole symmetry, for which we show that the OTOC can be measured without needing sign-reversal of the Hamiltonian. We describe a protocol to mitigate errors in the estimated OTOC, arising from the finite approximation of the system to the thermofield double state. We show that our protocol is also robust to main sources of decoherence in experiments.
The out-of-time-ordered correlator (OTOC) is a powerful tool for probing quantum information scrambling, a fundamental process by which local information spreads irreversibly throughout a quantum many-body system. Experimentally measuring the OTOC, however, is notoriously challenging due to the need for time-reversed evolution. Here, we present an experimental evaluation of the OTOC on a quantum computer, using three distinct protocols to address this challenge: the rewinding time method (RTM), the weak-measurement method (WMM), and the irreversibility-susceptibility method (ISM). Our experiments investigate the quantum dynamics of an XXZ spin-1/2 chain prepared in a thermal Gibbs state. As a key contribution, we provide the first experimental demonstration of the ISM, using the trapped-ion quantum computer, reimei. We also conduct a detailed comparative analysis of all three methods, revealing method-dependent behaviors in the measured OTOC. This work not only validates these protocols as practical tools for exploring quantum chaos on near-term hardware but also offers crucial insights into their respective advantages and limitations, providing a practical framework for future experimental investigations.
Out-of-time-ordered correlators (OTOCs) have been extensively studied in recent years as a diagnostic of quantum information scrambling. In this paper, we study quantum information-theoretic aspects of the regularized finite-temperature OTOC. We introduce analytical results for the bipartite regularized OTOC (BROTOC): the regularized OTOC averaged over random unitaries supported over a bipartition. We show that the BROTOC has several interesting properties, for example, it quantifies the purity of the associated thermofield double state and the operator purity of the analytically continued time-evolution operator. At infinite-temperature, it reduces to one minus the operator entanglement of the time-evolution operator. In the zero-temperature limit and for nondegenerate Hamiltonians, the BROTOC probes the groundstate entanglement. By computing long-time averages, we show that the equilibration value of the BROTOC is intimately related to eigenstate entanglement. Finally, we numerically study the equilibration value of the BROTOC for various physically relevant Hamiltonian models and comment on its ability to distinguish integrable and chaotic dynamics.
Out-of-time-ordered correlators (OTOCs) have been extensively studied in recent years as a diagnostic of quantum information scrambling. In this paper, we study quantum information-theoretic aspects of the regularized finite-temperature OTOC. We introduce analytical results for the bipartite regularized OTOC (BROTOC): the regularized OTOC averaged over random unitaries supported over a bipartition. We show that the BROTOC has several interesting properties, for example, it quantifies the purity of the associated thermofield double state and the operator purity of the analytically continued time-evolution operator. At infinite-temperature, it reduces to one minus the operator entanglement of the time-evolution operator. In the zero-temperature limit and for nondegenerate Hamiltonians, the BROTOC probes the groundstate entanglement. By computing long-time averages, we show that the equilibration value of the BROTOC is intimately related to eigenstate entanglement. Finally, we numerically study the equilibration value of the BROTOC for various physically relevant Hamiltonian models and comment on its ability to distinguish integrable and chaotic dynamics.
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.
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.